Showing papers on "Conjunctive normal form published in 1991"

Refinement types for ML

Abstract: Programming computers is a notoriously error-prone process. It is the job of the programming language designer to make this process more reliable. One approach to this is to impose some sort of typing discipline on the programs. In doing this, the programming language designer is immediately faced with a tradeoff: if the type system is too simple, it cannot accurately express important properties of the program; if it is too expressive, then mechanically checking or inferring the types becomes impractical. This thesis describes a type system called refinement types, which is an example of a new way to make this tradeoff, as well as a potentially useful system in itself. Refinement type inference requires programs to have types in two type systems: an expressive type inference system (intersection types with subtyping) and a relatively simple type system (basic polymorphic type inference). Refinement type inference inherits some properties from each of these: as in intersection types with subtyping, we can use the type system to do abstract interpretation; as in basic polymorphic type inference, refinement type inference is decidable (preliminary experiments suggest refinement type inference may be practical as well). We have implemented refinement type inference for a subset of Standard ML to test these ideas. We have added new syntax, called rectype declarations, to allow the programmer to specify relevant domains for the abstract interpretation. A prototype implementation of refinement type inference can do some interesting case analysis for Standard ML programs; for example, if the programmer uses a rectype declaration to declare interest in whether a boolean expression is in conjunctive normal form (CNF), refinement type inference can efficiently prove that a function for converting boolean expressions to CNF does indeed always return a boolean expression in CNF. Rectype declarations and refinement type inference seem flexible and efficient enough to practically enforce many other useful program properties as well.

A logic for natural language.

Abstract: This paper describes a language called LN whose structure mirrors tilat of natural language. LN is characterized by absence of variables and individual constants. Singular predicates assume the role of both individual constants and free variables. The role of bound variables is played by predicate functors called "selection operators." Like natural languages, LN is implicitly manysorted. LN does not have an identity relation. Its expressive power lies between the predicate calculus without identity and the predicate calculus with identity. The loss in expressiveness relative to the predicate calculus with identity however is not significant. Deduction in LN is intended to parallel reasoning in natural language, and therefore is termed "surface reasoning." In contrast to deduction in a disparate underlying logic such as clausal form, each step of a proof in [,N has a direct counterpart in the surface language. A sound and complete axiomatization is given. Derived rules, corresponding to monotonicity and conservativity of quantifiers and to unification and resolution in conventionallogic, are presented. Several problems are worked to illustrate reasoning in LN.

Optimizing the clausal normal form transformation

Abstract: Resolution based theorem proving systems require the conversion of predicate logic formulae into clausal normal form. The multiplication from disjunctive into conjunctive forms in general produces a lot of tautologous and subsumed clauses, which is relatively hard to recognize in later stages of the proof. In this paper an algorithm is presented that avoids the generation of redundant clauses. It is based on the generation of paths through a matrix and produces the set of prime implicants of the original formula.

Resolution and Path Dissolution in Multi-Valued Logics

Abstract: Path dissolution is an inferencing mechanism for classical logic that efficiently generalizes the method of analytic tableaux. Two features that both methods enjoy are (in the propositional case) strong completeness and the ability to produce a list of essential models (satisfying interpretations) of a formula. The latter feature is particularly valuable in a setting in which one wishes to make use of satisfying interpretations rather than merely to determine whether any exist.

Improving tableau deductions in multiple-valued logics

Abstract: Path dissolution is an efficient generalization of the method of analytic tableaux. Both methods feature (in the propositional case) strong completeness, the lack of reliance upon conjunctive normal form (CNF), and the ability to produce a list of essential models (satisfying interpretations) of a formula. Dissolution can speed up every step in a tableau deduction in classical logic. The authors consider means for adapting both techniques to multiple-valued logics, and show that the speed-up theorem applies in this more general setting. These results are pertinent for modeling uncertainty and commonsense reasoning. >

A note on the normal form of closed formulas of interpretability logic

Abstract: Each closed (i.e. variable free) formula of interpretability logic is equivalent in ILF to a closed formula of the provability logic G, thus to a Boolean combination of formulas of the form □n⊥.

