Under consideration for publication in Theory and Practice of Logic Programming

1

arXiv:1006.3039v2 [] 20 Jun 2010

Concurrent Goal-Based Execution of Constraint
Handling Rules
Edmund S. L. Lam
School of Computing, National University of Singapore
S16 Level 5, 3 Science Drive 2, Singapore 117543
lamsoonl@comp.nus.edu.sg

Martin Sulzmann
Informatik Consulting Systems AG
martin.sulzmann@gmail.com
submitted 5 December 2008; revised 18 January 2010; accepted 31 May 2010

Abstract
We introduce a systematic, concurrent execution scheme for Constraint Handling Rules
(CHR) based on a previously proposed sequential goal-based CHR semantics. We establish
strong correspondence results to the abstract CHR semantics, thus guaranteeing that any
answer in the concurrent, goal-based CHR semantics is reproducible in the abstract CHR
semantics. Our work provides the foundation to obtain efficient, parallel CHR execution
schemes.
KEYWORDS: multi-set rewriting, constraints, concurrency

1 Introduction
Rewriting is a powerful discipline to specify the semantics of programming languages and to perform automated deduction. There are numerous flavors of rewriting such as term, graph rewriting etc. Our focus here is on exhaustive, forward
chaining, multi-set constraint rewriting as found in Constraint Handling Rules
(CHR) (Frühwirth 1998) which are used in a multitude of applications such as
general purpose constraint programming, type system design, agent specification
and planning etc (Frühwirth 2006). Rewriting steps are specified via CHR rules
which replace a multi-set of constraints matching the left-hand side of a rule (also
known as rule head) by the rule’s right-hand side (also known as rule body).
CHR support a very fine-grained form of concurrency. CHR rules can be applied
concurrently if the rewriting steps they imply do not interfere with each other. An
interesting feature of CHR is that the left-hand side of a CHR rule can have a mix
of simplified and propagated constraint patterns. This provides the opportunity for
further concurrency. We can execute CHR rules concurrently as long as only their
propagated parts overlap.
The fact that the abstract CHR semantics is highly concurrent has so far not been

2

Edmund S. L. Lam and Martin Sulzmann

exploited in any major CHR implementation. Existing implementations are specified by highly deterministic semantics (Duck et al. 2004; De Koninck et al. 2008)
which support efficient and systematic but inherently single-threaded execution
schemes (Duck 2005; Schrijvers 2005). Our goal is to develop a systematic, yet
concurrent, semantics which can be efficiently executed in parallel on a multi-core
architecture. In the CHR context, there is practically no prior work which addresses
this important issue.
Specifically, we make the following contributions:
• We develop a novel goal-based concurrent CHR semantics.
• We verify that our semantics respects the abstract CHR semantics by establishing precise correspondence results.
• We examine which existing sequential CHR optimizations carry over to the
concurrent setting.
Section 4 contains the details. A concrete parallel implementation derived from our
concurrent semantics is studied elsewhere (Sulzmann and Lam 2008). Section 5.2
provides a summary.
The upcoming section gives an overview of our work. Section 3 reviews the abstract CHR semantics. Section 5 discusses prior work on execution schemes for CHR
and production rule systems which are a related rewriting mechanism. Section 6
concludes.
2 Overview
We first motivate concurrent execution of CHR rules via a few examples. Then, we
review existing deterministic CHR execution schemes which are the basis for our
concurrent goal-based CHR semantics.
2.1 CHR and Concurrency
Figures 1 and 2 contain several examples of CHR rules and derivations. We adopt
the convention that lower-case symbols refer to variables and upper-case symbols
refer to constraints. The notation rulename@ assigns distinct labels to CHR rules.
The first example simulates a simple communication channel. The Get(x) constraint represents the action of writing a value from the communication channel
into the variable x , while the P ut(y) constraint represents the action of putting the
value y into the channel. The interaction between both constraints is specified via
the CHR rule get which specifies the replacement of constraints matching Get(x)
and P ut(y) by x = y. The point to note is that in contrast to Prolog, we use
matching and not unification to trigger rules.
For example, the constraint store {Get(m), P ut()} matches the left-hand side of
the get rule by instantiating x by m and y by 1. Hence, {Get(m), P ut(1)} rewrites
to the answer {m = 1}. We write {Get(m), P ut(1)} ֌get {m = 1} to denote
this derivation step. Similarly, we find that {Get(n), P ut(8)} ֌get {n = 8}. Rules
can be applied concurrently as long as they do not interfere. In our case, the two

Concurrent Goal-Based Execution of Constraint Handling Rules

3

Communication channel:
get@Get(x ), P ut(y) ⇐⇒ x = y
{Get(m), P ut(1)} ֌get {m = 1} k {Get(n), P ut(8)} ֌get {n = 8}
{Get(m), P ut(1), Get(n), P ut(8)} ֌∗ {m = 1, n = 8}
Greatest common divisor:
gcd@Gcd(0) ⇐⇒ T rue
gcd@Gcd(n)\Gcd(m) ⇐⇒ m ≥ n&&n > 0 | Gcd(m − n)
{Gcd(3), Gcd(9)} ֌gcd2 {Gcd(3), Gcd(6)}
k
{Gcd(3), Gcd(3)} ֌gcd2 {Gcd(3), Gcd(0)}
{Gcd(3), Gcd(3), Gcd(9)}

֌gcd2,gcd2
֌∗

{Gcd(3), Gcd(0), Gcd(6)}
{Gcd(3)}

{Gcd(3), Gcd(3), Gcd(9)} ֌∗ {Gcd(3)}

Fig. 1. Communication channel and greatest common divisor

derivations above can be concurrently executed, indicated by the symbol k, and
we can straightforwardly combine both derivations which leads to the final answer
{m = 1, n = 8}. We write ֌∗ to denote exhaustive rule application.
The answer {m = 8, n = 1} is also possible but the CHR rewrite semantics is
committed-choice. We can guarantee a unique answer if the CHR rules are confluent
which means that rewritings applicable on overlapping constraint sets are always
joinable. In general, (non)confluence is of no concern to us here and is left to the programmer (if desired). We follow here the abstract CHR semantics (Frühwirth 1998)
(formally defined in Section 3) which is inherently indeterministic. Rewrite rules
can be applied in any order and thus CHR enjoy a high degree of concurrency.
The key to concurrency in CHR is monotonicity which guarantees that CHR
executions remain valid if we include a larger context (i.e. store). The following
result has been formally verified in (Abdennadher et al. 1999),
Theorem 1 (Monotonicity of CHR)
For any sets of CHR constraints A,B and S , if A ֌∗ B then A ⊎ S ֌∗ B ⊎ S
An immediate consequence of monotonicity is that concurrent CHR executions
are sound in the sense that their effect can be reproduced using an appropriate
sequential sequence of execution steps. Thus, we can derive the following rule:
S ⊎ S1 ֌∗ S ⊎ S2

S ⊎ S3 ֌∗ S ⊎ S4

(Concurrency)
S ⊎ S1 ⊎ S3 ֌∗ S ⊎ S2 ⊎ S4

4

Edmund S. L. Lam and Martin Sulzmann
merge1@Leq(x , a)\Leq(x , b) ⇐⇒ a < b | Leq(a, b)
merge2@M erge(n, a), M erge(n, b) ⇐⇒ a < b | Leq(a, b), M erge(n + 1, a)
Shorthands: L = Leq and M = M erge

֌merge2
֌merge2
֌merge2
֌merge1
֌merge1

M (1, a), M (1, c), M (1, e), M (1, g)
M (2, a), M (1, c), M (1, e), L(a, g)
M (2, a), M (2, c), L(a, g), L(c, e)
M (3, a), L(a, g), L(c, e), L(a, c)
M (3, a), L(a, c), L(c, g), L(c, e)
M (3, a), L(a, c), L(c, e), L(e, g)

k

֌∗

֌merge2
֌merge1
֌merge1
֌merge1
֌merge1
֌merge1
֌merge1

֌∗

M (1, b), M (1, d ), M (1, f ), M (1, h)
M (3, b), L(b, d ), L(d , f ), L(f , h)

M (3, a), L(a, c), L(c, e), L(e, g), M (3, b), L(b, d ), L(d , f ), L(f , h)
M (4, a), L(a, c), L(a, b), L(c, e), L(e, g), L(b, d ), L(d , f ), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, e), L(e, g), L(b, d ), L(d , f ), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(c, e), L(e, g), L(d , f ), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(d , e), L(e, g), L(d , f ), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(d , e), L(e, f ), L(e, g), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(d , e), L(e, f ), L(f , g), L(f , h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(d , e), L(e, f ), L(f , g), L(g, h)

M (1, a), M (1, c), M (1, e), M (1, g), M (1, b), M (1, d ), M (1, f ), M (1, h)
M (4, a), L(a, b), L(b, c), L(c, d ), L(d , e), L(e, f ), L(f , g), L(g, h)

Fig. 2. Merge sort

In (Frühwirth 2005), the above is referred to as ”Strong Parallelism of CHR”.
However, we prefer to use the term ”concurrency” instead of ”parallelism”. In the
CHR context, concurrency means to run a CHR program (i.e. a set of CHR rules)
by using concurrent execution threads.
Let’s consider the second CHR example from Figure 1 which computes the greatest common divisor among a set of numbers by applying Euclid’s algorithm. The
left-hand side of rule gcd is interesting because it uses a mix of simplified and
propagated constraint patterns. We replace (simplify) Gcd(m) by Gcd(m − n) but
keep (propagate) Gcd(n) if the guard m ≥ n&&n > 0 holds. For example, we
find that {Gcd(), Gcd()} ֌gcd2 {Gcd(), Gcd()} and {Gcd(), Gcd()} ֌gcd2
{Gcd(), Gcd()}. The point to note is the above rule applications only overlap on
the propagated part. Hence, we can execute both rewrite derivations simultaneously
{Gcd(), Gcd()Gcd()} ֌2×gcd2 {Gcd(), Gcd(), Gcd()}

Concurrent Goal-Based Execution of Constraint Handling Rules

5

Our last example in Figure 2 is a CHR encoding of the well-known merge sort
algorithm. To sort a sequence of (distinct) elements e1 , · · ·, em where m is a power
of 2, we apply the rules to the initial constraint store
M erge(, e ), · · ·, M erge(, em )
Constraint M erge(n, e) refers to a sorted sequence of numbers at level n whose
smallest element is e. Constraint Leq(a, b) denotes that a is less than b. Rule
merge initiates the merging of two sorted lists and creates a new sorted list at
the next level. The actual merging is performed by rule merge. Sorting of sublists
belonging to different mergers can be performed simultaneously. See the example
derivation in Figure 2 where we simultaneously sort the characters a, c, e, g and
b, d , f , h.
2.2 Goal-Based CHR Execution
Existing CHR implementation employ a more systematic CHR execution model
where rules are triggered based on a set of available goals. The idea behind a goalbased CHR execution model is to separate the constraint store into two components:
a set of goal constraints (constraints yet to be executed) and the actual constraint
store (constraints that were executed). Previously, in the abstract semantics transitions ֌A are among states Store whereas in the goal-based semantics we find now
transitions ֌G among states of the form hGoals | Storei. Only goal constraints
can trigger rules by searching for store constraint to build a complete match for a
rule head, thus allowing for execution of the rule.
Below, we give a goal-based execution of the earlier communication buffer example.
get @Get(x ), Put(y) ⇐⇒ x = y

(D1 Activate)
(D2 Drop)
(D3 Activate)
(D4 Drop)
(D5 Activate)
(D6 Fire get)
(D7 Activate
(D8 Fire get)
(D9 Solve)
(D10 Solve)

֌G
֌G
֌G
֌G
֌G
֌G
֌G
֌G
֌G
֌G

h{Get(x1 ), Get(x2 ), Put(1), Put(2)} | {}i
h{Get(x1 )#1, Get(x2 ), Put(1), Put(2)} | {Get(x1 )#1}i
h{Get(x2 ), Put(1), Put(2)} | {Get(x1 )#1}i
h{Get(x2 )#2, Put(1), Put(2)} | {Get(x1 )#1, Get(x2 )#2}i
h{Put(1), Put(2)} | {Get(x1 )#1, Get(x2 )#2}i
h{Put(1)#3, Put(2)} | {Get(x1 )#1, Get(x2 )#2, Put(1)#3}i
h{Put(2), x1 = 1} | {Get(x2 )#2}i
h{Put(2)#3, x1 = 1} | {Get(x2 )#2, Put(2)#3}i
h{x1 = 1, x2 = 2} | {}i
h{x2 = 2} | {x1 = 1}i
h{} | {x1 = 1, x2 = 2}i

We label the x th derivation step by a label Dx . Let’s walk through each of the
individual goal-based execution steps. Initially, all constraints are kept in the set
of goals. At this point, all of the goals are inactive. Execution of goals proceeds in
two stages: (1) Activation and (2a) rule execution, or (2b) dropping of goals. In the
first stage, we activate a goal. In general, the order in which goals are activated is
arbitrary. For concreteness, we assume a left-to-right activation order.
Hence, we first activate Get(x1 ) in derivation step (D1). Active goals carry a

6

Edmund S. L. Lam and Martin Sulzmann

unique identifier, a distinct integer number. Besides assigning numbers to active
goals, we also put them into the store. For instance, after activating Get (x1 ), we
have Get (x1 )#1 in both the goals and the store. 1
Active goals like Get (x1 )#1 are executed by trying to build a complete match for
a rule head with matching partner constraints in the store. Since there are no other
constraints in the store, we cannot match Get (x1 )#1 with the get rule. Therefore
we drop Get (x1 )#1 in step (D2). Dropping of a goal means the goal is removed from
the set of goals but of course the (now inactive) goal is still present in the store.
Step (D3) and (D4) are similar but executed on goal Get (x2 ). Then, we activate
Get (x2 ) and find that Get (x2 )#2 cannot build a complete match of the get rule,
thus it is dropped too.
Next, we activate Put (1) (Step D5). Constraint Put (1)#3 can match with either
Get (x1 )#1 or Get (x2 )#2 to form a complete instance of rule head of get . We pick
Get (x1 )#1 and fire the rule get , see step (D6). Step (D7) and (D8) perform similar
execution steps on Put (2) and the remaining stored constraint Get (x2 )#2. Finally,
we add the equations x1 = 1 and x2 = 2 into the store in steps (D9) and (D10).
Exhaustive application of this goal-based execution strategy then leads to a state
with no goals and a final store.
What we have described so far is essentially the execution scheme in which all
major CHR implementations are based on. The semantics of these implementations assume a deterministic activation policy. For example, goals are kept in a
stack (Duck et al. 2004) or priority queue (De Koninck et al. 2008). This of course
implies a strictly sequential execution scheme.
To obtain a systematic, yet concurrent, CHR execution scheme we adapt the goalbased CHR semantics as follows. Several active goal constraints can simultaneously
seek for partner constraints in the store to fire a rule instance. In the extreme case,
all goal constraints could be activated at once. However, we generally assume that
the number of active goals are bounded by n where n corresponds to the the number
of actual threads available to the run-time system (for example, processor cores).
Figure 3 shows a sample concurrent goal-based CHR derivation. We assume two
concurrent threads, referred to as a and b, each thread executes the standard goalδ

based derivation steps. The novelty is that each goal-based derivation step ֌G
now records its effect on the store. The effect δ represents the sets of constraints in

1

Numbered constraints also disambiguate multiple copies in the store but this is rather a sideeffect. The main purpose of numbering constraints is to indicate activation and retain the link
between active goal constraints and their stored copy.

