Victor W. Marek

IJCAI Conference 2017 Conference Paper

Solving Very Hard Problems: Cube-and-Conquer, a Hybrid SAT Solving Method

• Marijn J. H. Heule
• Oliver Kullmann
• Victor W. Marek

A recent success of SAT solving has been the solution of the boolean Pythagorean Triples problem [Heule et al. , 2016], delivering the largest proof yet, of 200 terabytes in size. We present this and the underlying paradigm Cube-and-Conquer, a powerful general method to solve big SAT problems, based on integrating the “old” and “new” methods of SAT solving.

AAAI Conference 2004 Conference Paper

Logic Programs with Abstract Constraint Atoms

• Victor W. Marek
• Miroslaw Truszczynski

We propose and study extensions of logic programming with constraints represented as generalized atoms of the form C(X), where X is a finite set of atoms and C is an abstract constraint (formally, a collection of sets of atoms). Atoms C(X) are satisfied by an interpretation (set of atoms) M, if M ∩ X ∈ C. We focus here on monotone constraints, that is, those collections C that are closed under the superset. They include, in particular, weight (or pseudo-boolean) constraints studied both by the logic programming and SAT communities. We show that key concepts of the theory of normal logic programs such as the one-step provability operator, the semantics of supported and stable models, as well as several of their properties including complexity results, can be lifted to such case.

I&C Journal 2004 Journal Article

Ultimate approximation and its application in nonmonotonic knowledge representation systems

• Marc Denecker
• Victor W. Marek
• Mirosław Truszczyński

In this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way. The key concept is that of an approximating operator, a monotone operator on the product bilattice, which gives approximate information on the original operator in an intuitive and well-defined way. With any given approximating operator our theory associates several different types of fixpoints, including the Kripke–Kleene fixpoint, stable fixpoints, and the well-founded fixpoint, and relates them to fixpoints of operators being approximated. Compared to our earlier work on approximation theory, the contribution of this paper is that we provide an alternative, more intuitive, and better motivated construction of the well-founded and stable fixpoints. In addition, we study the space of approximating operators by means of a precision ordering and show that each lattice operator O has a unique most precise—we call it ultimate—approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We then discuss applications of these results in logic programming.

AIJ Journal 2003 Journal Article

Uniform semantic treatment of default and autoepistemic logics

• Marc Denecker
• Victor W. Marek
• Mirosław Truszczyński

We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise one-to-one correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.

KR Conference 2002 Conference Paper

Ultimate approximations in nonmonotonic knowledge representation systems

• Marc Denecker
• Victor W. Marek
• Miroslaw Truszczynski

AIJ Journal 1999 Journal Article

Computing with default logic

• Paweł Cholewiński
• Victor W. Marek
• Mirosław Truszczyński
• Artur Mikitiuk

Default logic was proposed by Reiter as a knowledge representation tool. In this paper, we present our work on the Default Reasoning System, DeReS, the first comprehensive and optimized implementation of default logic. While knowledge representation remains the main application area for default logic, as a source of large-scale problems needed for experimentation and as a source of intuitions needed for a systematic methodology of encoding problems as default theories we use here the domain of combinatorial problems. To experimentally study the performance of DeReS we developed a benchmarking system, the TheoryBase. The TheoryBase is designed to support experimental investigations of nonmonotonic reasoning systems based on the language of default logic or logic programming. It allows the user to create parameterized collections of default theories having similar properties and growing sizes and, consequently, to study the asymptotic performance of nonmonotonic systems under investigation. Each theory generated by the TheoryBase has a unique identifier, which allows for concise descriptions of test cases used in experiments and, thus, facilitates comparative studies. We describe the TheoryBase in this paper and report on our experimental studies of DeReS performance based on test cases generated by the TheoryBase.

TCS Journal 1998 Journal Article

Revision programming

• Victor W. Marek
• Mirosz.xl;law Truszczyński

In this paper we introduce revision programming — a logic-based framework for describing constraints on databases and providing a computational mechanism to enforce them. Revision programming captures those constraints that can be stated in terms of the membership (presence or absence) of items (records) in a database. Each such constraint is represented by a revision rule α ← α 1, …, α k, where α and all ga i are of the form in(a) and out(b). Collections of revision rules form revision programs. Similarly as logic programs, revision programs admit both declarative and imperative (procedural) interpretations. In our paper, we introduce a semantics that reflects both interpretations. Given a revision program, this semantics assigns to any database B a collection (possibly empty) of P-justified revisions of B. The paper contains a thorough study of revision programming. We exhibit several fundamental properties of revision programming. We study the relationship of revision programming to logic programming. We investigate complexity of reasoning with revision programs as well as algorithms to compute P-justified revisions. Most importantly from the practical database perspective, we identify two classes of revision programs, safe and stratified, with a desirable property that they determine for each initial database a unique revision.