Set Of Clauses Research Articles

A simplifier for propositional formulas with many binary clauses.

Deciding whether a propositional formula in conjunctive normal form is satisfiable (SAT) is an NP-complete problem. The problem becomes linear when the formula contains binary clauses only. Interestingly, the reduction to SAT of a number of well-known and important problems--such as classical AI planning and automatic test pattern generation for circuits--yields formulas containing many binary clauses. In this paper we introduce and experiment with 2-SIMPLIFY, a formula simplifier targeted at such problems. 2-SIMPLIFY constructs the transitive closure of the implication graph corresponding to the binary clauses in the formula and uses this graph to deduce new unit literals. The deduced literals are used to simplify the formula and update the graph, and so on, until stabilization. Finally, we use the graph to construct an equivalent, simpler set of binary clauses. Experimental evaluation of this simplifier on a number of bench-mark formulas produced by encoding AI planning problems prove 2-SIMPLIFY to be a useful tool in many circumstances.

A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution

We prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form (DNFs) with small terms. We use this to prove lower bounds for the Res(k) propositional proof system, an extension of resolution which works with k-DNFs instead of clauses. We also obtain an exponential separation between depth d circuits of bottom fan-in k and depth d circuits of bottom fan-in k + 1. Our results for Res(k) are as follows: The 2n to n weak pigeonhole principle requires exponential size to refute in Res(k) for $k \leq \sqrt{\log n / \log \log n } $. For each constant k, there exists a constant w > k so that random w-CNFs require exponential size to refute in Res(k). For each constant k, there are sets of clauses which have polynomial size Res(k + 1) refutations but which require exponential size Res(k) refutations.

Querying several conflicting databases

This paper addresses the problem of querying several databases considered as a whole. Assuming that the different databases share a common data description language, the problem that arises is to consistently answer queries even if the database contents are contradictory. The main contribution of this paper is the specification of a query-evaluator for answering closed and open general queries addressed to several databases and for providing explanations about the results. For doing so, we first specify a query-evaluator, in logic, which assumes that the databases are propositional ones and do not contain disjunctions. Then we extend it to first-order databases defined by an extensional part (sets of positive or negative facts) and an intensional part (sets of first order clauses). We also show that the query-evaluator corresponds to a majority merging of the databases.

Toward reasoning with unified modeling language diagrams based on extensible markup language declarative description theory

A practical framework for representing knowledge and reasoning in the domain of Unified Modeling Language (UML) is proposed. In this framework, graphical diagrams in a UML model are encoded as Extensible Markup Language (XML)/Metadata Interchange (XMI) elements, which are regarded as facts about a specific problem instance in a knowledge base, and the general knowledge on UML, such as inherent interrelationships among diagram components and implicit properties of diagrams, is represented as a set of XML definite clauses. Equivalent transformation (ET) is used as a fundamental computation mechanism for reasoning with the represented diagrams. © 2004 Wiley Periodicals, Inc.

SATCHMOREBID: SATCHMO(RE) with BIDirectional relevancy

SATCHMORE was introduced as a mechanism to integrate relevancy testing with the model-generation theorem prover SATCHMO. This made it possible to avoid invoking some clauses that appear in no refutation, which was a major drawback of the SATCHMO approach. SATCHMORE relevancy, however, is driven by the entire set of negative clauses and no distinction is accorded to the query negation. Under unfavorable circumstances, such as in the presence of large amounts of negative data, this can reduce the efficiency of SATCHMORE. In this paper we introduce a further refinement of SATCHMO called SATCHMOREBID: SATCHMORE with BIDirectional relevancy. SATCHMOREBID uses only the negation of the query for relevancy determination at the start. Other negative clauses are introduced on demand and only if a refutation is not possible using the current set of negative clauses. The search for the relevant negative clauses is performed in a forward chaining mode as opposed to relevancy propagation in SATCHMORE which is based on backward chaining. SATCHMOREBID is shown to be refutationally sound and complete. Experiments on a prototype SATCHMOREBID implementation point to its potential to enhance the efficiency of the query answering process in disjunctive databases.

Fast algorithms for revision of some special propositional knowledge bases

In this paper, the computational complexity of propositional clause set counter-factuals is discussed. It is shown that the computational complexity of propositional clause set counterfactuals is at the second level of the polynomial hierarchy, and that the computational complexity of propositional Horn clause set counterfactuals is at the first level of the polynomial hierarchy. Furthermore, some polynomial algorithms are presented for some special propositional clause set, such as the unique satisfiable clause set and the clause set of which only one subset is minimally inconsistent with the input clause whose inconsistency check can be solved in polynomial time.