Concurrent Goal-Based Execution of Constraint Handling Rules

7

Short hands: G = Get P = Put
h{G(x1 ), G(x2 ), P(1), P(2)} | {}i
{}\{}
֌G h{G(x1 )#1, G(x2 ), P(1), P(2)} | {G(x1 )#1}i
||
{}\{}
֌G h{G(x1 ), G(x2 )#2, P(1), P(2)} | {G(x2 )#2}i

(D1a Activate)

(D1b Activate)

h{G(x1 ), G(x2 ), P(1), P(2)} | {}i
(D1a || D1b)

{}\{}
֌||G

(D2a Drop)

(D2b Drop)

h{G(x1 )#1, G(x2 )#2, P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i

{}\{}
֌G h{G(x2 )#2, P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i
||
{}\{}
֌G h{G(x1 )#1, P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i

h{G(x1 )#1, G(x2 )#2, P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i
(D2a || D2b)

{}\{}
֌||G

(D3a Activate)

(D3b Activate)

h{P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i
{}\{}
֌G h{P(1)#3, P(2)} | {G(x1 )#1, G(x2 )#2, P(1)#3}i
||
{}\{}
֌G h{P(1), P(2)#4} | {G(x1 )#1, G(x2 )#2, P(2)#4}i

h{P(1), P(2)} | {G(x1 )#1, G(x2 )#2}i
(D3a || D3b)

{}\{}
֌||G

h{P(1)#3, P(2)#4} | {G(x1 )#1, G(x2 )#2, P(1)#3, P(2)#4}i

δ1
֌G h{x1 = 1, P(2)#4} | {G(x2 )#2, P(2)#4}i
||
δ2
(D4b Fire get )
֌G h{P(1)#3, x2 = 2} | {G(x1 )#1, P(1)#3}i
where δ1 = {}\{G(x1 )#1, P(1)#3} δ2 = {}\{G(x2 )#2, P(1)#4}
(D4a Fire get )

h{P(1)#3, P(2)#4} | {G(x1 )#1, G(x2 )#2, P(1)#3, P(2)#4}i
δ
֌||G
h{x1 = 1, x2 = 2} | {}i
where δ = {}\{G(x1 )#1, P(1)#3, G(x2 )#2, P(1)#4}
{}\{}
{}\{}
֌G h{x2 = 2} | {x1 = 1}i
||
(D5b Solve)
֌G h{x1 = 1} | {x2 = 2}i

(D4a || D4b)

(D5a Solve)

h{x1 = 1, x2 = 2} | {}i
(D5a || D5b)

{}\{}
֌||G

h{} | {x1 = 1, x2 = 2}i

Fig. 3. Example of concurrent goal-based CHR derivation

the store which were propagated or simplified. Goal-based derivation steps can be
executed concurrently if their effects are not in conflict.
δ1

hG1 | HS 1 ∪ HS 2 ∪ S i ֌||G hG1′ | HS 2 ∪ S i
δ2

(Goal-Concurrency)

hG2 | HS 1 ∪ HS 2 ∪ S i ֌||G hG2′ | HS 1 ∪ S i
δ1 = HP 1 \HS 1 δ2 = HP 2 \HS 2
HP 1 ⊆ S HP 2 ⊆ S δ = HP 1 ∪ HP 2 \HS 1 ∪ HS 2
hG1 ⊎ G2 ⊎ G | HS 1 ∪ HS 2 ∪ S i
δ
֌||G

hG1′ ⊎ G2′ ⊎ G | S i

The (Goal-Concurrency) rule, abbreviated (k G), states that two goal-derivations
are not in conflict if their simplification effects are disjoint and the propagated
effects are present in the joint store. We will provide more explanations later. Let’s
continue with our example.

8

Edmund S. L. Lam and Martin Sulzmann

Each thread activates one of the two Get goals (Steps D1a and D1b). Since both
steps involve no rule application, side-effects are empty ({}\{}). Both steps are
{}\{}

executed concurrently denoted by the concurrent derivation step (D1a||D2a) ֌||G .
Concurrent goal-based execution threads operate on a shared store and their effects
will be immediately made visible to other threads. This is important to guarantee
exhaustive rule firings.
In the second step (D2a||D2b), both active goals are dropped because there is
no complete match for any rule head yet. Next, steps (D3a) and (D3b) activate
the last two goal constraints, Put (1) and Put (2). Each active constraint can match
with either of the two Get constraints in the store. We assume that active constraint Put (1)#3 in step (D4a) matches with Get (x1 )#1, while Put (2)#4 in step
(D4b) matches with Get (x2 )#2, corresponding to the side-effects δ1 and δ2 . This
guarantees that steps (D4a) and (D4b) operates on different (non-conflicting) parts
of the store. Thus, we can execute them concurrently which yields step (D4a||D4b).
Their side-effects are combined as δ. Finally, in step (D5a||D5b) we concurrently
solve the two remaining equations by adding them into the store and we are done.
The correctness of our concurrent goal-based semantics is established by showing
that all concurrent derivations can be replicated by sequential goal-based executions. We also prove that there is a correspondence between our goal-based CHR semantics with the abstract CHR semantics. This proof generalizes from (Duck 2005)
which shows a correspondence between the refined CHR operational semantics and
abstract semantics. There are a number of subtle points we came across when developing the concurrent variant of the goal-based semantics. We will postpone a
discussion of these issues, as well as a complete formalization of the concurrent
goal-based semantics until Section 4. Next, we formally introduce the details of the
abstract CHR semantics.
3 Constraint Handling Rules
Figure 4 reviews the essentials of the abstract CHR semantics (Frühwirth 1998).
The general form of CHR rules contains propagated heads HP and simplified heads
HS as well as a guard tg
r @HP \HS ⇐⇒ tg | B
In CHR terminology, a rule with simplified heads only (HP is empty) is referred to
as a simplification rule, a rule with propagated heads only (HS is empty) is referred
to as a propagation rule. The general form is referred to as a simpagation rule.
CHR rules manipulate a global constraint store which is a multi-set of constraints.
We execute CHRs by exhaustive rewriting of constraints in the store with respect
to the given rule system (a finite set of CHR rules), via the derivations ֌. To avoid
ambiguities, we annotate derivations of the abstract semantics with A.
Rule (Rewrite) describes application of a CHR rule r at some instance φ. We simply (remove from the store) the matching copies of φ(HS ) and propagate (keep in the
store) the matching copies of φ(HP ). But this only happens if the instantiated guard
φ(tg ) is entailed by the equations present in the store S , written Eqs(S ) |= φ(tg ).

Concurrent Goal-Based Execution of Constraint Handling Rules

9

Notations:
⊎
Multi-set union
|=
Theoretic entailment
φ
Substitution
a
Set/List of a’s
CHR Syntax:
Functions
Constants
Terms
Predicates
Equations
CHR constraints
Constraints
CHR Guards
CHR Heads
CHR Body
CHR Rule
CHR Store
CHR Program

f ::= + |>| && | · · ·
v ::= 1 | true | · · ·
t ::= x | f t
p ::= Get | Put | · · ·
e ::= t = t
c ::= p(t)
b ::= e | c
tg ::= t
H ::= c
B ::= b
R ::= r @H \H ⇐⇒ tg | B
S ::= b
P ::= R

Abstract Semantics Rules:

(Rewrite)

∃φ

Store ֌A Store

(r @HP \HS ⇐⇒ tg | B ) ∈ P such that
Eqs(S ) |= φ ∧ tg φ(HP ⊎ HS ) = HP′ ⊎ HS′
HP′ ⊎ HS′ ⊎ S ֌A HP′ ⊎ φ(B ) ⊎ S
S ⊎ S1 ֌∗A S ⊎ S2

S ⊎ S3 ֌∗A S ⊎ S4

(Concurrency)
S ⊎ S1 ⊎ S3 ֌∗A S ⊎ S2 ⊎ S4
S ֌A S ′

S ֌A S ′

S ′ ֌∗A S ′′

(Closure)
S ֌∗A S ′

S ֌∗A S ′′

where Eqs(S ) = {e | e ∈ S , e is an equation}

Fig. 4. Abstract CHR semantics

In case of a propagation rule we need to avoid infinite re-propagation. We refer to (Abdennadher 1997; Duck 2005) for details. Rule (Concurrency), introduced
in (Frühwirth 2005), states that rules can be applied concurrently as long as they
simplify on non-overlapping parts of the store.
Definition 3.1 (Non-overlapping Rule Application)
Two applications of the rule instances r @HP \HS ⇐⇒ tg | B and r ′ @HP′ \HS′ ⇐⇒

10

Edmund S. L. Lam and Martin Sulzmann
Notations:
⊎
Multi-set union
∪
Set union
|=
Theoretic entailment
φ
Substitution
a
Set/List of a’s
CHR Syntax:
Functions
Constants
Terms
Predicates
Equations
CHR Constraints
Constraints
CHR Guards
CHR Heads
CHR Body
CHR Rule
CHR Program
Num Constraint
Goal Constraint
Stored Constraint
CHR Num Store
CHR Goals
CHR State
Side Effects

f ::= + |>| && | · · ·
v ::= 1 | true | · · ·
t ::= x | f t
p ::= Get | Put | · · ·
e ::= t = t
c ::= p(t)
b ::= e | c
tg ::= t
H ::= c
B ::= b
R ::= r @H \H ⇐⇒ tg | B
P ::= R
nc ::= c#i
g ::= c | e | nc
sc ::= nc | e
Sn ::= sc
G ::= g
σ ::= hG, Sni
δ ::= Sn \Sn

Fig. 5. CHR Goal-based Syntax

tg′ | B ′ in store S are said to be non-overlapping if and only if they simplify unique
parts of S (i.e. HS , HS′ ⊆ S and HS ∩ HS′ = ∅).
The two last (Closure) rules simply specify the transitive application of CHR rules.
4 Concurrent Goal-Based CHR Operational Semantics
We present the formal details of the concurrent goal-based CHR semantics. Figure
5 describes the necessary syntactic extensions. Because constraints in the store
now have unique identifiers, we treat the store as a set (as opposed to a multiset)
and use set union ∪. Goals are still treated as multi-sets because they can contain
multiple copies of (un-numbered) CHR constraints. The actual semantics is given
in two parts. Figure 6 describes the single-step execution part whereas Figure 7
introduces the concurrent execution part. The first part is a generalization of an
earlier goal-based description (Duck 2005) whereas the second (concurrent) part is
novel.
We first discuss the single-step derivation steps in Figure 6. A derivation step
δ

σ ֌G σ ′ maps the CHR state σ to σ ′ with some side-effect δ. δ represents the con-

Concurrent Goal-Based Execution of Constraint Handling Rules

11

δ

hGoal | Storei ֌G hGoal | Storei
W = WakeUp(e, Sn)
(Solve)
W \{}

h{e} ⊎ G | Sni ֌G hW ⊎ G | {e} ∪ Sni
i is a fresh identifier
(Activate)
{}\{}

h{c} ⊎ G | Sni ֌G h{c#i } ⊎ G | {c#i } ∪ Sni
(r @HP′ \HS′ ⇐⇒ tg | B ′ ) ∈ P such that
∃φ Eqs(Sn) |= φ ∧ tg φ(HP′ ) = DropIds(HP )
φ(HS′ ) = φ({c} ⊎ DropIds(HS ))
δ = HP \{c#j } ∪ HS

(Simplify)

h{c#j } ⊎ G | {c#j } ∪ HP ∪ HS ∪ Sni
δ

֌G

hφ(B ′ ) ⊎ G | HS ∪ Sni

(r @HP′ \HS′ ⇐⇒ tg | B ′ ) ∈ P such that
∃φ Eqs(Sn) |= φ ∧ tg φ(HS′ ) = DropIds(HS )
φ(HP′ ) = φ({c} ⊎ DropIds(HP ))
δ = {c#j } ∪ HP \HS

