Model generation can be regarded as a special case of the Constraint Satisfaction Problem (CSP). It has many applications in AI, computer science and mathematics. In this paper, we describe SEM, a System for Enumerating finite Models of first-order many-sorted theories. To the best of our knowledge, SEM outperforms any other finite model generation system on many test problems. The high performance of SEM relies on the following two techniques: (a) an efficient implementation of constraint propagation which requires little dynamic allocation of storage; (b) a powerful heuristic which eliminates many isomorphic partial models during the search. We will present the basic algorithm of SEM along with these two techniques. Our experimental results show that general purpose finite model generators are indeed useful in many applications. 1

We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.

This paper shows a way of using such resources to solve hard problems.

The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis-Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis-Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis-Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.

Abstract. The key design challenges in the construction of a SAT-based relational model finder are described, and novel techniques are proposed to address them. An efficient model finder must have a mechanism for specifying partial solutions, an effective symmetry detection and breaking scheme, and an economical translation from relational to boolean logic. These desiderata are addressed with three new techniques: a symmetry detection algorithm that works in the presence of partial solutions, a sparse-matrix representation of relations, and a compact representation of boolean formulas inspired by boolean expression diagrams and reduced boolean circuits. The presented techniques have been implemented and evaluated, with promising results. 1

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

The HR program forms concepts and makes conjectures in domains of pure mathematics and uses theorem prover OTTER and model generator MACE to prove or disprove the conjectures. HR measures properties of concepts and assesses the theorems and proofs involving them to estimate the interestingness of each concept and employ a best first search. This approach has led HR to the discovery of interesting new mathematics and enables it to build theories from just the axioms of finite algebras.

Deduction-based software component retrieval uses preand postconditions as indexes and search keys and an automated theorem prover (ATP) to check whether a component matches. This idea is very simple but the vast number of arising proof tasks makes a practical implementation very hard. We thus pass the components through a chain of filters of increasing deductive power. In this chain, rejection filters based on signature matching and model checking techniques are used to rule out non-matches as early as possible and to prevent the subsequent ATP from "drowning." Hence, intermediate results of reasonable precision are available at (almost) any time of the retrieval process. The final ATP step then works as a confirmation filter to lift the precision of the answer set. We implemented a chain which runs fully automatically and uses MACE for model checking and the automated prover SETHEO as confirmation filter. We evaluated the system over a medium-sized collection of components. The resul...

The Davis-Putnam method is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently using the trie data structure for propositional clauses by presenting seven implementations of the method. We propose a new technique for implementing unit propagation whose complexity is sublinear to the number of occurrences of the variable in the input. We present the performance of our programs on some quasigroup problems. The efficiency of our programs allowed us to solve some open quasigroup problems.

We perform a comprehensive theoretical and experimental analysis of the use of all-different constraints. We prove that generalized arcconsistency on such constraints lies between neighborhood inverse consistency and, under a simple restriction, path inverse consistency on the binary representation of the problem. By generalizing the arguments of Kondrak and van Beek, we prove that a search algorithm that maintains generalized arc-consistency on alldifferent constraints dominates a search algorithm that maintains arc-consistency on the binary representation. Our experiments show the practical value of achieving these high levels of consistency. For example, we can solve almost all benchmark quasigroup completion problems up to order 25 with just a few branches of search. These results demonstrate the benefits of using non-binary constraints like all-different to identify structure in problems.