A Resolution Calculus for First-order Schemata

The satisfiability (SAT) problem is an important problem of automated reasoning. In the past decades, many methods of SAT are proposed, such as method based on resolution, method based on tableau and method based on extension rule. Based on the study of extension rule, we find that the SAT problem can be solved with hitting set algorithms. If we can find a hitting set of a clause set, and if there is not any pair of complementary literals in the hitting set, the clause set is satisfiable. Algorithm BHS based on Boolean algebra proposed by Jiang is an efficient hitting set algorithm. SSBF proposed in this paper is a revised algorithm of BHS that can be used to judge the satisfiability of a clause set. The test result shows that SSBF is an efficient algorithm.

The satisfiability problem is always a core problem in artificial intelligence and has been concerned for decades. Achievements in the research of the satisfiability problem have already been used in the research of other areas, such as computer aided design, database system. Now, there are many methods for the satisfiability problem, such as methods based on resolution, methods based on tableau, methods based on extension rule. By the study of extension rule, a conclusion can be drawn that the satisfiability of a clause set can be judged by the existence of a hitting set, which containing no complementary pairs of literals. With this conclusion, the algorithm CBHST (complementary binary hitting set tree) is designed. CBHST judge the existence of a hitting set without complementary pairs of literals in a binary tree. If the hitting set exists, CBHST will give the result that the clause set is satisfiable. Finally, CBHST is compared with an algorithm based on resolution DR and an algorithm based on extension rule IER. The test data shows that CBHST is efficient.

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

Query-answering (QA) problems have attracted wider interest recently, owing partly to emerging applications involving integration between rules and ontologies in the Semantic Web's layered architecture. Success of using satisfiability (SAT) solvers as low-level solvers for dealing with several kinds of problems motivates us to apply SAT solvers to a class of QA problems. To find all elements of an answer set, a variant of a usual SAT solver, called an all-solution satisfiability (All-SAT) solver, is used for determining all models of a given input set of propositional clauses. Using an All-SAT solver, we propose a procedure for solving QA problems. The procedure generates input for an All-SAT solver from a high-level description of a given QA problem by problem transformation in a clause space that includes set-bounded variables. Based on the equivalent transformation principle, the correctness of the proposed procedure is shown.

We address the problem of verifying that the functions of a program meet their contracts, specified by pre/postconditions. We follow an approach based on constrained Horn clauses (CHCs) by which the verification problem is reduced to the problem of checking satisfiability of a set of clauses derived from the given program and contracts. We consider programs that manipulate algebraic data types (ADTs) and a class of contracts specified by catamorphisms, that is, functions defined by simple recursion schemata on the given ADTs. We show by several examples that state-of-the-art CHC satisfiability tools are not effective at solving the satisfiability problems obtained by direct translation of the contracts into CHCs. To overcome this difficulty, we propose a transformation technique that removes the ADT terms from CHCs and derives new sets of clauses that work on basic sorts only, such as integers and booleans. Thus, when using the derived CHCs there is no need for induction rules on ADTs. We prove that the transformation is sound, that is, if the derived set of CHCs is satisfiable, then so is the original set. We also prove that the transformation always terminates for the class of contracts specified by catamorphisms. Finally, we present the experimental results obtained by an implementation of our technique when verifying many non-trivial contracts for ADT manipulating programs.

Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, for example, automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of first-order clauses. The approach builds upon i>positive unit hyperresolution (i>PUHR) i>tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux depth-first relying on a i>complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MM-SATCHMO, of this procedure is given, and its performance on benchmark suites is reported. The second minimal model generation procedure performs a breadth-first, constrained expansion of PUHR (complement) tableaux. Both procedures are optimal in the sense that each minimal model is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense: The depth-first minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all finite. The breadth-first minimal model generation procedure computes all finite minimal Herbrand models of the set of clauses under consideration. The proposed procedures are compared with related work in terms of both principles and performance on benchmark problems.

Binary-clause reasoning has been shown to reduce the size of the search space on many satisfiability problems, but has often been so expensive that run-time was higher than that of a simpler procedure that explored a larger space. The method of Sharir for detecting strongly connected components in a directed graph can be adapted to performing “lean” resolution on a set of binary clauses. Beyond simply detecting unsatisfiability, the goal is to find implied equivalent literals, implied unit clauses, and implied binary clauses.