(Propagate)

h{c#j } ⊎ G | {c#j } ∪ HP ∪ HS ∪ Sni
δ

֌G

hφ(B ′ ) ⊎ {c#j } ⊎ G | {c#j } ∪ HP ∪ Sni

(Simplify) and (Propagate) does not apply on c#j in Sn
(Drop)
{}\{}

h{c#j } ⊎ G | Sni ֌G hG | Sni
where

Eqs(S )
DropIds(Sn)
WakeUp(e, Sn)

=
=
=

{e | e ∈ S , e is an equation}
{c | c#i ∈ Sn} ⊎ {e | e ∈ Sn, e is an equation}
{c#i | c#i ∈ Sn ∧ φ m.g.u. of Eqs(Sn)∧
θ m.g.u. of Eqs(Sn ∪ {e}) ∧ φ(c) 6= θ(c)}

Fig. 6. Goal-Based CHR Semantics (Single-Step Execution)

straints that where propagated or simplified during rule application. Hence derivation steps that do not involve rule application ((Activate) and (Drop)) contain no
side-effects (i.e. {}\{}). We will omit side-effects δ as and when it is not relevant
to our discussions. We ignore the (Solve) step for the moment. In (Activate), we
activate a goal CHR constraint by assigning it a fresh unique identifier and adding
it to the store. Rewrite rules are executed in steps (Simplify) and (Propagate). We

12

Edmund S. L. Lam and Martin Sulzmann

distinguish whether the rewrite rule is executed on a simplified or propagated active
(goal) constraint c#i. For both cases, we seek for the missing partner constraints in
the store for some matching substitution φ. The auxiliary function DropIds ignores
the unique identifiers of numbered constraints. They do not matter when finding a
rule head match. The guard tg must be entailed by the primitive (here equational)
store constraints under the substitution φ.
In case of a simplified goal, step (Simplify), we apply the rule instance of r by
deleting all simplified matching constraints HS and adding the rule body instance
φ(B ) into the goals. Since c#i is simplified, we drop c#i from the goals as it does
not exist in the store any more. In case of a propagated goal, step (Propagate), c#i
remains in the goal set as well in the store and thus can possibly fire further rules
HP \HS

instances. For both (Simplify) and (Propagate) derivation step, say σ ֌G σ ′ , we
record as side-effect the numbered constraints in the store that were propagated
(HP ) or simplified (HS ) during the derivation step. We will elaborate on the purpose
of side-effects when we introduce the concurrent part of the semantics.
In step (Drop), we remove an active constraint from the set of goals, if the
constraint failed to trigger any CHR rule.
Rule (Solve) moves an equation goal e into the store and wakes up (reactivates)
any numbered constraint in the store which can possibly trigger further CHR rules
due to the presence of e. Here is a simple example to show why reactivation is
necessary.
r 1@A(x ), B (x ) ⇐⇒ C (x )
h{a = 2} | {A(a)#1, B (2)#2}i
(Solve)
(Simp r 1)
···

{A(2)#1}\{}
֌G
{}\{A(2)#1,B (2)#2}
֌G

h{A(2)#1} | {A(2)#1, B (2)#2, a = 2}i
h{C (2)} | {a = 2}i

For clarity, we normalize all constraints in the store once an equation is added.
Prior to addition of a = 2, A(a)#1, B (2)#2 cannot fire rule r 1. After adding a = 2
however, we can normalize A(a)#1 to A(2)#2, which can now fire r 1 with B (2)#2.
To guarantee exhaustive rule firings, we reactivate A(2)#2 by adding it back to the
set of goals. WakeUp(e, Sn) represents a conservative approximation of the to be
reactivated constraints (Duck 2005). Note that we treat reactivated constraints as
propagated constraints in the side-effects.
Figure 7 presents the concurrent part of the goal-based operational semantics.
In the (Lift) step, we turn a sequential goal-based derivation into a concurrent
derivation. Note that side-effects are retained. Step (Goal Concurrency) joins together two concurrent derivations operating on a shared store, if their rewriting
side-effects δ1 and δ2 are non-overlapping as defined below.
Definition 4.1 (Non-overlapping Rewriting Side-Effects)
Two rewriting side-effects δ1 = HP 1 \HS 1 and δ2 = HP 2 \HS 2 are said to be nonoverlapping, if and only if HS 1 ∩ (HP 2 ∪ HS 2 ) = {} and HS 2 ∩ (HP 1 ∪ HS 1 ) = {}
Concurrent derivations with non-overlapping side-effects essentially simplify distinct constraints in the store, as well as propagate constraints which are not simpli-

Concurrent Goal-Based Execution of Constraint Handling Rules

13

δ

hGoal | Storei ֌||G hGoal | Storei
δ

hG | Sni ֌G hG ′ | Sn ′ i
(Lift)
δ

hG | Sni ֌||G hG ′ | Sn ′ i
δ1

hG1 | HS 1 ∪ HS 2 ∪ S i ֌||G hG1′ | HS 2 ∪ S i
δ2

(Goal Concurrency)

hG2 | HS 1 ∪ HS 2 ∪ S i ֌||G hG2′ | HS 1 ∪ S i
δ1 = HP 1 \HS 1 δ2 = HP 2 \HS 2
HP 1 ⊆ S HP 2 ⊆ S δ = HP 1 ∪ HP 2 \HS 1 ∪ HS 2
hG1 ⊎ G2 ⊎ G | HS 1 ∪ HS 2 ∪ S i
δ
֌||G

hG1′ ⊎ G2′ ⊎ G | S i

δ

σ ֌||G σ ′

δ

σ ֌||G σ ′

σ ′ ֌∗||G σ ′′

(Closure)
σ ֌∗||G σ ′

σ ֌∗||G σ ′′

Fig. 7. Goal-Based CHR Semantics (Concurrent Part)

fied by one another. The (Goal Concurrency) step expresses non-overlapping sideeffects by structurally enforcing that simplified constraints HS 1 and HS 2 match
distinct parts of the store, while propagated constraints HP 1 and HP 2 are found
in the shared part of the store S not modified by both concurrent derivations. In
the resulting concurrent derivation, the side-effects δ1 and δ2 are composed by the
union of the propagate and simplify components respectively, forming δ.
An immediate consequence is that we can execute k derivations concurrently by
stacking them together as long as all side-effects are mutually non-overlapping. The
following lemma summarizes this observation.

Lemma 1 (k -Concurrency)

14

Edmund S. L. Lam and Martin Sulzmann

For any finite k of mutually non-overlapping concurrent derivations,
HP1 \HS 1

hG1 | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hG1′ | {} ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i
···
HPi \HSi

hGi | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hGi′ | HS 1 ∪ · · ∪{} ∪ · · ∪HSk ∪ S i
···
HPk \HSk

hGk | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hGk′ | HS 1 ∪ · · ∪HSi ∪ · · ∪{} ∪ S i
HP 1 ⊆ S · ·HPi ⊆ S · ·HPk ⊆ S
δ = HP 1 ∪ · · ∪HPi ∪ · · ∪HPk \HS 1 ∪ · · ∪HSi ∪ · · ∪HSk
hG1 ⊎ · · ⊎Gi ⊎ · · ⊎Gk ⊎ G | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i
δ
֌||G

hG1′ ⊎ · · ⊎Gi′ ⊎ · · ⊎Gk′ ⊎ G | S i

we can decompose this into k −1 applications of the (pair-wise) (Goal Concurrency)
derivation step.
The (Closure) step defines transitive application of the concurrent goal-based
derivation. Because side-effect labels are only necessary for the (Goal Concurrency)
step, we drop the side-effects in transitive derivations.
Any concurrent goal-based derivation can be reproduced in the abstract CHR
semantics. This correspondence result is important to make use of the concurrent
goal-based semantics as a more systematic execution scheme for CHR. We will
formally verify this as well as other results in the up-coming Section 4.2. First, we
give an in-depth discussion of the more subtle aspects of the concurrent goal-based
semantics.
4.1 Discussion
Most of the issues we encounter are related to the problem of exhaustive rule firings.
For brevity, we omit side-effects in derivation steps in the following examples as they
do not matter.
Goal Storage, Shared Store and Single-Step Execution: Each of these issues
affect (exhaustive) rule firings. We first consider goal storage. Suppose we would
only store goals after execution (rule head matching). That is, we do not add the
goals into the store during (Activate) step, but only during the (Drop) step.
i is a fresh identifier
(Activate’)
h{c} ⊎ G | Sni ֌G h{c#i } ⊎ G | Sni
(Simplify) and (Propagate) does not apply on c#i in Sn
(Drop’)
h{c#i } ⊎ G | Sni ֌G hG | {c#i } ∪ Sni

Then, for the CHR program
r 1@A(x ), B (y) ⇐⇒ C (x , y)

Concurrent Goal-Based Execution of Constraint Handling Rules

15

we obtain the following derivation
h{A(1), B (2)} | {}i
(Activate’)
(Activate’)

h{A(1)} | {}i ֌||G h{A(1)#1} | {}i
||
h{B (2)} | {}i ֌||G h{B (2)#2} | {}i

h{A(1), B (2)} | {}i ֌||G h{A(1)#1, B (2)#2} | {}i
(Drop’)
(Drop’)

h{A(1)#1} | {}i ֌||G h{} | {A(1)#1}i
||
h{B (2)#2} | {}i ֌||G h{} | {B (2)#2}i

h{A(1)#1, B (2)#2} | {}i ֌||G h{} | {A(1)#1, B (2)#2}i

Initially both goals A(1) and B (2) are concurrently activated. Since (Activate’)
does not store goals immediately, both active goals are not visible to each other
in the store. Hence, we wrongfully apply the (Drop’) step for both goals. However,
there is clearly a complete rule head match A(1)#1, B (2)#2.
Next, we investigate the shared store issue. Suppose we allow for concurrent
executions on (non-shared) split stores. Then, the following derivation is possible.
r 1@A, B ⇐⇒ C

r 2@D, E ⇐⇒ F

(Drop) h{A#3} | {A#3, E #2}i ֌||G h{} | {A#3, E #2}i
(Drop) h{D#4} | {B #1, D#4}i ֌||G h{} | {B #1, D#4}i
h{A#3, D#4} | {A#3, B #1, D#4, E #2}i ֌||G h{} | {A#3, B #1, D#4, E #2}i

The resulting store is a final store, there are no more goals left. However, if we
consider the entire store {A#3, E #2, B #1, D #4}, it is clearly that goal A#3 can
execute rule r and goal D #4 can execute rule r. We conclude that splitting of
the store leads to ”stuck” states. We fail to exhaustively fire CHR rules.
For similar reasons, we demand that when joining concurrent executions, each
individual execution can only make a single-step. Otherwise, we encounter again a
stuck state.
r 1@A, B ⇐⇒ C
(P 1) h{A} | {}i ֌||G h{A#2} | {A#2}i ֌||G h{} | {A#2}i
(P 2) h{B } | {}i ֌||G h{B #3} | {B #3}i ֌||G h{} | {B #3}i
h{A, B } | {}i ֌∗||G h{} | {A#2, B #3}i
The sequence of derivation steps (P1) first activates A which is then dropped.
Similarly, (P2) activates B which is then dropped as well which then leads to
the stuck state h{} | {A#2, B #3}i. We clearly missed to fire rule r. This shows
that single-step concurrent execution are essential to guarantee that newly added

16

Edmund S. L. Lam and Martin Sulzmann

constraints are visible to all concurrent active goals, hence we have exhaustive rule
firings in the goal-based semantics.
The underlying reason for non-exhaustive firing of rules is that the goal-based
semantics is not monotonic in its store argument. However, execution is monotonic
in the goal argument which leads us to the next issue.

Lazy Matching and Asynchronous Goal Execution: When executing goals,
we lazily compute only matches that contain the specific goal and immediately
apply such matches without concerning any further matches. For instance consider
the following CHR program and goal-based derivation:

r 0@A(x ), B (y) ⇐⇒ D(x , y)

֌G

h{A(1)#4} ⊎ {A(2), A(3)} | {B (2)#1, B (3)#2, B (4)#3, A(1)#4}i
h{D(1, 2)} ⊎ {A(2), A(3)} | {B (3)#2, B (4)#3}i

We have applied the rule instance A(1)#4, B (2)#1 independently of the existence
of the other goals (i.e. {A(2), A(3)}). In the literature, such a matching scheme is
known as a lazy matching scheme, and often implemented by variants of the LEAPS
algorithm (D.P. Miranker and Gadbois 1990).
Lazy matching in the goal-based semantics is possible only because the goal-based
semantics is monotonic with respect to the set of goals. The following illustrates
this monotonicity property of goals:

֌G

֌G

h{A(1)#4} | {B (2)#1, B (3)#2, B (4)#3, A(1)#4}i
h{D(1, 2)} | {B (3)#2, B (4)#3}i

h{A(1)#4} ⊎ {A(2), A(3)} | {B (2)#1, B (3)#2, B (4)#3, A(1)#4}i
h{D(1, 2)} ⊎ {A(2), A(3)} | {B (3)#2, B (4)#3}i
hG | Sni ֌G hG ′ | Sn ′ i
hG ⊎ G ′′ | Sni ֌G hG ′ ⊎ G ′′ | Sn ′ i

The above property essentially states that we can execute goals G without prior
knowledge of goals G ′′ . Because of monotonicity, we are guaranteed that future
executions of G ′′ will not invalidate them.