A model elimination calculus for generalized clauses

Abstract: Generalized clauses differ from (ordinary) clauses by allowing conjunctions of literals in the role of (ordinary) literals, i.e. they are disjunctions of conjunctions of simple literals. An advantage of this clausal form is that implications with conjunctive conclusions or disjunctive premises are not split into multiple clauses. An extension of Lovelands model elimination calculus [Loveland, 1969a, Loveland, 1978] is presented able to deal with such generalized clauses. Furthermore we describe a method for generating lemmas that correspond to valid instances of conjunctive conclusions. Using these lemmas it is possible to avoid multiple proofs of the premises of implications with conjunctive conclusions.

Complexity of sentences over number rings

Abstract: Let I be an algebraic integer ring. The decision problem of the solvability of diophantine equations with parameters in I is proved to be co-NP-complete. However, the decision problem of the solvability of diophantine equations with parameters in all algebraic integer rings is in P. Let $\varphi (x.y)$ be a quantifier-free arithmetical formula in conjunctive normal form or disjunctive normal form. The decision problem of the sentences of the form $\exists x\forall y\varphi (x,y)$, true in I, is NP-complete. Then the decision problem of the sentences of the form $\forall x\exists y\varphi (x,y)$, true in I, is co-NP-complete, whereas the decision problem of the sentences of the form $\forall x\exists y\varphi (x,y)$, true in all algebraic integer rings, is in P. Some other related decision problems are also proven to be in P.

Operational issues in automated theorem proving using matings

Abstract: Much research in automated theorem proving has focused on improving the efficiency of procedures based on Robinson's resolution principle. The mating paradigm for automated theorem provers was proposed by Andrews and a similar approach called the connection method was suggested by Bibel to avoid converting a well-formed formula (wff) to clause form, which introduces redundancy and impedes analysis of the logical structure of the wff. In this thesis, we address various operational issues that arise in implementations of mating search and discuss techniques for improving the performance of mating search when searching for a refutation of a wff of first-order logic; for example, we address two crucial issues that arise in the search for refutations and are inadequately handled in current implementations: when and how to expand the search space. Two of these techniques--path-focused duplication and path enumeration--that have been implemented significantly improve the performance of search for refutations in TPS. Some other of these strategies significantly improve the performance of a propositional calculus prover that is based on the mating method; In fact, our performance on the pigeon hole problems is comparable to that of the best provers in the field. All our techniques are applicable to any wff W of first-order logic; we neither assume that W is in conjunctive normal form nor convert W to conjunctive normal form. We also discuss modifications of the unification algorithms that ensure the acyclicity of the imbedding relation, which provides an alternative to Skolemization. We incorporate the essential ideas of Cox's work on intelligent backtracking into mating search. This is a non trivial task, since an effective strategy must also efficiently handle the dynamically growing and shrinking search space introduced by path-focused duplication. Murray and Rosenthal introduced path dissolution inference rule, which removes unsatisfiable paths from a wff and is strongly complete. In this thesis, we provide an alternate characterization of this rule, which simplifies a criterion that determines the applicability of path-dissolution as well as the exposition of the path-dissolution inference rule.

A Clausal Form for the Completion of Logic Programs.

Abstract: The Clark completion of a program is a way of making explicit the inferences which may be made from the program using the Negation as Failure (NAF) rule. This may be thought of as adding negative information to the program in such a way that an atom fails ii its negation is derivable from the completion of the program. We show how the completion process may be extended to hereditary Harrop formulae, a class of formulae that properly includes Horn clauses, and, more importantly, that the completion of a hereditary Harrop formulae program may be expressed as a hereditary Harrop formulae program. In this way the richer framework of hereditary Harrop formulae allows the completion to be given in a more explicit form than that of Clark. This also allows the completion to be seen as a set of clauses, i.e. a program, rather than just as a formula of rst-order logic. With the restriction that programs are locally consistent, we show that the completion behaves as expected, in that the computational properties of the program and its completion correspond precisely.

