3264852
Reasoning with higher-order abstract syntax in a logical framework
Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higher-order abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution, is difficult to treat in proofs involving induction. In this article, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed λ-terms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of definition is a proof-theoretic device that allows certain theories to be treated as "closed" or as defining fixed points. We explore the difficulties of formal meta-theoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent trade-off between the benefits of higher-order abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF:=. We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.
100764
Foundational aspects of syntax
A large variety of computing systems, such as compilers, interpreters, static analyzers, and theorem provers, need to manipulate syntactic objects like programs, types, formulas, and proofs. A common characteristic of these syntactic objects is that they contain variable binders, such as quantifiers, scoping operators, and parameters. The presence of binders complicates formal specifications and symbolic processing. Consider, for example, a function definition of the form
7600116
Cut-elimination for a logic with definitions and induction
Abstract In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic FOλ Δ N that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cut-elimination theorem for FOλ Δ N , adapting a technique due to Tait and Martin-Lof. This cut-elimination proof is technically interesting and significantly extends previous results of this kind.
14544430
Specifying Theorem Provers in a Higher-Order Logic Programming Language
Since logic programming systems directly implement search and unification and since these operations are essential for the implementation of most theorem provers, logic programming languages should make ideal implementation languages for theorem provers. We shall argue that this is indeed the case if the logic programming language is extended in several ways. We present an extended logic programming language where first-order terms are replaced with simply-typed λ-terms, higher-order unification replaces first-order unification, and implication and universal quantification are allowed in queries and the bodies of clauses. This language naturally specifies inference rules for various proof systems. The primitive search operations required to search for proofs generally have very simple implementations using the logical connectives of this extended logic programming language. Higher-order unification, which provides sophisticated pattern matching on formulas and proofs, can be used to determine when and at what instance an inference rule can be employed in the search for a proof. Tactics and tacticals, which provide a framework for high-level control over search, can also be directly implemented in this extended language. The theorem provers presented in this paper have been implemented in the higher-order logic programming language λProlog.
5180180
Logic programming in a fragment of intuitionistic linear logic
The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction, linear implication, universal quantification, the of course exponential, and the constants for the empty context and for the erasing contexts. It is shown that the logic has a goal-directed interpretation. It is also shown that the nondeterminism that results from the need to split contexts in order to prove a multiplicative conjunction can be handled by viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). Examples taken from theorem proving, natural language parsing, and database programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately.<<ETX>>
14782174
Uniform Proofs as a Foundation for Logic Programming
Abstract Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 (1991) 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.
16186126
Representing and Reasoning with Operational Semantics
The operational semantics of programming and specification languages is often presented via inference rules and these can generally be mapped into logic programming-like clauses. Such logical encodings of operational semantics can be surprisingly declarative if one uses logics that directly account for term-level bindings and for resources, such as are found in linear logic. Traditional theorem proving techniques, such as unification and backtracking search, can then be applied to animate operational semantic specifications. Of course, one wishes to go a step further than animation: using logic to encode computation should facilitate formal reasoning directly with semantic specifications. We outline an approach to reasoning about logic specifications that involves viewing logic specifications as theories in an object-logic and then using a meta-logic to reason about properties of those object-logic theories. We motivate the principal design goals of a particular meta-logic that has been built for that purpose.
13298925
Collection analysis for Horn clause programs
We consider approximating data structures with collections of the items that they contain. For examples, lists, binary trees, tuples, etc, can be approximated by sets or multisets of the items within them. Such approximations can be used to provide partial correctness properties of logic programs. For example, one might wish to specify than whenever the atom sort(t,s) is proved then the two lists t and s contain the same multiset of items (that is, s is a permutation of t). If sorting removes duplicates, then one would like to infer that the sets of items underlying t and s are the same. Such results could be useful to have if they can be determined statically and automatically. We present a scheme by which such collection analysis can be structured and automated. Central to this scheme is the use of linear logic as a computational logic underlying the logic of Horn clauses
657782
Experiences with an Architecture for Intelligent, Reactive Agents
This paper briefly describes a robot architecture that has been under development for the last eight years. This architecture uses several levels of abstraction and description languages that are compatible between levels. The makeup of the architecture helps to coordinate planful activities with real-time behaviors for dealing with dynamic environments. In recent years, many architectures have been created with similar attributes. The two features that distinguish this architecture from most of those are: 1) a variety of useful software tools have been created to help implement this architecture on multiple real robots; and 2) this architecture, or parts of it have been implemented on over half a dozen very different robot systems using a variety of processors, operating systems, effectors and sensor suites.
7547345
Path Planning Through Time and Space in Dynamic Domains
Realistic robot problems involve navigating the robot through time as well as space. The obstacles that a robot must avoid and the pathways on which it travels are subject to changes throughout time. These changes can occur in a predictable or unpredictable fashion. This paper presents an integrated route planning and spatial representation system that allows paths to be calculated in dynamic domains. The path planner finds the "best route" through a given n-dimensional space. The "best route" is defined as the path through space-time with the best score as determined by a set of user-defined evaluation functions. The algorithm takes into account the following: Capabilities of the robot executing generated plans, traversability of space, and interactions with both predictable and unpredictable dynamic objects. The route planning algorithm is highly parallel and can be run on an SIMD (Single Instruction Multiple Data) machine in 0(p) time, where p is the length of a path. In domains with unpredictable processes, this system may be run in an incremental fashion, allowing the robot to create real-time routes along the optimal path as described by its current information. This paper will discuss: Spatial representation, an SIMD algorithm for route planning in a dynamic domain, and results from an implementation on a traditional computer architecture.
839746
Roadmap for enhanced languages and methods to aid verification
This roadmap describes ways that researchers in four areas---specification languages, program generation, correctness by construction, and programming languages---might help further the goal of verified software. It also describes what advances the "verified software" grand challenge might anticipate or demand from work in these areas. That is, the roadmap is intended to help foster collaboration between the grand challenge and these research areas.A common goal for research in these areas is to establish language designs and tool architectures that would allow multiple annotations and tools to be used on a single program. In the long term, researchers could try to unify these annotations and integrate such tools.