Concurrent Goal-Based Execution of Constraint Handling Rules

17

Monotonicity of the goals also allows us to execute goals asynchronously. For
instance, consider the following:
r 1@A(x ), B (y) ⇐⇒ C (x , y)
h{A(1)#1} | {A(1)#1, B (2)#2} ∪ {A(3)#3, B (4)#4}i
δ1
֌||G

h{C (1, 2)} | {} ∪ {A(3)#3, B (4)#4}i
h{A(3)#3} | {A(1)#1, B (2)#2} ∪ {A(3)#3, B (4)#4}i

δ2

֌||G h{C (3, 4)} | {A(1)#1, B (2)#2} ∪ {}i
δ1 = {}\{A(1)#1, B (2)#2} δ2 = {}\{A(3)#3, B (4)#4}
δ = {}\{A(1)#1, B (2)#2, A(3)#3, B (4)#4}
h{A(1)#1} ⊎ {A(3)#3} | {A(1)#1, B (2)#2} ∪ {A(3)#3, B (4)#4}i
δ
֌||G

h{C (1, 2)} ⊎ {C (3, 4)} | {} ∪ {}i

The above describes the concurrent execution of goals A(1)#1 and A(3)#3. Notice that in the derivations of the premise, we can ignore all goals which are not
relevant to the derivation. For instance, execution of A(1)#1 does not need goal
A(3)#3 to be visible, hence the goals effectively executes asynchronously. Goals do
however, implicitly ”synchronize” via the shared store. Namely, concurrent derivations must be chosen such that rewrite side-effects involve distinct parts of the
store.
4.2 Correspondence Results
We formally verify that the concurrent goal-based semantics is in exact correspondence to the abstract CHR semantics when it comes to termination and exhaustive
rule firings. Detailed proofs are given in the appendix. In the main text, we provide key lemmas and proof sketches. We first introduce some elementary definitions
before stating the formal results.
Definitions: The first two definitions concern the abstract CHR semantics. A store
is final if no further rules are applicable.
Definition 4.2 (Final Store)
A store S is known as a final store, denoted FinalA (S ) if and only if no more CHR
rules applies on it (i.e. ¬∃S ′ such that S ֌A S ′ ).
A CHR program terminates if all derivations lead to a final store in a finite
number of states.
Definition 4.3 (Terminating CHR Programs)
A CHR program P is said to be terminating, if and only if for any CHR store S ,
all derivations starting from S are finite.
Next, we introduce some definitions in terms of the goal-based semantics. In an
initial state, all constraints are goals and the store is empty. Final states are states

18

Edmund S. L. Lam and Martin Sulzmann

which no longer have any goals. We will prove the exhaustiveness of the goalbased semantics by proving a correspondence between final stores in the abstract
semantics and final states of the goal-based semantics
Definition 4.4 (Initial and Final CHR States)
An initial CHR state is a CHR state of the form hG | {}i where G contains no
numbered constraints (c#n), while a final CHR state is of the form h{} | Sni
A state is reachable if there exists a (sequential) goal-based sequence of derivations to this state. We write ֌∗G to denote the transitive closure of ֌G .
Definition 4.5 (Sequentially Reachable CHR states)
For any CHR program P, a CHR state hG ′ | Sn ′ i is said to be sequentially reachable
by P if and only if there exists some initial CHR state hG | {}i such that hG |
{}i ֌∗G hG ′ | Sn ′ i.
4.2.1 Correspondence of Derivations
We build a correspondence between the abstract semantics and the concurrent goalbased semantics. We begin with Theorem 2, which states the correspondence of the
(sequential) goal-based semantics.
Theorem 2 (Correspondence of Sequential Derivations)
For any reachable CHR state hG | Sni, CHR state hG ′ | Sn ′ i and CHR program P,
if
then

hG | Sni ֌∗G hG ′ | Sn ′ i
(NoIds(G) ⊎ DropIds(Sn)) = (NoIds(G ′ ) ⊎ DropIds(Sn ′ )) ∨
(NoIds(G) ⊎ DropIds(Sn)) ֌∗A (NoIds(G ′ ) ⊎ DropIds(Sn ′ ))

where NoIds = {c | c ∈ G, c is a CHR constraint} ⊎ {e | e ∈ G, e is an equation}
The above result guarantees that any sequence of sequential goal-based derivations starting from a reachable CHR state either yields equivalent CHR abstract
stores (due to goal-based behavior not captured by the abstract semantics, namely
(Solve) (Activate), (Drop)) or corresponds to a derivation in the abstract semantics
(due to rule application). A goal-based semantics state hG | Sni is related to an
abstract semantics store by removing all numbered constraints in G and unioning
it with constraints in Sn without their identifiers. The theorem and its proof is a
generalization of an earlier result given in (Duck 2005).
We formalize the observation that the goal context can be extended without
interfering with previous goal executions.
Lemma 2 (Monotonicity of Goals in Goal-based Semantics)
For any goals G,G ′ and G ′′ and CHR store Sn and Sn ′ , If hG | Sni ֌∗G hG ′ | Sn ′ i
then hG ⊎ G ′′ | Sni ֌∗G hG ′ ⊎ G ′′ | Sn ′ i.
Next, we state that given any goal-based derivation with side-effects δ, we can
safely ignore any constraints (represented by S2 ) in the store which is not part of
δ.

Concurrent Goal-Based Execution of Constraint Handling Rules

19

Lemma 3 (Isolation of Goal-based Derivations)
HP \HS

If hG | HP ∪ HS ∪ S1 ∪ S2 i ֌G hG ′ | HP ∪ S1′ ∪ S2 i
HP \HS

then hG | HP ∪ HS ∪ S1 i ֌G hG ′ | HP ∪ S1′ i
Lemma 3 can be straight-forwardly extended to multiple derivation steps. This
is stated in Lemma 4.
Lemma 4 (Isolation of Transitive Goal-based Derivations)
If hG | HP ∪ HS ∪ S1 ∪ S2 i ֌∗G hG ′ | HP ∪ S1′ ∪ S2 iwith side-effects δ = HP \HS then
hG | HP ∪ HS ∪ S1 i ֌∗G hG ′ | HP ∪ S1′ i
The next states that any concurrent derivation starting from a reachable CHR
state can be replicated by a sequence of sequential goal-based derivations. Lemma
5 is the first step to prove the correspondence of concurrent goal-based derivations.
Lemma 5 (Sequential Reachability of Concurrent Derivation Steps)
For any sequentially reachable CHR state σ, CHR state σ ′ and rewriting side-effects
δ

δ if σ ֌||G σ ′ then σ ′ is sequentially reachable, σ ֌∗G σ ′ with side-effects δ.
Proof
(Sketch) Via Lemma 1, we can always reduce k mutually non-overlapping concurrent derivations into several applications of the (Goal Concurrency) step. Hence we
can prove Lemma 5 by structural induction over the concurrent goal-based derivation steps (Lift) and (Goal Concurrency) where we use Lemmas 2 and 4 to show
that concurrent derivations can always be replicated by a sequence of sequential
goal-based derivations.
Theorem 3 (Sequential Reachability of Concurrent Derivations)
For any initial CHR state σ, CHR state σ ′ and CHR Program P, if σ ֌∗||G σ ′ then
σ ֌∗G σ ′ .
The above follows directly from Lemma 5 by converting each single step concurrent derivation into a sequence of sequential derivations, and showing their composibility.
From Theorem 2 and 3, we have the following corollary, which states the correspondence between concurrent goal-based CHR derivations and abstract CHR
derivations.
Corollary 1 (Correspondence of Concurrent Derivations)
For any reachable CHR state hG | Sni, CHR state hG ′ | Sn ′ i and CHR program P,
if
then

hG | Sni ֌∗||G hG ′ | Sn ′ i
(NoIds(G) ⊎ DropIds(Sn)) = (NoIds(G ′ ) ⊎ DropIds(Sn ′ )) ∨
(NoIds(G) ⊎ DropIds(Sn)) ֌∗A (NoIds(G ′ ) ⊎ DropIds(Sn ′ ))

where NoIds = {c | c ∈ G, c is a CHR constraint} ⊎ {e | e ∈ G, e is an equation}

20

Edmund S. L. Lam and Martin Sulzmann
4.2.2 Correspondence of Termination

We show that all derivations from an initial state to final states in the concurrent
goal-based semantics correspond to some derivation from a store to a final store in
the abstract semantics. We first define rule head instances:
Definition 4.6 (Rule head instances)
For any CHR state σ = hG, Sni and CHR program P, any (HP ∪ HS ) ⊆ Sn is
known as a rule head instance of σ, if and only if ∃(r @HP′ \HP′ ⇐⇒ tg | B ) ∈ P,∃φ
Eqs(Sn) |= φ ∧ tg and φ(HP′ ⊎ HS′ ) = DropIds(HP ∪ HS ).
Definition 4.7 (Active rule head instances)
For any CHR state σ = hG, Sni and CHR program P, a rule head instance H of
σ is said to be active if and only if there exists at least one c#i ∈ G such that
c#i ∈ H .
Rule head instances (Definition 4.6) are basically minimal subsets of the store
which matches a rule head. Active rule head instance (Definition 4.7) additional
have at least one of it is numbered constraint c#i in the goals as well. Therefore, by
the definition of the goal-based semantics, active rule head instances will eventually
be triggered by either the (Simplify) or (Propagate) derivation steps.
Lemma 6 (Rule instances in reachable states are always active)
For any reachable CHR state hG | Sni, any rule head instance H ⊆ Sn must be
active. i.e. ∃c#i ∈ H such that c#i ∈ G.
Lemma 6 shows that all rule head instances in reachable states are always active.
This means that by applying the semantics steps in any way, we must eventually
apply the rule head instances as long as all it is constraints remain in the store.
Theorem 4 states that termination of a concurrent goal-based derivation corresponds to termination in the abstract semantics. This is of course, provided that
the CHR program is terminating.
Theorem 4 (Correspondence of Termination)
For any initial CHR state hG, {}i, final CHR state h{}, Sni and terminating CHR
program P,
if hG | {}i ֌∗||G h{} | Sni
then G ֌∗A DropIds(Sn) and FinalA (DropIds(Sn))
We prove this theorem by first using Theorem 3 which guarantees that a concurrent goal-based derivation from an initial state to a final state corresponds to
some abstract semantics derivation. We next show that final states correspond to
final stores in the abstract semantics. This is done by contradiction, showing that
assuming otherwise contradicts with Lemma 6.

Concurrent Goal-Based Execution of Constraint Handling Rules

21

4.3 Concurrent CHR Optimizations
In the sequential setting, there exist a wealth of optimizations (Duck 2005; Schrijvers 2005;
Sneyers et al. 2005) to speed up the execution of CHR. Fortunately, many of these
methods are still largely applicable to our concurrent goal-based variant as we discuss in the following. For the remainder, we assume that each goal (thread) tries
the CHR rules from top-to-bottom to match the rule execution order assumed in
(Duck 2005; Schrijvers 2005; Sneyers et al. 2005).
Basic constraint indexing like lookups via hashtables are still applicable with
minor adaptations. For instance, the underlying hashtable implementation must be
thread safe. Consider the following example:
r 0@A(x , y), B (x ), C (y) ⇐⇒ x > y | D (x , y)
Suppose we have the active constraint A(1, 2)#n. To search for a partner constraint of the form B (1)#m and C (2)#p, standard CHR compilation techniques
would optimize with indexing (hashtables) which allows constant time lookup for
these constraints. The use of such indexing techniques is clearly applicable in a
concurrent goal execution setting as long as concurrent access of the indexing data
structures are handled properly. For example, we can possibly have a concurrent
active constraint A(1, 3)#q which will compete with A(1, 2)#n for a matching
partner B (1)#m. As such, hashtable implementations that facilitate such indexing
must be able to be accessed and modified concurrently.
Guard optimizations/simplifications aim at simplifying guard constraints by replacing guard conditions with equivalent but simplified forms. Since guards are
purely declarative, they are not influenced by concurrently executing goal threads
(i.e. CHR rules). Hence, all existing guard optimizations carry over to the concurrent setting.
The join order of a CHR rule determines the order in which partner constraints
are searched to execute a rule. The standard CHR optimization known as optimal
join-ordering and early guard scheduling (Duck 2005) aims at executing goals with
the most optimal order of partner constraints lookup and guard testing. By optimal,
we refer to maximizing the use of constant time index lookup. Considering the same
CHR rule (r 0) above, given the active constraint B (x ), an optimal join-ordering
is to lookup for A(x , y), schedule guard x > y, then lookup for C (y). Since our
concurrent semantics does not restrict the order in which partner constraints are
matched, optimal join ordering and early guard scheduling are still applicable.
Another set of optimizations tries to minimize the search for partner constraints
by skipping definitely failing searches. Consider the following example:
r 1@A ⇐⇒ · · ·
r 2@A, B ⇐⇒ · · ·
If the active goal A cannot fire rule (r1) then we cannot fire rule (r2) either. Hence,
after failing to fire rule (r1) we can drop goal A. Thus, we optimize away some
definitely failing search. This statement is immediately true in the sequential setting
where no other thread affects the constraint store. The situation is different in a
concurrent setting where some other thread may have added in between the missing

22

Edmund S. L. Lam and Martin Sulzmann

constraint A. Then, even after failing to fire (r1) we could fire rule (r2). However, we
can argue that the optimization is still valid for this example. We will not violate the
important condition to execute CHR rules exhaustively because the newly added
constraint A will eventually be executed by a goal thread which then fires rule (r1).
Hence, the only concern is here that the optimization leads to indeterminism in the
execution order of CHR rules which is anyway unavoidable in a concurrent setting.
Yet there are existing optimizations which are not applicable in the concurrent
setting. For example, continuation optimizations (Duck 2005; Schrijvers 2005) are
not entirely applicable. Consider the following CHR rule:
r 4@A(x ), A(y) ⇐⇒ x == y | · · ·
Given an active constraint A(1)#n, fail continuation optimization will infer that
if we fail to fire the rule with A(1)#n matching A(x ), there is no point trying to
match it with A(y) because it will most certainly fail as well, assuming that the
store never changes. In a concurrent goal execution setting, we cannot assume that
the store never changes (while trying to execute a CHR). For instance, after failing
to trigger the rule by matching A(1)#n with A(x ), suppose that a new active goal
A(1)#m is added to the store concurrently. Now when we match A(1)#n to A(y)
we can find match the partner A(1)#m with A(x ), hence breaking the assumptions
of the fail continuation optimization.
Late (also known as delayed) storage optimization (Duck 2005) aims at delaying
the storage of a goal g, until the latest point of its execution where g is possibly a
partner constraint of another active constraint. Consider the following example:
r 1@P1 =⇒ Q
r 2@P2 , T1 ⇐⇒ R
r 3@P3 , R1 ⇐⇒ True
r 4@P4 =⇒ S
r 5@P5 , S1 ⇐⇒ True
Note to distinguish the rule heads, we annotate each rule head with a subscript
integer (eg. Px ). With late storage analysis techniques described in (Duck 2005),
we can delay storage of an active constraint P until just before the execution of the
body of r 4. This is because the execution of goal S (obtained from firing of r 4) can
possibly trigger r 5. While this is safe in the sequential goal execution scheme, it
is possible that rule matches are missing in the concurrent goal execution setting.
Consider the case where we have some simultaneously active goals P #n and T #m.
Since P #n is only stored when its execution has reached r 4, the match r 2 can be
missed entirely by both active parallel goals P #n and T #m. Specifically, this
happens if goal T #m is activated only after P #n has tried matching with P2
(of r 2), but completes goal execution (by trying T1 of r 2, and failing to match)
before goal P #n is stored. Hence, we conclude that we cannot safely implement
late storage in the concurrent setting.
5 Related Work
We review prior work on execution schemes for CHR and production rule systems.