Average case analysis of a k-CNF learning algorithm

Abstract: Author(s): Hirschberg, Daniel S.; Pazzani, Michael J. | Abstract: We present an approach to modeling the average case behavior of an algorithm for learning Conjunctive Normal Form (CNF, i.e., conjunctions of disjunctions). Our motivation is to predict the expected error of the learning algorithm as a function of the number of training examples. We evaluate the average case model by comparing the error predicted by the model to the actual error obtained by running the leaming algorithm, and show how the analysis can lead to insight into the behavior of the algorithm and the factors that affect the error.

Algebra of Normal Forms

Abstract: Summary. We mean by a normal form a finite set of ordered pairs of subsets of a fixed set that fulfils two conditions: elements of it consist of disjoint sets and elements of it are incomparable w.r.t. inclusion. The underlying set corresponds to a set of propositional variables but is arbitrary. The correspodents to a normal form of a formula, e.g. a disjunctive normal form, is as follows. The normal form is the set of disjuncts and a disjunct is an ordered pair consisting of the sets of propostional variables that occur in the non-negated and negated disjunct. The requirement that the element of a normal form consists of disjoint sets means that contradictory disjuncts have been removed, and the second condition means that the absorption law has been used to shorten the normal form. We construct a lattice h , ⊔, ⊓i , where a ⊔b = µ(a ∪b) and a ⊓ b = µc, c being the set of all pairs h X1 ∪ Y1,X2 ∪ Y2i , h X1,X2i ∈ a and h Y1,Y2i ∈ b, which consist of disjoint sets. µa denotes here the set of all minimal, w.r.t. inclusion, elements of a. We prove that the lattice of normal forms over a set defined in this way is distributive and that ∅ is the minimal element of it.

Some Aspects of the Probabilistic Behavior of Variants of Resolution

Abstract: The SAT-Problem fot boolean formulas in conjunctive normal form often arises in the area of artificial intelligence, its solution is known as mechanical theorem proving. Most mechanical theorem provers perform this by using a variant of the resolution principle. The time and space complexity of resolution strongly depends on the class of formulas. Horn formulas, where each clause may contain at most one positive literal, represent the most important class where resolution proofs with polynomial length exist.

Incomplete Information and Uncertainty

Abstract: In this paper we examine the problem of representing different types of incomplete data in an information system. Clausal form logic is used as a representation language and we look at the consequences of the subsequent increase in expressive power. We propose a method of improving the system’s response to a query by giving an uncertainty measure based on entropy.

Complexity results for the default- and the autoepistemic logic

Abstract: The default logic (Rei 80) and the autoepistemic logic (Mo 85) are non-monotonic logics. In default logic a default theory consists of a set of propositions sentences W and a set of defaults D. A sentence is defined to be derivable from a given default theory if it belongs to an extension of the default theory. A central question is: Given a default theory Δ = (D, W) and a sentence β, is there an extension of Δ which contains β?

On Conjunctive Normal Form Satisfiability

Abstract: This paper focuses on algorithms that solve CSAT (conjunctive normal form satisfiability) by searching for a satisfying truth assignment for the given formula F. We identify four basic ways to improve the basic search procedure: constraint propagators, simplifying transformations, heuristics, and other miscellaneous improvements. In each of these categories, we survey the existing improvements and suggest new ones. We lower the average time it takes to perform the simplest kind of constraint propagation from O(L) to O(L/P), where L is the length of F and P is the number of propositions in F; this is optimal. We lower the current upper bound for CSAT from O(20.128 L) to O(20.128 (L-N)), where N is the number of clauses in F. Finally, we experimentally determine the fastest possible algorithm with respect to each of the basic improvements we consider. Comments University of Pennsylvania Department of Computer and Information Sciences Technical Report No. MSCIS-91-94. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/339 On Conjunctive Normal Form Satisfiability