Complexity of manipulation and bribery in premise-based judgment aggregation with simple formulas

Input resolution with paramodulation is a theorem proving procedure complete for sets of unit clauses with equality. This procedure recommends itself because it is easy to implement, and several implementations are in use in more general theorem proving programs. In this note we show that input resolution with paramodulation requires, in the worst case, proofs of exponential length even though the satisfiability problem for sets of unit clauses can be solved in polynomial time.

MAX SAT (the maximum satisfiability problem) is stated as follows: given a set of clauses with weights, find a truth assignment that maximizes the sum of the weights of the satisfied clauses. In this paper, we consider approximation algorithms for MAX SAT proposed by Yannnkakis and Goemans-Williamson and present an approximation algorithm which is an improvement of Yannakakis' algorithm. Although Yannakakis' original algorithm has no better performance guarantee than Goemans-Williamson, our improved algorithm has a better performance guarantee and leads to a 0.770 approximation algorithm.

A new algebraic model, called the generalized satisfiability problem (GSP) model, is introduced for representing and solving combinatorial problems. The GSP model is an alternative to the common method in the literature of representing such problems as language-recognition problems. In the GSP model, a problem instance is represented by a set of variables together with a set of terms, and the computational objective is to find a certain sum of products of terms over a commutative semiring. The model is general enough to express all the standard problems about sets of clauses and generalized clauses, all nonserial optimization problems, and all $\{ 0,1\} $-linear programming problems. The model can also describe many graph problems, often in a very direct structure-preserving way. Two important properties of the model are the following;1. In the GSP model, one can naturally discuss the structure of individual problem instances. The structure of a GSP instance is displayed in a “structure tree.” The smaller the “weighted depth” or “channelwidth” of the structure tree for a GSP instance, the faster the instance can be solved by any one of several generic algorithms. 2. The GSP model extends easily so as to apply to hierarchically specified problems and enables solutions to instances of such problems to be found directly from the specification rather than from the (often exponentially) larger specified object.

Improved Approximation Algorithms for MAX SAT

The cycle double cover conjecture is a famous longstanding unsolved conjecture in graph theory. It is related and can be reduced to the compatible circuit decomposition problem. Recently Fleischner et al. (2018) provided a sufficient condition for a compatible circuit decomposition, which is called SUD-\(K_5\)-minor freeness. In a previous work we developed an abstract mathematical model for finding SUD-\(K_5\)-minors and based on the model a mixed integer linear program (MIP). In this work we propose a respective boolean satisfiability (SAT) model and compare it with the MIP model in computational tests. Non-trivial symmetry breaking constraints are proposed, which improve the solving times of both models considerably. Compared to the MIP model the SAT approach performs significantly better. We use the faster algorithm to further test graphs of graph theoretic interest and were able to get new insights. Among other results we found snarks with 30 and 32 vertices that do not contain a perfect pseudo-matching, that is a spanning subgraph consisting of \(K_2\) and \(K_{1,3}\) components, whose contraction leads to a SUD-\(K_5\)-minor free graph.

Abstraction based approaches like ProVerif are very efficient in protocol verification, but have a limitation in dealing with stateful protocols. A number of extensions have been proposed to allow for a limited amount of state information while not destroying the advantages of the abstraction method. However, the extensions proposed so far can only deal with a finite amount of state information. This can in many cases make it impossible to formulate a verification problem for an unbounded number of agents and one has to rather specify a fixed set of agents. Our work shows how to overcome this limitation by abstracting state into countable families of sets. We can then formalize a problem with unbounded agents, where each agent maintains its own set of keys. Still, our method does not loose the benefits of the abstraction approach, in particular, it translates a verification problem to a set of first-order Horn clauses that can then be efficiently verified with tools like ProVerif.

In the proposed data model for XML databases, an XML element is directly represented as a ground (variable-free) XML expression—a generalization of an XML element by incorporation of variables for representation of implicit information and enhancement of its expressive power—while a collection of XML documents as a set of ground expressions, each describing an XML element in the documents. Relationships among elements in the collection as well as integrity constraints are formalized as XML clauses. An XML database, consisting of: (i) a document collection (or an extensional database), (ii) a set of relationships (or an intensional database) and (iii) a set of integrity constraints, is therefore modeled as an XML declarative description comprising a set of ground XML expressions and XML clauses. Its semantics is a set of ground XML expressions, which are explicitly described by the extensional database or implicitly derived from the intensional database and satisfy all the specified set of constraints.KeywordsDescription LogicIntegrity ConstraintUnit ClauseImplicit InformationSemistructured DataThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.