Concurrent Goal-Based Execution of Constraint Handling Rules

23

5.1 CHR Execution Schemes
There exists a wealth of prior work on the semantics of CHR. We refer to (Sneyers et al. )
for a comprehensive summary. Our focus here is on the operational CHR semantics
and we briefly review the most relevant works.
The theoretical (a.k.a. high-level) operational semantics (Frühwirth 1998) is derived from the abstract semantics and inherits its high degree of indeterminism.
The theoretical semantics has been mainly used for the study of high-level properties such as confluence (Abdennadher 1997; Abdennadher et al. 1999). Confluence analysis has been exploited to study the degree of concurrency in CHR programs (Frühwirth 2005; Schrijvers and Sulzmann 2008). None of these works however provide direct glues how to systematically execute concurrent programs.
In (Duck et al. 2004; De Koninck et al. 2008; De Koninck et al. 2008) some systematic, highly deterministic semantics have been developed to achieve efficient implementation schemes. However, these semantics are inherently single-threaded. Our
motivation is to obtain systematic yet concurrent semantics which led us to develop
the goal-based concurrent semantics presented in this paper. In the special case of a
single goal thread, our semantics is equivalent to the refined operational semantics
given in (Duck et al. 2004; De Koninck et al. 2008; De Koninck et al. 2008).
There are only few works which explore different semantics, other than the theoretical or abstract semantics, to address concurrency. The work in (Sarna-Starosta and Ramakrishnan 2007)
adopts a set-based semantics and supports tabled, possible concurrent, rule execution. This execution scheme is not applicable to CHR programs in general which
usually assume a multi-set based semantics. The recent work in (Betz et al. 2009)
takes a new stab at concurrency by introducing the notion of persistent constraints.
The idea is to split the store into linear (multi-set like) and persistent (set like) constraints. We are not aware of any evidence which shows that this approach supports
effective concurrency in practice. Our approach leads to an efficient parallel implementation as we explain in the next section.

5.2 From Concurrent to Parallel CHR Execution
In our earlier works (Lam and Sulzmann 2007; Sulzmann and Lam 2008) we have
developed a parallel CHR implementation scheme based on an informally described
concurrent goal-based execution scheme, see Section 3 in (Sulzmann and Lam 2008).
The present works provides a concise formal treatment of the implemented concurrent goal-based execution scheme. In our implementation, multiple threads,
each executing a unique CHR goal, are executed in parallel on multiple processor
cores. Parallel goal executions are largely asynchronous, only implicitly synchronizing via the shared constraint store. Atomic CHR execution is guaranteed via advanced synchronization primitives such as Software Transactional Memory. We refer
to (Sulzmann and Lam 2008) for a thorough description of the more subtle implementation details. Our experimental results reported in (Sulzmann and Lam 2008)
show that we achieve good scalability when the number of processor cores increases.
The overhead of the parallel implementation is fairly minor compared to a single-

24

Edmund S. L. Lam and Martin Sulzmann
Parallel Matching (Match)

ii

Parallel Conflict Resolution (CR)


Parallel Rule Application (Act)

Fig. 8. Parallel Production Rule Execution Cycles

threaded implementations thanks to the use of lock-free algorithms. Optimization
methods applicable in the concurrent/parallel setting are discussed in the earlier
Section 4.3.

5.3 Parallel Production Rule Systems
Parallel execution models of forward chaining production rule based languages (e.g.
OPS5 (Forgy and McDermott 1977)) have been widely studied in the context of
production rule systems. A production rule system is defined by a set of multiheaded production rules (analogous to CHR rules) and a set of assertions (analogous
to the CHR store). Production rule systems are richer than the CHR language,
consisting of user definable execution strategies and negated rule heads. This makes
parallelizing production rule execution extremely difficult, because rule application
is not monotonic (rules may not be applied in a larger context). As such, many
previous works in parallel production rule systems focuses on efficient means of
maintaining correctness of parallel rule execution (e.g. data dependency analysis
(Ishida 1991), sequential to parallel program transformation (Gamble 1990)), with
respect to such user specified execution strategies. These works can be classified
under two approaches, namely synchronous and asynchronous parallel production
systems.
For synchronous parallel production systems (e.g. UMPOPS (Gupta et al. 1988)),
multiple processors/threads run in parallel. They are synchronized by execution cycles of the production systems. Figure 8 illustrates the production cycle of a typical production rule system, consisting of three execution phases. In the (Match)
phase, all rule matches are computed. Conflict resolution (CR) involves filtering
out matches that do not conform to the user specified rule execution strategy,
while (Act) applies the rule matches that remains (known as the eligible set) after
the (CR) phase. By synchronizing parallel rule execution in production cycles, a
larger class of user specified execution strategies can be supported since execution
is staged.
Matching in synchronous production rule systems often use some variant of the
RETE network (Forgy 1982). RETE is an incremental matching algorithm where

Concurrent Goal-Based Execution of Constraint Handling Rules
r 1@A(x )\B (x ), C (y) ⇐⇒ D(x , y)

25

{A(1), A(2), B (1), B (2), C (3)}
Entrance

mmm
mmm
m
m
mm
vv mm
m
A(x) M1



PPP
PPP
PPP
PPP
''


B(x) M2

mm
mmm
m
m
mmm
m
vv mm

C(y) M3













 

A(x),B(x) M4

QQQ
QQQ
QQQ
QQQ
((

A(x),B(x),C(y) M5



r1
M1 = {A(1), A(2)} M2 = {B (1), B (2)} M3 = {C (3)}
M4 = {{A(1), B (1)}, {A(2), B (2)}}
M5 = {{A(1), B (1), C (3)}, {A(2), B (2), C (3)}}

Fig. 9. Example of a RETE network, in CHR context

matching is done eagerly (data driven) in that each newly added assertion (constraint in CHR context) triggers computation of all it is possible matches to rule
heads. Figure 9 illustrates a RETE network (acyclic graph), described in CHR
context. Root node is the entrance where new constraints are added. Intermediate
nodes with single output edges are known as alpha nodes. Intermediate nodes with
two output edges are beta nodes, representing joins between alpha nodes. Each
alpha node is associated with a set of constraint matching its pattern, while a beta
node is associated with a set of partial/complete matches. Parallel implementation
of RETE (Mahajan and Kumar 1990) allows distinct parts of the network to be
computed in parallel.
The most distinct characteristic of RETE is that partial matches are computed
and stored. This and the eager nature of RETE matching is suitable for production rule systems as assertions (constraints) are propagated (not deleted) by default. Hence computing all matches rarely results in redundancy. Traditional CHR
systems do not advocate this eager matching scheme because doing so results to
many redundancies, due to overlapping simplified matching heads. Eager matching algorithms is also proved in (D.P. Miranker and Gadbois 1990) to have a larger
asymptotic worst-case space complexity than lazy matching algorithms.
In (Miranker 1990), the matching algorithm TREAT is proposed. TREAT is
similar to RETE, except it does not store partial matches. TREAT performs better

26

Edmund S. L. Lam and Martin Sulzmann

than RETE if the overhead of maintaining and storing partial matches outweighs
that of re-computing partial matches.
Asynchronous parallel production rule systems (e.g. Swarm (Gamble 1990), CREL
(Miranker et al. 1989)) introduce parallel rule execution via asynchronously running processors/threads. In such systems, rules can fire asynchronously (not synchronized by production cycles), hence enforcing execution strategies is more difficult and limited. Similar to implementations of goal based CHR semantics rule
matching is such systems often use a variant of the LEAPS (D.P. Miranker and Gadbois 1990)
lazy matching algorithm.
5.3.1 Observations
Staging executions in synchronous parallel production rule systems allows for flexibility in imposing execution strategies, but at a cost. In (Neiman 1991), synchronous
execution of UMPOPS production rule system is shown to be less efficient than
asynchronous execution. Hence it is clear that synchronous systems will only be
necessary if we wish to impose some form of execution strategies on top of the
abstract CHR semantics (e.g. rule-priority, refined operational semantics). We are
interested in concurrent CHR semantics on the abstract CHR semantics. Its nondeterminism and monotonicity property provides us with the flexibility to avoid
executing threads in strict staging cycles. Thus our approach is very similar to
asynchronous parallel production rule systems.
Lazy matching in single-threaded CHR execution is the best choice, since we
only ever have one thread of execution and wish to avoid computing overlapping
(redundant) rule head matches. No doubt that in a parallel setting, eager matching
(like RETE, TREAT) may be more optimal if the executed CHR program consist
of rules with more propagated heads. This is because we compute more matches
in parallel with brute force (find all match) parallelism and we can get away with
less redundancy. Yet to cater for the general case (more simplified heads), we again
choose lazy matching.
We therefore conclude that the goal-based execution model of CHR is still the
ideal choice for a parallel implementation of the abstract CHR semantics. For CHR
with rule priorities or refined CHR operational semantics, a variant of the synchronous parallel production rule execution is a possible choice. We leave this topic
for future work.
6 Conclusion
We have introduced a novel concurrent goal-based CHR semantics which is inspired
by traditional single-threaded (sequential) goal-based CHR execution models. Existing CHR semantics aim at introducing specific execution strategies (e.g. ordered
goal execution, rule priorities) on top of the CHR abstract semantics, hence adding
more determinism. In contrast, the concurrent goal-based CHR semantics exploits
the inherent non-deterministic and concurrent abstract CHR semantics, while introducing a systematic goal-based execution strategy. We have shown that all concur-

Concurrent Goal-Based Execution of Constraint Handling Rules

27

rent derivations can be replicated in the sequential goal-based semantics and that
there is a correspondence between the sequential goal-based semantics and the abstract CHR semantics. Thus, establishing correctness of our concurrent goal-based
CHR semantics. Our semantics provides the basis for an efficient parallel CHR
implementation. The details are studied elsewhere (Sulzmann and Lam 2008).
An interesting question is how our concurrent semantics can help to parallelize an
existing single-threaded semantics such as (Duck et al. 2004). We leave the study
of this issue for future work.

Acknowledgments
We thank the reviewers for their helpful comments on a previous version of this
paper.

References
Abdennadher, S. 1997. Operational semantics and confluence of constraint propagation
rules. In Proc. of CP’97. LNCS. Springer-Verlag, 252–266.
Abdennadher, S., Fruhwirth, T., and Meuss, H. 1999. Confluence and semantics of
constraint simplification rules. Constraints Journal 4.
Betz, H., Raiser, F., and Frühwirth, T. 2009. Persistent constraints in Constraint
Handling Rules. In WLP ’09: Proc. 23rd Workshop on (Constraint) Logic Programming.
To appear.
De Koninck, L., Stuckey, P., and Duck, G. 2008. Optimizing compilation of CHR
with rule priorities. In Proc. of FLOPS’08. LNCS, vol. 4989. Springer-Verlag, 32–47.
D.P. Miranker, D. Brant, B. L. and Gadbois, D. 1990. On the performance of
lazy matching in production systems. In In proceedings of International Conference on
Artificial Intelligence AAAI. 685–692.
Duck, G. J. 2005. Compilation of Constraint Handling Rules. Ph.D. thesis, The University of Melbourne.
Duck, G. J., Stuckey, P. J., de la Banda, M. J. G., and Holzbaur, C. 2004. The
refined operational semantics of Constraint Handling Rules. In Proc of ICLP’04. LNCS,
vol. 3132. Springer-Verlag, 90–104.
Forgy, C. 1982. Rete: A fast algorithm for the many patterns/many objects match
problem. Artif. Intell. 19, 1, 17–37.
Forgy, C. and McDermott, J. P. 1977. Ops, a domain-independent production system
language. In IJCAI. 933–939.
Frühwirth, T. 1998. Theory and practice of constraint handling rules. Journal of Logic
Programming, Special Issue on Constraint Logic Programming 37, 1-3, 95–138.
Frühwirth, T. 2005. Parallelizing union-find in Constraint Handling Rules using confluence analysis. In Proc. of ICLP’05. LNCS, vol. 3668. Springer-Verlag, 113–127.
Frühwirth, T. 2006. Constraint handling rules: the story so far. In Proc. of PPDP ’06.
ACM Press, 13–14.
Gamble, R. F. 1990. Transforming rule-based programs: from the sequential to the
parallel. In IEA/AIE ’90: Proceedings of the 3rd international conference on Industrial
and engineering applications of artificial intelligence and expert systems. ACM, New
York, NY, USA, 854–863.

