12061015
Automatic Invention of Integer Sequences
We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.
5289412
Theories of Abstraction
universitt a di genova facoltt a di ingegneria dipartimento informatica sistemistica telematica Abstract This article surveys the eld of Artiicial Intelligence for theories of abstraction. We identify several unifying themes and argue that, even though these theories make diierent representational choices, they are all trying to capture the same intuitive ideas. Progress has also been made in understanding the reduction in search brought by abstraction. In this paper, we demonstrate how these theories of abstraction have helped comprehend the advantages abstraction provides. Finally, several diierent systems with abstraction have been implemented, tested and used. We focus on how the theories of abstraction have been innuenced by and have themselves innuenced the development of such systems.
18410341
A Theory of Abstraction
Abstract Informally , abstraction can be described as the process of mapping a representation of a problem onto a new representation. The aim of this paper is to propose the beginnings of a theory of reasoning with abstraction which captures and generalizes most previous work in the area. The theory allows us to study the properties of abstraction mappings and provides the foundations for the mechanization of abstraction inside an abstract proof checker.
7417163
Easy Problems are Sometimes Hard
We present a detailed experimental investigation of the easy-hard-easy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be orders of magnitude more difficult than the worst problems in problem classes and regions of the phase transition considered hard. These difficult problems are either hard unsatisfiable problems or are satisfiable problems which give a hard unsatisfiable subproblem following a wrong split. Whilst these hard unsatisfiable problems may have short proofs, these appear to be difficult to find, and other proofs are long and hard.
15125535
The Hardest Random SAT Problems
We describe a detailed experimental investigation of the phase transition for several different classes of satisfiability problems including random k-SAT, the constant probability model, and encodings of k-colourability and the independent set problem. We show that the constant probability model has been prematurely dismissed in favour of the random k-SAT model. We also show that for each of these problem class the conventional picture of easy-hard-easy behaviour is inadequate. In each of the problem classes, although median problem difficulty shows an easy-hard-easy pattern, there is also a region of very variable problem difficulty. Within this region, we have found problems orders of magnitude harder than those in the middle of the phase transition. These extraordinary problems can easily dominate the mean problem difficulty. We report experimental evidence which strongly suggests that this behaviour is due to a “constraint gap”, a region where the number of constraints on variables is minimal while simultaneously the depth of search required to solve problems is maximal. We also report results suggesting that better algorithms will be unable to eliminate this constraint gap and hence will continue to find very difficult problems in this region. Finally, we report an interesting correlation between these variable regions and a peak in the number of prime implicates. We predict that these extraordinarily hard problems will be of considerable use in analysing and comparing the performance of satisfiability algorithms.
2156641
Multiset Ordering Constraints
We identify a new and important global (or nonbinary) constraint which ensures that the values taken by two vectors of variables, when viewed as multisets, are ordered. This constraint is useful for a number of different applications including breaking symmetry and fuzzy constraint satisfaction. We propose and implement a linear time algorithm for enforcing generalised arc-consistency on such a multiset ordering constraint. Experimental results show considerable promise.
14691140
Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs
A graph will be assumed to be finite and unoriented, with no loops or multiple edges; if multiple edges are to be allowed, the term multigraph will be used. A graph or multigraph. will be called k-connected if at least k vertices and their incident edges must be removed to disconnect it (a complete graph is considered to be k-connected for any k). A block (respectively, multiblock) is a 2-connected graph (respectively, multigraph) with at least 2 vertices, and a brick is a 3-connected graph with at least 4 vertices. A Iabelling of a graph or multigraph with n vertices is a l-l correspondence from the set { 1, 2,..., n} onto the set of its vertices. Let A(x, y) be the mixed exponential generating function C,,, An,,,,x”.vmln!, where A,,, is the number of labelled graphs with n vertices and m edges, and let C(x, y) and B(x, y) be analogous generating functions which count labelled connected graphs and labelled blocks, respectively. The following formulae, due to Riddell [lo], appear in one-variable form in [6, pp. 3-111:
28528633
Total Vertex Enumeration in Rooted Planar Maps
15854568
Enumeration of eulerian and unicursal planar maps
Sum-free enumerative formulae are derived for several classes of rooted planar maps with no vertices of odd valency (eulerian maps) and with two vertices of odd valency (unicursal maps). As corollaries we obtain simple formulae for the numbers of unrooted eulerian and unicursal planar maps. Also, we obtain a sum-free formula for the number of rooted bi-eulerian (eulerian and bipartite) maps and some related results.
44916266
Inversion of Cycle Index Sum Relations for 2- and 3-Connected Graphs
Abstract Algebraic inversion of cycle index sum relations is employed to derive new algorithms for counting unlabeled 2-connected graphs, homeomorphically irreducible 2-connected graphs, and 3-connected graphs. The new algorithms are significantly more efficient than earlier ones, both asymptotically and for modest values of the order. Tables of computed results are included.
3106079
Efficient Exploration With Latent Structure
When interacting with a new environment, a robot can improve its online performance by efficiently exploring the effects of its actions. The efficiency of exploration can be expanded significantly by modeling and using latent structure to generalize experiences. We provide a theoretical development of the problem of exploration with latent structure, analyze several algorithms and prove matching lower bounds. We demonstrate our algorithmic ideas on a simple robot car repeatedly traversing a path with two different surface properties.