Stratified resolution

We introduce a calculus of stratified resolution, in which special attention is paid to clauses that “define” relations. If such clauses are discovered in the initial set of clauses, they are treated using the rule of definition unfolding, i.e. the rule that replaces defined relations by their definitions. Stratified resolution comes with a powerful notion of redundancy: a clause to which definition unfolding has been applied can be removed from the search space. To prove the completeness of stratified resolution with redundancies, we use a novel combination of Bachmair and Ganzinger’s model construction technique and a hierarchical construction of orderings and least fixpoints.

Model building with ordered resolution: extracting models from saturated clause sets

A method is proposed to transform automatically some satisfiable clause sets S into clause sets S′ having a unique Herbrand model (w.r.t. a given signature) which also satisfies S. These clause sets with only one model can be used as representations of (in general infinite) Herbrand models of the initial set of clauses. Existing theorem provers may be used to evaluate literals and clauses in the models thus represented (using the standard proof by consistency mechanism). We also prove that for some classes of clauses, the extracted model can also be represented by a tree automaton on tuples of finite trees. This entails in particular that the evaluation of arbitrary function-free formulae in the represented models is decidable.

A calculus combining resolution and enumeration for building finite models

A calculus is proposed for simultaneous search for refutations and models for sets of clauses. It combines existing resolution-based model building approaches with enumeration techniques that are usually restricted to tableau-based theorem provers or finite model builders. The method is sound, refutationally complete, and builds models for any satisfiable set of clauses having a finite model. It strictly enlarges the scope of resolution-based model building, by allowing one to build models for sets of clauses for which resolution does not terminate.

TinkerType: a language for playing with formal systems

TinkerType is a pragmatic framework for compact and modular description of formal systems (type systems, operational semantics, logics, etc.). A family of related systems is broken down into a set of clauses – individual inference rules – and a set of features controlling the inclusion of clauses in particular systems. Simple static checks are used to help maintain consistency of the generated systems. We present TinkerType and its implementation and describe its application to two substantial repositories of typed lambda-calculi. The first repository covers a broad range of typing features, including subtyping, polymorphism, type operators and kinding, computational effects, and dependent types. It describes both declarative and algorithmic aspects of the systems, and can be used with our tool, the TinkerType Assembler, to generate calculi either in the form of typeset collections of inference rules or as executable ML typecheckers. The second repository addresses a smaller collection of systems, and provides modularized proofs of basic safety properties.

Extracting models from clause sets saturated under semantic refinements of the resolution rule

We present an algorithm that—given a set of clauses S saturated under some semantic refinements of the resolution calculus—automatically constructs a Herbrand model M of S . M is represented by a set of atoms with equality and disequality constraints interpreted over the finite tree algebra, hence the problem of evaluating first-order formulae in M is decidable.

On good algorithms for determining unsatisfiability of propositional formulas

This paper is concerned with an algorithm that provides short certificates of unsatisfiability with high probability when an input I is a random instance of 3-SAT. Rather than build a refutation DAG, the algorithm finds bounds on n I (true) , the number of variables that must be set to true, and n I (false) , the number that must be set to false, if all clauses of I are to be satisfied. If the sum n I (true)+n I (false) is greater than the number of variables in I then I must be unsatisfiable. Bounds on n I (true) and n I (false) may be found by throwing out all clauses except those having only negative or only positive literals and finding n I +(true) for the remaining positive clause set I + and n I − (false) for the remaining negative clause set I − . These questions can alternatively be stated as 3-hitting set problems on I + and I − separately. It is shown that a good enough approximation algorithm for 3-hitting set can determine useful bounds on n I (true) and n I (false) (high probability of success for large enough constant ratio of clauses to variables). Although a good enough algorithm seems evasive, one that comes fairly close is proposed and analyzed.

A relevance restriction strategy for automated deduction

Identifying relevant clauses before attempting a proof may lead to more efficient automated theorem proving. Relevance is here defined relative to a given set of clauses S and one or more distinguished sets of support T. The role of a set of support T can be played by the negation of the theorem to be proved or the query to be answered in S which gives the refutation search goal orientation. The concept of relevance distance between two clauses C and D of S is defined using various metrics based on the properties of paths connecting C to D. This concept is extended to define relevance distance between a clause and a set (or multiple sets) of support. Informally, the relevance distance reflects how closely two clauses are related. The relevance distance to one or more support sets is used to compute a relevance set R, a subset of S that is unsatisfiable if and only if S is unsatisfiable. R is computed as the set of clauses of S at distance less than n from one or more support sets; if n is sufficiently large then R is unsatisfiable if S is. If R is much smaller than S, a refutation from R may be obtainable in much less time than a refutation from S. R must be efficiently computable to achieve an overall efficiency improvement. Different relevance metrics are defined, characterized and related. The tradeoffs between the amount of effort invested in computing a relevance set and the resulting gains in finding a refutation are addressed. Relevance sets may be utilized with arbitrary complete theorem proving strategies in a completeness-preserving manner. The potential of the advanced relevance techniques for various applications of theorem proving is discussed