28

Edmund S. L. Lam and Martin Sulzmann
Sequential Goal-based Semantics k -closure
δ

(k-Step)

σ

֌0G

σ ֌G σ ′

σ ′ ֌kG σ ′′

σ
σ ֌k+1
σ ′′
G

Concurrent Goal-based Semantics k -closure
δ

σ ֌||G σ ′
(k-Step)

σ

֌0||G

σ ′ ֌k||G σ ′′

σ
σ ֌k+1
σ ′′
||G

Fig. 10. k -closure derivation steps

Gupta, A., Forgy, C., Kalp, D., Newell, A., and Tambe, M. 1988. Parallel ops5 on
the encore multimax. In ICPP (1). 71–280.
Ishida, T. 1991. Parallel rule firing in production systems. IEEE Transactions on Knowledge and Data Engineering 3, 1, 11–17.
Lam, E. S. L. and Sulzmann, M. 2007. A concurrent Constraint Handling Rules implementation in Haskell with software transactional memory. In Proc. of ACM SIGPLAN
Workshop on Declarative Aspects of Multicore Programming (DAMP’07). 19–24.
Mahajan, M. and Kumar, V. K. P. 1990. Efficient parallel implementation of rete
pattern matching. Comput. Syst. Sci. Eng. 5, 3, 187–192.
Miranker, D. P. 1990. TREAT: a new and efficient match algorithm for AI production
systems. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA.
Miranker, D. P., Kuo, C., and Browne, J. C. 1989. Parallelizing transformations for
a concurrent rule execution language. Tech. rep., Austin, TX, USA.
Neiman, D. E. 1991. Control issues in parallel rule-firing production systems. In in
Proceedings of National Conference on Artificial Intelligence. 310–316.
Sarna-Starosta, B. and Ramakrishnan, C. R. 2007. Compiling constraint handling
rules for efficient tabled evaluation. In Proc. of PADL’07. LNCS, vol. 4354. Springer,
170–184.
Schrijvers, T. 2005. Analyses, optimizations and extensions of Constraint Handling
Rules: Ph.D. summary. In Proc. of ICLP’05. LNCS, vol. 3668. Springer-Verlag, 435–
436.
Schrijvers, T. and Sulzmann, M. 2008. Transactions in constraint handling rules. In
Proc. of ICLP’08. LNCS, vol. 5366. Springer, 516–530.
Sneyers, J., Schrijvers, T., and Demoen, B. 2005. Guard and continuation optimization for occurrence representations of chr. In Proc. of ICLP’05. LNCS, vol. 3668.
Springer-Verlag, 83–97.
Sneyers, J., Van Weert, P., Schrijvers, T., and De Koninck, L. As time goes by:
Constraint handling rules - a survey of chr research from 1998 to 2007. To appear in
TPLP.
Sulzmann, M. and Lam, E. S. L. 2008. Parallel execution of multi-set constraint rewrite
rules. In Proc. of PPDP’08. ACM Press, 20–31.

Concurrent Goal-Based Execution of Constraint Handling Rules

29

7 Proofs
In this section, we provide the proofs of the Lemmas and Theorems discussed in
this paper. Because many of our proofs rely on inductive steps on the derivations,
we define k -step derivations to facilitate the proof mechanisms. Figure 10 shows
δ

k -step derivations of the sequential goal-based derivations ֌G and the concurrent
δ

goal-based derivations ֌||G .
7.0.2 Proof of Correspondence of Derivations
Theorem 2 (Correspondence of Sequential Derivations) For any reachable
CHR state hG | Sni, CHR state hG ′ | Sn ′ i and CHR Program P,
if
then

hG | Sni ֌∗G hG ′ | Sn ′ i
(NoIds(G) ⊎ DropIds(Sn)) = (NoIds(G ′ ) ⊎ DropIds(Sn ′ )) ∨
(NoIds(G) ⊎ DropIds(Sn)) ֌∗A (NoIds(G ′ ) ⊎ DropIds(Sn ′ ))

where NoIds = {c | c ∈ G, c is a CHR constraint} ⊎ {e | e ∈ G, e is an equation}
Proof
We prove that for all finite n and reachable states hG | Sni, hG ′ | Sn ′ i, hG | Sni ֌nG
hG ′ | Sn ′ i either yields equivalent abstract stores or corresponds to some abstract
semantics derivation. We prove by induction on the derivation steps n. Showing that
goal-based derivation of any finite n steps satisfying one of the following conditions:
• (C1) (NoIds(G) ⊎ DropIds(Sn)) = (NoIds(G ′ ) ⊎ DropIds(Sn ′ ))
• (C2) (NoIds(G) ⊎ DropIds(Sn)) ֌∗A (NoIds(G ′ ) ⊎ DropIds(Sn ′ ))
We have the following axioms, by definition of the functions NoIds and DropIds,
for any goals G or store Sn:
•
•
•
•
•
•

(a1) For any equation e, NoIds({e} ⊎ G) = {e} ⊎ NoIds(G)
(a2) For any equation e, DropIds({e} ∪ Sn) = {e} ⊎ DropIds(Sn)
(a3) For any numbered constraint c#i, NoIds({c#i} ⊎ G) = NoIds(G)
(a4) For any numbered constraint c#i, DropIds({c#i} ∪ Sn) = {c} ⊎ DropIds(Sn)
(a5) For any CHR constraint c, NoIds({c} ⊎ G) = {c} ⊎ NoIds(G)
(a6) For any store Sn ′ , DropIds(Sn ∪ Sn ′ ) = DropIds(Sn) ⊎ DropIds(Sn ′ )
(a1) and (a2) are so because NoIds and DropIds have no effect on equations. (a3)
is true because NoIds is defined to drop numbered constraints. (a4) is true because
DropIds is defined to remove identifier components of numbered constraints. We
have (a5) because NoIds has no effect on CHR constraints. By definition of DropIds,
(a6) is true.
Base case: We consider hG | Sni ֌0G hG ′ | Sn ′ i. By definition of ֌0G , we
have G = G ′ and Sn = Sn ′ . Hence (NoIds(G) ⊎ DropIds(Sn)) = (NoIds(G ′ ) ⊎
DropIds(Sn ′ )) and we are done.
Inductive case: We assume that the theorem is true for some finite k > 0, hence
hG | Sni ֌kG hG ′ | Sn ′ i have some correspondence with the abstract semantics.

30

Edmund S. L. Lam and Martin Sulzmann

We now prove that by extending these k derivations with another step, we preδ

serve correspondence, namely hG | Sni ֌kG hG ′ | Sn ′ i ֌G hG ′′ | Sn ′′ i has a
correspondence with the abstract semantics. We prove this by considering all possible form of derivation step, step k + 1 can take:
δ

• (Solve) k + 1 step is of the form h{e} ⊎ G ′′′ | Sn ′ i ֌G hW ⊎ G ′′′ | {e} ∪ Sn ′ i such
that for some G ′′′ and W
G ′ = {e} ⊎ G ′′′ , G ′′ = W ⊎ G ′′′ and Sn ′′ = {e} ∪ Sn ′

(asolve )

where e is an equation, W = WakeUp(e, Sn) contains only goals of the form c#i.
This is because (Solve) only wakes up stored numbered constraints. Hence,
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ ) =
=
=
=
=

NoIds(W ⊎ G ′′′ ) ⊎ DropIds({e} ∪ Sn ′ ) (asolve )
NoIds(G ′′′ ) ⊎ DropIds({e} ∪ Sn ′ ) (a3)
NoIds(G ′′′ ) ⊎ {e} ⊎ DropIds(Sn ′ ) (a2)
NoIds({e} ⊎ G ′′′ ) ⊎ DropIds(Sn ′ ) (a1)
NoIds(G ′ ) ⊎ DropIds(Sn ′ ) (asolve )

Hence we can conclude that the evaluated store of derivation step k +1 is equivalent
to abstract store of evaluated store of step k , therefore satisfying condition (C1).
δ

• (Activate) k +1 step is of the form h{c}⊎G ′′′ | Sn ′ i ֌G h{c#i}⊎G ′′′ | {c#i}∪Sn ′ i
such that for some G ′′′
G ′ = {c} ⊎ G ′′′ , G ′′ = {c#i} ⊎ G ′′′ and Sn ′′ = {c#i} ∪ Sn ′

(aact )

Hence,
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ )
= NoIds({c#i} ⊎ G ′′′ ) ⊎ DropIds({c#i} ∪ Sn ′ )
= NoIds(G ′′′ ) ⊎ DropIds({c#i} ∪ Sn ′ )
= NoIds(G ′′′ ) ⊎ {c} ⊎ DropIds(Sn ′ )
= NoIds({c} ⊎ G ′′′ ) ⊎ DropIds(Sn ′ )
= NoIds(G ′ ) ⊎ DropIds(Sn ′ )

(aact )
(a3)
(a4)
(a5)
(aact )

Hence we can conclude that evaluated store of derivation step k + 1 is equivalent
to abstract store of evaluated store of step k , therefore satisfying condition (C1).
δ

• (Simplify) k + 1 step is of the form h{c#i} ⊎ G ′′′ | HP ∪ {c#i} ∪ HS ∪ Sn ′′′ i ֌G
hB ⊎ G ′′′ | HP ∪ Sn ′′′ i for some HP ,HS and B such that for some G ′′′ and Sn ′′′
Sn ′ = HP ∪ {c#i} ∪ HS ∪ Sn ′′′ , Sn ′′ = HP ∪ Sn ′′′ ,
G ′ = {c#i} ⊎ G ′′′ and G ′′ = B ⊎ G ′′′
and there exists a CHR rule

r @HP′ \HS′
φ(HS′ )
′

DropIds({c#i} ∪ HS ) =
Eq(Sn ′′′ ) |= φ ∧ tg B = φ(B )

(a1simp )

′

⇐⇒ tg | B such that exists φ where
DropIds(HP ) = φ(HP′ )
(a2simp )

Hence,
NoId (G ′ ) ⊎ DropIds(Sn ′ )
= NoIds({c#i } ⊎ G ′′′ ) ⊎ DropIds(HP ∪ {c#i } ∪ HS ∪ Sn ′′′ )
= NoIds(G ′′′ ) ⊎ DropIds(HP ∪ {c#i } ∪ HS ∪ Sn ′′′ )
= NoIds(G ′′′ ) ⊎ DropIds(HP ) ⊎ DropIds({c#i } ∪ HS ) ⊎ DropIds(Sn ′′′ )
= NoIds(G ′′′ ) ⊎ φ(HP′ ) ⊎ φ(HS′ ) ⊎ DropIds(Sn ′′′ )

(a1simp )
(a3 )
(a6 )
(a2simp )

Concurrent Goal-Based Execution of Constraint Handling Rules

31

By definition of the abstract semantics and a2simp , we know that we have the rule
application φ(HP′ )∪φ(HS′ ) ֌A φ(B ′ ) Therefore, by monotonicity of CHR rewriting
(Theorem 1)
NoId (G ′ ) ⊎ DropIds(Sn ′ )
=
NoIds(G ′′′ ) ⊎ φ(HP′ ) ⊎ φ(HS′ ) ⊎ DropIds(Sn ′′′ )
֌A NoIds(G ′′′ ) ⊎ φ(B ′ ) ⊎ DropIds(Sn ′′′ )
=
NoIds(φ(B ′ ) ⊎ G ′′′ ) ⊎ DropIds(Sn ′′′ )
=
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ )

(Theorem1)
(a1), (a3)
(a1simp )

Therefore, we have NoId (G) ⊎ DropIds(Sn) ֌∗A NoId (G ′ ) ⊎ DropIds(Sn ′ ) ֌A
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ ), such that the k + 1 goal-based derivation step satisfy
condition (C2).
δ

• (Propagate) k + 1 step is of the form h{c#i} ⊎ G ′′′ | HP ∪ {c#i} ∪ HS ∪ Sn ′′′ i ֌G
hB ⊎ {c#i} ⊎ G ′′′ | HP ∪ {c#i} ∪ Sn ′′′ i for some HP ,HS and B such that for some
G ′′′ and Sn ′′′
Sn ′ = HP ∪ {c#i} ∪ HS ∪ Sn ′′′ , Sn ′′ = HP ∪ {c#i} ∪ Sn ′′′ ,
G ′ = {c#i} ⊎ G ′′′ and G ′′ = B ⊎ {c#i} ⊎ G ′′′

(a1prop )

and there exists a CHR rule r @HP′ \HS′ ⇐⇒ tg | B ′ such that exists φ where
DropIds(HS ) = φ(HS′ ) DropIds({c#i} ∪ HP ) = φ(HP′ )
Eq(Sn ′′′ ) |= φ ∧ tg B = φ(B ′ )

(a2prop )

Hence,
NoId (G ′ ) ⊎ DropIds(Sn ′ )
= NoIds({c#i } ⊎ G ′′′ ) ⊎ DropIds(HP ∪ {c#i } ∪ HS ∪ Sn ′′′ )
= NoIds(G ′′′ ) ⊎ DropIds(HP ∪ {c#i } ∪ HS ∪ Sn ′′′ )
= NoIds(G ′′′ ) ⊎ DropIds({c#i } ∪ HP ) ⊎ DropIds(HS ) ⊎ DropIds(Sn ′′′ )
= NoIds(G ′′′ ) ⊎ φ(HP′ ) ⊎ φ(HS′ ) ⊎ DropIds(Sn ′′′ )

