Alan M. Frisch

Breaking Row and Column Symmetries in Matrix Models

We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.

Global Constraints for Lexicographic Orderings

We propose some global constraints for lexicographic orderings on vectors of variables. These constraints are very useful for breaking a certain kind of symmetry arising in matrices of decision variables. We show that decomposing such constraints carries a penalty either in the amount or the cost of constraint propagation. We therefore present a global consistency algorithm which enforces a lexicographic ordering between two vectors of n variables in O(nb) time, where b is the cost of adjusting the bounds of a variable. The algorithm can be modified very slightly to enforce a strict lexicographic ordering. Our experimental results on a number of domains (balanced incomplete block design, social golfer, and sports tournament scheduling) confirm the efficiency and value of these new global constraints.

Anytime deduction for probabilistic logic

Abstract This paper proposes and investigates an approach to deduction in probabilistic logic, using as its medium a language that generalizes the propositional version of Nilssons probabilistic logic by incorporating conditional probabilities. Unlike many other approaches to deduction in probabilistic logic, this approach is based on inference rules and therefore can produce proofs to explain how conclusions are drawn. We show how these rules can be incorporated into an anytime deduction procedure that proceeds by computing increasingly narrow probability intervals that contain the tightest entailed probability interval. Since the procedure can be stopped at any time to yield partial information concerning the probability range of any entailed sentence, one can make a tradeoff between precision and computation time. The deduction method presented here contrasts with other methods whose ability to perform logical reasoning is either limited or requires finding all truth assignments consistent with the given sentences.

Essence: A constraint language for specifying combinatorial problems

Essence is a formal language for specifying combinatorial problems in a manner similar to natural rigorous specifications that use a mixture of natural language and discrete mathematics. Essence provides a high level of abstraction, much of which is the consequence of the provision of decision variables whose values can be combinatorial objects, such as tuples, sets, multisets, relations, partitions and functions. Essence also allows these combinatorial objects to be nested to arbitrary depth, providing for example sets of partitions, sets of sets of partitions, and so forth. Therefore, a problem that requires finding a complex combinatorial object can be specified directly by using a decision variable whose type is precisely that combinatorial object.

The substitutional framework for sorted deduction: fundamental results on hybrid reasoning

Abstract Researchers in artificial intelligence have recently been taking great interest in hybrid representations, among them sorted logics—logics that link a traditional logical representation to a taxonomic (or sort) representation such as those prevalent in semantic networks. This paper introduces a general framework—the substitutional framework—for integrating logical deduction and sortal deduction to form a deductive system for sorted logic. This paper also presents results that provide the theoretical under-pinnings of the framework. A distinguishing characteristic of a deductive system that is structured according to the substitutional framework is that the sort subsystem is invoked only when the logic subsystem performs unification, and thus sort information is used only in determining what substitutions to make for variables. Unlike every other known approach to sorted deduction, the substitutional framework provides for a systematic transformation of unsorted deductive systems into sorted ones.

A Re‐Evaluation of Story Grammars

Black and Wilensky (1979) have made serious methodological errors in analyzing story grammars, and in the process they have committed additional errors in applying formal language theory. Our arguments involve clarifying certain aspects of knowledge representation crucial to a proper treatment of story understanding. Particular criticisms focus on the following shortcomings of their presentation: 1) an erroneous statement from formal language theory, 2) misapplication of formal language theory to story grammars, 3) unsubstantiated and doubtful analogies with English grammar, 4) various non sequiturs concerning the generation of non-stories, 5) a false claim based on the artificial distinction between syntax and semantics, and 6) misinterpretation of the role of story grammars in story understanding. We conclude by suggesting appropriate criteria for the evaluation of story grammars.

Propagation algorithms for lexicographic ordering constraints

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finite-domain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq.

CGRASS: a system for transforming constraint satisfaction problems

Experts at modelling constraint satisfaction problems (CSPs) carefully choose model transformations to reduce greatly the amount of effort that is required to solve a problem by systematic search. It is a considerable challenge to automate such transformations and to identify which transformations are useful. Transformations include adding constraints that are implied by other constraints, adding constraints that eliminate symmetrical solutions, removing redundant constraints and replacing constraints with their logical equivalents. This paper describes the CGRASS (Constraint Generation And Symmetry-breaking) system that can improve a problem model by automatically performing transformations of these kinds. We focus here on transforming individual CSP instances. Experiments on the Golomb ruler problem suggest that producing good problem formulations solely by transforming problem instances is, generally, infeasible. We argue that, in certain cases, it is better to transform the problem class than individual instances and, furthermore, it can sometimes be better to transform formulations of a problem that are more abstract than a CSP.

Knowledge Retrieval as Limited Inference

Artificial intelligence reasoning systems commonly employ a knowledge base module that stores a set of facts expressed in a representation language and provides facilities to retrieve these facts. A retriever could range from a simple pattern matcher to a complete logical inference system. In practice, most fall in between these extremes, providing some forms of inference but not others. Unfortunately, most of these retrievers are not precisely defined.

Constraints for breaking more row and column symmetries

Constraint programs containing a matrix of two (or more) dimensions of decision variables often have row and column symmetries: in any assignment to the variables the rows can be swapped and the columns can be swapped without affecting whether or not the assignment is a solution. This introduces an enormous amount of redundancy when searching a space of partial assignments. It has been shown previously that one can remove consistently some of these symmetries by extending such a program with constraints that require the rows and columns to be lexicographically ordered. This paper identifies and studies the properties of a new additional constraint—the first row is less than or equal to all permutations of all other rows—that can be added consistently to break even more symmetries. Two alternative implementations of this stronger symmetry-breaking method are investigated, one of which employs a new algorithm that in time linear in the size of the matrix enforces the constraint that the first row is less than or equal to all permutations of all other rows. It is demonstrated experimentally that our method for breaking more symmetries substantially reduces search effort.