International Agreements on Product Standards: An Incomplete-Contracting Theory

We propose a theory of international agreements on product standards. The key feature of the model is that agreements are viewed as incomplete contracts. In particular, these do not specify standards for products that may arise in the future. One potential remedy to contractual incompleteness is a dispute settlement procedure (DSP) that provides arbitration in states of the world that are not covered by the ex ante agreement. We identify conditions under which a DSP can provide ex-ante efficiency gains, and examine how these gains depend on the fundamentals of the problem. Another potential remedy to contractual incompleteness is given by rigid rules, i.e. rules that are not product-specific. We argue that the nondiscrimination rule is the only rule of this kind that increases ex ante efficiency for any probability distribution over potential products. Finally we show that, under relatively weak conditions, the optimal ex-ante agreement is structured in three parts: (i) a set of clauses that specify standards for existing products; (ii)a rigid nondiscrimination rule, and (iii) a dispute settlement procedure. Although the model focuses on the case of product standards, the analysis suggests a more general incomplete-contracting theory of trade agreements.

A programmable approach to maintenance of a finite knowledge base

In this paper, we present a programmable method of revising a finite clause set. We first present a procedure whose formal parameters are a consistent clause set Γ and a clause A and whose output is a set of minimal subsets of Γ which are inconsistent with A. The maximal consistent subsets can be generated from all minimal inconsistent subsets. We develop a prototype system based on the above procedure, and discuss the implementation of knowledge base maintenance. At last, we compare the approach presented in this paper with other related approaches. The main characteristic of the approach is that it can be implemented by a computer program.

A data model for XML databases

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. Axioms and relationships among elements in the collection as well as structural and integrity constraints are formalized as XML clauses. An XML database, consisting of: (i) a document collection (or an extensional database), (ii) a set of axioms and relationships (or an intensional database), (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 database, based on the defined intensional database, and satisfy all the specified set of constraints. Thus, selective and complex queries, formulated as sets of XML clauses, about information satisfying specific criteria and possibly implicit in the database, become expressible and computable. The model thereby serves as an effective and well-founded XML database management framework with succinct representational and operational uniformity, reasoning capabilities as well as complex and deductive query supports.

Consistent subsets of inconsistent systems: structure and behaviour

A large Knowledge System operating for a long time almost inevitably becomes ‘polluted’ by wrong data that make the system inconsistent. Despite this fact, a sizeable part of the system remains unpolluted, and retains useful information. It is widely adopted that a maximally consistent subset of a system (mc-subset) contains a significant portion of unpolluted data. So, determining mc-subsets is a necessary step towards reasoning with inconsistent knowledge. We consider extensions of the MAX-SAT problem, investigate characteristic features of mc-subsets, present algorithms for computing all or major mc-subsets of inconsistent sets of clauses, and, report results of experiments evaluating parameters of mc-subsets.

Resolution Proofs of Matching Principles

This paper discusses current techniques for proving lower bounds on the size of resolution proofs from sets of clauses expressing assertions about matchings in bipartite graphs (a class including the well-known pigeon-hole clauses). The techniques are illustrated by demonstrating an improved lower bound for some examples discussed by Kleine Buning. The final section discusses some problems where current techniques appear inadequate, and new ideas seem to be required.

Building Infinite Models for Equational Clause Sets: Constructing Non-Ambiguous Formulae

Building Infinite Models for Equational Clause Sets: Constructing Non-Ambiguous Formulae

The phase transition in random horn satisfiability and its algorithmic implications

AbstractLet c > 0 be a constant, and Φ be a random Horn formula with n variables and m = c · 2n clauses, chosen uniformly at random (with repetition) from the set of all nonempty Horn clauses in the given variables. By analyzing PUR, a natural implementation of positive unit resolution, we show that limn→∞ Pr(Φ is satisfiable) = 1 − F(e−c), where F(x) = (1 − x)(1 − x2)(1 − x4)(1 − x8) …. Our method also yields as a byproduct an average‐case analysis of this algorithm. Published 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 483–506, 2002