(a1prop )
(a3 )
(a6 )
(a2prop )

By definition of the abstract semantics and a2simp , we know that we have the rule
application φ(HP′ )∪φ(HS′ ) ֌A φ(B ′ ) Therefore, by monotonicity of CHR rewriting
(Theorem 1)
NoId (G ′ ) ⊎ DropIds(Sn ′ )
=
NoIds(G ′′′ ) ⊎ φ(HP′ ) ⊎ φ(HS′ ) ⊎ DropIds(Sn ′′′ )
֌A NoIds(G ′′′ ) ⊎ φ(B ′ ) ⊎ DropIds(Sn ′′′ )
=
NoIds(φ(B ′ ) ⊎ G ′′′ ) ⊎ DropIds(Sn ′′′ )
=
NoIds(φ(B ′ ) ⊎ {c#i } ⊎ G ′′′ ) ⊎ DropIds(Sn ′′′ )
=
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ )

(Theorem1)
(a1), (a5)
(a3)
(a1prop )

Therefore, we have NoId (G) ⊎ DropIds(Sn) ֌∗A NoId (G ′ ) ⊎ DropIds(Sn ′ ) ֌A
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ ), such that the k + 1 goal-based derivation step satisfy
condition (C2).
δ

• (Drop) k + 1 step is of the form h{c#i} ⊎ G ′′ | Sn ′ i ֌G h{G ′′ | Sn ′ i such that for
some G ′′′
G ′′ = {c#i} ⊎ G ′ and Sn ′ = Sn ′′

(adrop )

Hence,
NoIds(G ′′ ) ⊎ DropIds(Sn ′′ ) =
=

NoIds({c#i} ⊎ G ′ ) ⊎ DropIds(Sn ′ ) (adrop )
NoIds(G ′ ) ⊎ DropIds(Sn ′ )
(a3)

32

Edmund S. L. Lam and Martin Sulzmann

Hence we can conclude that evaluated store of derivation step k + 1 is equivalent
to abstract store of evaluated store of step k , therefore satisfying condition (C1).
Considering all forms of k + 1 derivation steps, (Solve), (Activate) and (Drop)
satisfies condition bf (C 1), while (Simplify) and (Propagate) satisfy condition (C2).
Hence we can conclude that Theorem 2 holds.
Lemma 1 (k -Concurrency) For any finite k of mutually non-overlapping concurrent derivations,
HP1 \HS 1

hG1 | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hG1′ | {} ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i
··
HPi \HSi

hGi | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hGi′ | HS 1 ∪ · · ∪{} ∪ · · ∪HSk ∪ S i
··
HPk \HSk

hGk | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i ֌||G hGk′ | HS 1 ∪ · · ∪HSi ∪ · · ∪{} ∪ S i
HP 1 ⊆ S · ·HPi ⊆ S · ·HPk ⊆ S
δ = HP 1 ∪ · · ∪HPi ∪ · · ∪HPk \HS 1 ∪ · · ∪HSi ∪ · · ∪HSk
hG1 ⊎ · · ⊎Gi ⊎ · · ⊎Gk ⊎ G | HS 1 ∪ · · ∪HSi ∪ · · ∪HSk ∪ S i
δ
֌||G

hG1′ ⊎ · · ⊎Gi′ ⊎ · · ⊎Gk′ ⊎ G | S i

we can decompose this into k −1 applications of the (pair-wise) (Goal Concurrency)
derivation step.
Proof
We prove the soundness of k -concurrency by showing that k mutually non-overlapping
concurrent derivation can be decomposed into k −1 applications of (Goal Concurrency)
step. We prove by induction on the number of concurrent derivations k .
Base case: k = 2. 2-concurrency immediately corresponds to (Goal Concurrency)
rule, hence it is true by definition.
Inductive case: We assume that for j > 2 and j < k , we can decompose
j mutually non-overlapping concurrent derivations. into j − 1 applications of the
(Goal Concurrency) step. We now consider j + 1 mutually non-overlapping concurrent derivations. Because all derivations are non-overlapping, we can compose
any two derivations amongst these j + 1 into a single concurrent step via the
(Goal Concurrency) rule. We pick any two concurrent derivations, say the j th and
(j + 1)th (Note that by symmetry, this choice is arbitrary):
HPj \HSj

hGj | HS 1 ∪ · · ∪HSj ∪ HSj +1 ∪ S i ֌||G hGj′ | HS 1 ∪ · · ∪{} ∪ HSj +1 ∪ S i
hGj +1 | HS 1 ∪ · · ∪HSj ∪ HSj +1 ∪ S i

HPj +1 \HSj +1

HPj ⊆ S

֌||G

hGj′+1 | HS 1 ∪ · · ∪HSj ∪ {} ∪ S i

HPj +1 ⊆ S

By applying the above two non-overlapping derivations with an instance of the
(Goal Concurrency) rule, we have:
HPj ′ \HSj ′

hGj ′ | HS 1 ∪ · · ∪HSj ′ ∪ S i ֌||G hGj′′ | HS 1 ∪ · · ∪{} ∪ S i
where Gj ′ = Gj ⊎ Gj +1 Gj′′ = Gj′ ⊎ Gj′+1
HSj ′ = HSj ∪ HSj +1 HPj ′ = HPj ∪ HPj +1

Concurrent Goal-Based Execution of Constraint Handling Rules

33

Hence we have reduced j + 1 non-overlapping concurrent derivations into j
non-overlapping concurrent derivations by combining via the (Goal Concurrency)
derivation step.
HP1 \HS 1

hG1 | HS 1 ∪ · · ∪HSj ′ ∪ S i ֌||G hG1′ | {} ∪ · · ∪HSj ′ ∪ S i
··
HPj ′ \HSj ′

hGj ′ | HS 1 ∪ · · ∪HSj ′ ∪ S i ֌||G hGj′′ | HS 1 ∪ · · ∪{} ∪ S i
HP 1 ⊆ S · ·HPj ′ ⊆ S
δ = HP 1 ∪ · · ∪HPj ′ \HS 1 ∪ · · ∪HSj ′
hG1 ⊎ · · ⊎Gj ′ ⊎ G | HS 1 ∪ · · ∪HSj ′ ∪ S i
δ
֌||G

hG1′ ⊎ · · ⊎Gj′′ ⊎ G | S i

Hence, by our original assumption, the above is decomposable into j − 1 applications of the (Goal Concurrency) step. This implies that j + 1 concurrent derivations
are decomposable into j (Goal Concurrency) step.
Lemma 2 (Monotonicity of Goals in Goal-based Semantics) For any goals
G,G ′ and G ′′ and CHR store Sn and Sn ′ , if hG | Sni ֌∗G hG ′ | Sn ′ i then hG ⊎ G ′′ |
Sni ֌∗G hG ′ ⊎ G ′′ | Sn ′ i
Proof
We need to prove that for any finite k , if hG | Sni ֌kG hG ′ | Sn ′ i we can always
extend the goals with any G ′′ such that hG ⊎ G ′′ | Sni ֌kG hG ′ ⊎ G ′′ | Sn ′ i.
We prove this by induction on the number of derivation steps k , showing that
for any finite i ≤ k , goals are monotonic.
Base case: We consider hG | Sni ֌0G hG ′ | Sn ′ i. By definition of ֌0G , we have
G = G ′ and Sn = Sn ′ . Hence we immediately have hG ⊎ G ′′ | Sni ֌0G hG ′ ⊎ G ′′ |
Sn ′ i
Inductive case: We assume that the lemma is true for some finite i > 0, hence
hG | Sni ֌iG hG ′ | Sn ′ i is monotonic with respect to the goals.
We now prove that by extending these i derivations with another step, we still
δ

preserve monotonicity of the goals. Namely, if hG | Sni ֌iG h{g} ⊎ Gi | Sni i ֌G
δ

hGi+1 | Sni+1 i then hG ⊎ G ′′ | Sni ֌iG hGi ⊎ G ′′ | Sni i ֌G hGi+1 ⊎ G ′′ | Sni+1 i
We prove this by considering all possible form of derivation step, step i + 1th can
take:
• (Solve) Consider i + 1th derivation step of the form h{e} ⊎ Gi | Sni i ֌G hW ⊎ G |
{e} ∪ Sni i for some equation e and W = WakeUp(e, Sni ).
By definition, the (Solve) step only make reference to e and Sni , hence we can
extend Gi with any G ′′ without affecting the derivation step, i.e.
h{e} ⊎ Gi ⊎ G ′′ | Sni i ֌G hW ⊎ Gi ⊎ G ′′ | {e} ∪ Sni i
Hence, given our assumption that the first i derivations are monotonic with respect
to the goals, extending with a i + 1th (Solve) step preserves monotonicity of the
goals.

34

Edmund S. L. Lam and Martin Sulzmann

• (Activate) Consider i +1th derivation step of the form h{c}⊎Gi | Sni i ֌G h{c#j }⊎
Gi | {c#j } ∪ Sni i for some CHR constraint c, goals Gi and store Sni .
By definition, the (Activate) step only make reference to goal c, hence we can
extend Gi with any G ′′ without affecting the derivation step, i.e.
h{c} ⊎ Gi ⊎ G ′′ | Sni i ֌G h{c#j } ⊎ Gi ⊎ G ′′ | {c#j } ∪ Sni i
Hence, given our assumption that the first i derivations are monotonic with respect
to the goals, extending with a i + 1th (Activate) step preserves monotonicity of the
goals.
• (Simplify) Consider i + 1th derivation step of the form h{c#j } ⊎ Gi | {c#j } ⊎ HS ∪
Sni i ֌G hB ⊎ Gi | Sni i for some CHR constraints HS and body constraints B .
By definition, the (Simplify) step only make reference to goal c#j , and HS of the
store, hence we can extend Gi with any G ′′ without affecting the derivation step,
i.e.
h{c#j } ⊎ Gi ⊎ G ′′ | {c#j } ∪ HS ∪ Sni i ֌G hB ⊎ Gi ⊎ G ′′ | Sni i
Hence, given our assumption that the first i derivations are monotonic with respect
to the goals, extending with a i + 1th (Simplify) step preserves monotonicity of the
goals.
• (Propagate) Consider i +1th derivation step of the form h{c#j }⊎Gi | HS ∪Sni i ֌G
hB ⊎ {c#j } ⊎ Gi | Sni i for some CHR constraints HS and body constraints B .
By definition, the (Propagate) step only make reference to goal c#j , and HS of the
store, hence we can extend Gi with any G ′′ without affecting the derivation step,
i.e.
h{c#j } ⊎ Gi ⊎ G ′′ | HS ∪ Sni i ֌G hB ⊎ {c#j } ⊎ Gi ⊎ G ′′ | Sni i
Hence, given our assumption that the first i derivations are monotonic with respect
to the goals, extending with a i + 1th (Propagate) step preserves monotonicity of
the goals.
• (Drop) Consider i +1th derivation step of the form h{c#j }⊎Gi | Sni i ֌G hGi | Sni i
for some numbered constraint c#j .
By definition, the (Drop) step only make reference to goal c#j , while its premise
depend on Sni , hence we can extend goals Gi with any G ′′ without affecting the
derivation step, i.e.
h{c#j } ⊎ Gi ⊎ G ′′ | Sni i ֌G hGi ⊎ G ′′ | Sni i
Hence, given our assumption that the first i derivations are monotonic with respect
to the goals, extending with a i + 1th (Drop) step preserves monotonicity of the
goals.
Hence, with our assumption of monotonicity of goals for i steps, the goals are still
monotonic for i + 1 steps regardless of the form of the i + 1th derivation step.
HP \HS

Lemma 3 (Isolation of Goal-based Derivations) If hG | HP ∪ HS ∪ S1 ∪ S2 i ֌G
HP \HS

hG ′ | HP ∪ S1′ ∪ S2 i then hG | HP ∪ HS ∪ S1 i ֌G hG ′ | HP ∪ S1′ i

Concurrent Goal-Based Execution of Constraint Handling Rules

35

Proof
We need to show that for any goal-based derivation, we can omit any constraint of
the store which is not a side-effect of the derivation. To prove this, we consider all
possible forms of goal-based derivations:
• (Solve) Consider derivation of the form
W \{}

h{e} ⊎ G | W ∪ {} ∪ S1 ∪ S2 i ֌G hW ⊎ G | W ∪ {} ∪ {e} ∪ S1 ∪ S2 i
Since wake up side-effect is captured in W , we can drop S2 without affecting the
derivation. Hence we also have:
W \{}

h{e} ⊎ G | W ∪ {} ∪ S1 i ֌G hW ⊎ G | W ∪ {} ∪ {e} ∪ S1 i
• (Activate) Consider derivation of the form
{}\{}

h{c} ⊎ G | {} ∪ {} ∪ S1 ∪ S2 i ֌G h{c#i} ⊎ G | {} ∪ {} ∪ {c#i} ∪ S1 ∪ S2 i
Since (Activate) simply introduces a new constraint c#i into the store, we can drop
S2 without affecting the derivation. Hence we also have:
{}\{}

h{c} ⊎ G | {} ∪ {} ∪ S1 i ֌G h{c#i} ⊎ G | {} ∪ {} ∪ {c#i} ∪ S1 i
• (Simplify) Consider derivation of the form
HP \HS

h{c#i} ⊎ G | HP ∪ HS ∪ S1 ∪ S2 i ֌G hB ⊎ G | HP ∪ S1 ∪ S2 i
Since S2 is not part of the side-effects of this derivation, we can drop S2 without
affecting the derivation. Hence we also have:
HP \HS

h{c#i} ⊎ G | HP ∪ HS ∪ S1 i ֌G hB ⊎ G | HP ∪ S1 i
• (Propagate) Consider derivation of the form
HP \HS

h{c#i} ⊎ G | HP ∪ HS ∪ S1 ∪ S2 i ֌G hB ⊎ {c#i} ⊎ G | HP ∪ S1 ∪ S2 i
Since S2 is not part of the side-effects of this derivation, we can drop S2 without
affecting the derivation. Hence we also have:
HP \HS

h{c#i} ⊎ G | HP ∪ HS ∪ S1 i ֌G hB ⊎ {c#i} ⊎ G | HP ∪ S1 i
• (Drop) Consider derivation of the form
{}\{}

h{c#i} ⊎ G | {} ∪ {} ∪ S1 ∪ S2 i ֌G hG | {} ∪ {} ∪ S1 ∪ S2 i
(Drop) simply removes the goal c#i when no instances of (Simplify) or (Propagate)
can apply on it. Note that its premise references to the entire store, so removing
S2 may seems unsafe. But since removing constraints from the store will not cause
c#i to be applicable to any instances of (Simplify) or (Propagate), hence we also
have:
{}\{}

h{c} ⊎ G | {} ∪ {} ∪ S1 i ֌G hG | {} ∪ {} ∪ S1 i

36

Edmund S. L. Lam and Martin Sulzmann

Lemma 4 (Isolation of Transitive Goal-based Derivations) If hG | HP ∪ HS ∪ S1 ∪
S2 i ֌∗G hG ′ | HP ∪S1′ ∪S2 i with side-effects δ = HP \HS , then hG | HP ∪HS ∪S1 i ֌∗G
hG ′ | HP ∪ S1′ i
Proof
k

We need to prove that for all k , hG | HP ∪ HS ∪ S1 ∪ S2 i ֌G hG ′ | HP ∪ S1′ ∪ S2 i
with side-effects δ = HP \HS we can always safely omit affected potions of the store
from the derivation. We prove by induction on i ≤ k .
Base case: i = 1. Consider, hG | HP ∪ HS ∪ S1 ∪ S2 i ֌1G hG ′ | HP ∪ S1′ ∪ S2 i.
This corresponds to the premise in Lemma 3, hence we can safely omit S2 from the
derivation.
Inductive case: i > 1. we assume that for any hG | HPi ∪ HSi ∪ S1i ∪ S2i i ֌iG
′
hG ′ | HPi ∪ S1i
∪ S2i i with side-effects δi = HPi \HSi , we can safely omit S2i from
the derivation. Let’s consider a j = i + 1 derivation step from here, which contains
side-effects δj = HPj \HSj non-overlapping with δi . Hence HPj and HSj must be in
S2i (i.e. S2i = HPj ∪ HSj ∪ S1j ∪ S2j ).
֌iG
δj
֌G

hG | HPi ∪ HSi ∪ S1i ∪ HPj ∪ HSj ∪ S1j ∪ S2j i
′
hG ′ | HP ∪ S1i
∪ HPj ∪ HSj ∪ S1j ∪ S2j i
′
′
∪ S2j i
∪ HPj ∪ S1j
hG ′′ | HP ∪ S1i

Hence consider the following substitutions:
HP = HPi ∪ HPj
S1 = S1i ∪ S1j
δ = HP \HS

HS = HSi ∪ HSj
′
′
S1′ = S1i
∪ S1j

hG | HP ∪ S1′ ∪ S2j i with side-effects δ such
we have hG | HP ∪ HS ∪ S1 ∪ S2j i ֌i+1
G
that no constraints in S2j is in δ. Hence we can safely omit S2j from the derivation
and we have isolation for i + 1 derivations as well.
Lemma 5 (Sequential Reachability of Concurrent Derivation Steps) For any sequenδ

tially reachable CHR state σ, CHR state σ ′ and rewriting side-effects δ if σ ֌||G σ ′
then σ ′ is sequentially reachable, σ ֌∗G σ ′ with side-effects δ.
Proof
From the k -concurrency Lemma (Lemma 1) we showed that any finite k mutually
non-overlapping concurrent goal-based derivations can be replicated by nested application of the (Goal Concurrency) step. Hence, to prove sequential reachability
of concurrent derivations, we only need to consider the derivation steps (Lift) and
(Goal Concurrency) which sufficiently covers the concurrent behaviour of any k
concurrent derivations.
We prove by structural induction of the concurrent goal-based semantics derivation steps (Lift) and (Goal Concurrency).
• (Lift) is the base case. Application of (Lift) simply lifts a goal-based derivation
δ

δ

σ ֌G σ ′ into a concurrent goal-based derivation σ ֌||G σ ′ . Thus states σ ′ derived

Concurrent Goal-Based Execution of Constraint Handling Rules

37

δ

from the (Lift) step is immediately sequentially reachable since σ ֌G σ ′ implies
σ ֌∗G σ ′ .
• (Goal Concurrency)
δ1

(D1)

hG1 | HS 1 ∪ HS 2 ∪ S i ֌||G hG1′ | {} ∪ HS 2 ∪ S i

(D2)

hG2 | HS 1 ∪ HS 2 ∪ S i ֌||G hG2′ | HS 1 ∪ {} ∪ S i
δ1 = HP 1 \HS 1 δ2 = HP 2 \HS 2
⊆ S HP 2 ⊆ S δ = HP 1 ∪ HP 2 \HS 1 ∪ HS 2

δ2

HP 1

hG1 ⊎ G2 ⊎ G | HS 1 ∪ HS 2 ∪ S i
(C)

δ

֌||G

hG1′ ⊎ G2′ ⊎ G | S i

we assume that (D1) and (D2) are sequentially reachable. This means that we
have the following:
hG1 | HS 1 ∪ HS 2 ∪ S i ֌∗G hG1′ | {} ∪ HS 2 ∪ S i
with side-effects δ1 = HP 1 \HS 1 such that HP 1 ⊆ S

(aD1 )

hG2 | HS 1 ∪ HS 2 ∪ S i ֌∗G hG2′ | HS 1 ∪ {} ∪ S i
with side-effects δ2 = HP 2 \HS 2 such that HP 2 ⊆ S

(aD2 )

Since both derivations are by definition non-overlapping in side-effects, we can show
that (C) is sequentially reachable, using monotonicity of goals (Lemma 2) and
isolation of derivations (Lemma 3):
֌∗G
֌∗G

hG1 ⊎ G2 ⊎ G | HS 1 ∪ HS 2 ∪ S i
hG1′ ⊎ G2 ⊎ G | HS 2 ∪ S i
(Lemma2, aD1 )
hG1′ ⊎ G2′ ⊎ G | S i
(Lemma2, Lemma4, aD2 )

Hence, the above sequential goal-based derivation shows that (Goal Concurrency)
derivation step is sequentially reachable with side-effect δ.

Theorem 3 (Sequential Reachability of Concurrent Derivations) For any
initial CHR state σ, CHR state σ ′ and CHR Program P, if σ ֌∗||G σ ′ then σ ֌∗G σ ′ .
Proof
We prove that for all finite k number of concurrent derivation steps σ ֌k||G σ ′ , we
can find a corresponding sequential derivation sequence σ ֌∗G σ ′ .
Base case: k = 1. We consider σ ֌1||G σ ′ . From Lemma 5, we can conclude that
we have σ ֌∗G σ ′ as well.
Inductive case: k > 1. We consider σ ֌k||G σ ′ , assuming that it is sequentially
reachable, hence we also have σ ֌∗G σ ′ . We consider extending this derivation with
the k +1th step σ ′ ֌||G σ ′′ . By Lemma 5, we can conclude that the k +1th concurrent
derivation is sequential reachable, hence σ ′ ֌∗G σ ′′ . Hence we have σ ֌∗G σ ′ ֌∗G σ ′′
+1 ′′
showing that σ ֌k||G
σ is sequentially reachable.

38

Edmund S. L. Lam and Martin Sulzmann
7.0.3 Correspondence of Termination

Lemma 6 (Rule instances in reachable states are always active) For any
reachable CHR state hG | Sni, any rule head instance H ⊆ Sn must be active. i.e.
∃c#i ∈ H such that c#i ∈ G.
Proof
We will prove this for the sequential goal-based semantics. Since Theorem 3 states
all concurrent derivation is sequentially reachable, this Lemma immediately applies
to the concurrent goal-based semantics as well.
We prove that for all finite k derivations from any initial CHR state hG | {}i, i.e.
hG | {}i ֌kG hG ′ | Sn ′ i, all rule head instances H ⊆ Sn ′ has at least one c#i ∈ H
such that c#i ∈ G. We prove by induction on i < k that states reachable by i
derivations from an initial stage have the above property.
Base case: i = 0. Hence hG | {}i ֌0G hG ′ | Sn ′ i. By definition, G = G ′ and
Sn ′ = {}. Since Sn ′ is empty, the base case immediately satisfies the Lemma.
Inductive case: i > 0. We assume that for any hG | {}i ֌iG hG ′ | Sn ′ i, all
rule head instances H ⊆ Sn ′ is active, hence have at least one c#i ∈ H such that
c#i ∈ G ′ . We extend this derivation with an i + 1th step, hence hG | {}i ֌iG hG ′ |
δ

Sn ′ i ֌G hG ′′ | Sn ′′ i. We now prove that all rule head instances in Sn ′′ are active.
We consider all possible forms of this i + 1th derivation step. We omit side-effects.
• (Solve) i +1 derivation step is of the form h{e}⊎G ′′′ | Sn ′ i ֌G hW ⊎G ′′′ | {e}∪Sn ′ i
for some goals G ′′′ and W = WakeUp(e, Sn ′ ). Our assumption provides that all rule
head instances in Sn ′ are active. Introducing e into the store will possibly introduce
new rule head instances. This is because for some CHR rule (r @HP \HS ⇐⇒ tg |
B ) ∈ P since we may have a new φ such that Eqs({e} ∪ Sn ′ ) |= φ ∧ tg and
φ(HP ∪ HS ) ∈ Sn ′ . This means that there is at least one c#i in φ(HP ∪ HS )
which is further grounded by e. Thankfully, by definition of W = WakeUp(e, Sn ′ ),
we have c#i ∈ W . Hence new rule head instances will become active because of
introduction of W to the goals.
• (Activate) i + 1 derivation step is of the form h{c} ⊎ G ′′′ | Sn ′ i ֌G h{c#i} ⊎ G ′′′ |
{c#i}∪Sn ′ i. Our assumption provides that all rule head instances in Sn ′ are active.
By adding c#i to the store, we can possibly introduce new rule head instances
{c#i} ∪ H such that H ∈ Sn ′ . Since c#i is also retained as a goal, such new rule
head instances are active as well.
• (Simplify) i + 1 derivation step is of the form h{c#i} ⊎ G ′′′ | {c#i} ∪ HS ∪ Sn ′ i ֌G
hB ⊎ G ′′′ | Sn ′ i. Our assumption provides that all rule head instances in Sn ′ are
active. c#i has applied a rule instance, removing c#i and some HS from the store.
Since c#i is no longer in the store, we can safely remove c#i from the goals.
Removing HS from the store will only (possibly) remove other rule head instance
from the store. Hence rule head instances in Sn ′ still remain active.
• (Propagate) i +1 derivation step is of the form h{c#i}⊎G ′′′ | {c#i}∪HS ∪Sn ′ i ֌G
hB ⊎{c#i}⊎G ′′′ | {c#i}∪Sn ′ i. Our assumption provides that all rule head instances
in Sn ′ are active. c#i has applied a rule instance, removing some HS from the store.
Since c#i is still in the store, we cannot safely remove c#i from the goals, thus

Concurrent Goal-Based Execution of Constraint Handling Rules

39

it is retained. Removing HS from the store will only (possibly) remove other rule
head instance from the store. Hence rule head instances in Sn ′ , including those that
contains c#i, still remain active.
• (Drop) i + 1 derivation step is of the form h{c#i} ⊎ G ′′′ | Sn ′ i ֌G hG ′′′ | Sn ′ i. Our
assumption provides that all rule head instances in Sn ′ are active. Premise of the
(Drop) step demands that no (Simplify) and (Propagate) steps apply on c#i. This
means that c#i is not part of any rule head instances in Sn ′ . Hence we can safely
remove c#i from the goals without risking to deactivate any rule instances.
Hence (Solve) and (Activate) guarantees that new rule head instances become active, (Drop) safely removes a goal without deactivating any rule head instances and
(Simplify) and (Propagate) only removes constraint from the store. In all cases,
existing rule head instances remain active while new rule head instances become
active, thus we have proved the lemma.
Theorem 4 (Correspondence of Termination) For any initial CHR state
hG, {}i, final CHR state h{}, Sni and terminating CHR program P,
if hG | {}i ֌∗||G h{} | Sni
then G ֌∗A DropIds(Sn) and FinalA (DropIds(Sn))
Proof
We prove that for any concurrent derivation hG | {}i ֌∗||G h{} | Sni, we have a
corresponding abstract derivation G ֌∗A DropIds(Sn). Theorem 3 states that we
can replicate the above concurrent derivation, with a sequential derivation. Hence
we have hG | {}i ֌∗G h{} | Sni. By instantiating Theorem 2, we immediately have
G ֌∗A DropIds(Sn) from this sequential goal-based derivation.
Next we show that DropIds(Sn) is a final store (FinalA (DropIds(Sn))) with respect to some CHR program P. We prove by contradiction: Suppose DropIds(Sn)
is not a final store, hence h{} | Sni has at least one rule head instance H of P in
Sn which is not active, since the goals are empty. However, this contradicts with
Lemma 6, which states that all reachable states have only active rule instances.
Since h{} | Sni is sequentially reachable, it must be the case that Sn has no rule
head instances of P. Therefore DropIds(Sn) must be a final store.

