Evaluation of Queries under Closed-World Assumption

The minimal entailment Min has been characterized elsewhere by where Cn is the first-order consequence operation, P is a set of clauses (indefinite deductive data base; in short: a data base), is a clause (a query), and Pos is the set of positive (that is, bodiless) ground clauses. In this paper, we address the problem of the computational feasibility of criterion (1). Our objective is to find a query evaluation algorithm that decides P Min by what we call indefinite modeling, without actually computing all ground positive consequences of P and P {}. For this purpose, we introduce the concept of minimal indefinite Herbrand model MP of P, which is defined as the set of subsumption-minimal ground positive clauses provable from P. The algorithm first computes MP by finding the least fixed-point of an indefinite consequence operator TIP. Next, the algorithm verifies whether every ground positive clause derivable from MP {} by one application of the parallel positive resolution rule (in short: the PPR rule) is subsumed by an element of MP. We prove that the PPR rule, which can derive only positive clauses, is positively complete, that is, every positive clause provable from a data base P is derivable from P by means of subsumption and finitely many applications of PPR. From this we conclude that the presented algorithm is partially correct and that it eventually halts if both P and MP are finite. Moreover, we indicate how the algorithm can be modified to handle data bases with infinite indefinite Herbrand models. This modification leads to a concept of universal model that allows for nonground clauses in its Herbrand base and appears to be a good candidate for representation of indefinite deductive data bases.     

The Kreisel length-of-proof problem

A first-order system F has theKreisel length-of-proof property if the following statement is true for all formulas(x): If there is ak1 such that for alln0 there is a proof of(¯n) in F with at mostk lines, then there is a proof of x(x) in F. We consider this property for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata (including individual axioms) and a finite number of rules of inference. We prove that any usual Parikh system formulation of Peano arithmetic has the Kreisel length-of-proof property if the underlying logic of the system is formulated without a schema for universal instantiation in either one of two ways. (In one way, the formula to be instantiated is built up in steps, and in the other way, the term to be substituted is built up in steps.) Our method of proof uses techniques and ideas from unification theory.     

Automated Synthesis of Recursive Programs from a ∀∃ Logical Specification

The specification of a function is often given by a logical formula, called a -formula, of the following form: xy(x,y). More precisely, a specification is given in the context of a certain theory E and is stated by the judgment E xy (x,y).In this paper, we consider the case in which E is an equational theory. It is divided into two parts. In the first part, we develop a theory for the automated proof of such judgments in the initial model ofE . The validity in the initial model means that we consider not only equational theorems but also inductive ones. From our theory we deduce an automated method for the proof of a class of such judgments. In the second part, we present an automatedmethod for program synthesis. We show how the previous proof method can be used to generate a recursive program for a function f that satisfies a judgment E x (x, f(x)).We illustrate our method with the automated synthesis of some recursive programs on domains such as integers and lists. Finally, we describe our system LEMMA, which is an implementation in Common Lisp of these new methods.     

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs,reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on , , =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of , , =, {...}, , where {...} is the tuple operator and is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of , , =, {...}, , obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving is allowed.     

The Use of Certain Equations of Mathematical Physics in Optimization of a Function on a Set. I

The results of application of potential theory to optimization are used to extend the use of (Helmholtz) diffusion and diffraction equations for optimization of their solutions (x, ) with respect to both x, and . If the aim function is modified such that the optimal point does not change, then the function (x, ) is convex in (x, for small . The possibility of using heat conductivity equation with a simple boundary layer for global optimization is investigated. A method is designed for making the solution U(x,t) of such equations to have a positive-definite matrix of second mixed derivatives with respect to x for any x in the optimization domain and any small t < 0 (the point is remote from the extremum) or a negative-definite matrix in x (the point is close to the extremum). For the functions (x, ) and U(x,t) having these properties, the gradient and the Newton–Kantorovich methods are used in the first and second stages of optimization, respectively.     

Exploiting transition locality in automatic verification of finite-state concurrent systems

In this paper we show that statistical properties of the transition graph of a system to be verified can be exploited to improve memory or time performances of verification algorithms.We show experimentally that protocols exhibit transition locality. That is, with respect to levels of a breadth-first state space exploration, state transitions tend to be between states belonging to close levels of the transition graph. We support our claim by measuring transition locality for the set of protocols included in the Mur verifier distribution .We present a cache-based verification algorithm that exploits transition locality to decrease memory usage and a disk-based verification algorithm that exploits transition locality to decrease disk read accesses, thus reducing the time overhead due to disk usage. Both algorithms have been implemented within the Mur verifier.Our experimental results show that our cache-based algorithm can typically save more than 40% of memory with an average time penalty of about 50% when using (Mur) bit compression and 100% when using bit compression and hash compaction, whereas our disk-based verification algorithm is typically more than ten times faster than a previously proposed disk-based verification algorithm and, even when using 10% of the memory needed to complete verification, it is only between 40 and 530% (300% on average) slower than (RAM) Mur with enough memory to complete the verification task at hand. Using just 300 MB of memory our disk-based Mur was able to complete verification of a protocol with about 10⁹ reachable states. This would require more than 5 GB of memory using standard Mur .     

Definability of Polyadic Lifts of Generalized Quantifiers

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type < 1 >.We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.     

On problem transformability in VLSI

The two basic performance parameters that capture the complexity of any VLSI chip are the area of the chip,A, and the computation time,T. A systematic approach for establishing lower bounds onA is presented. This approach relatesA to the bisection flow, . A theory of problem transformation based on , which captures bothAT ² andA complexity, is developed. A fundamental problem, namely, element uniqueness, is chosen as a computational prototype. It is shown under general input/output protocol assumptions that any chip that decides ifn elements (each with (1+)lognbits) are unique must have =(nlogn), and thus, AT²=(n ²log² n), andA= (nlogn). A theory of VLSI transformability reveals the inherentAT ² andA complexity of a large class of related problems.This work was supported in part by the Semiconductor Research Corporation under contract RSCH 84-06-049-6.     

Directional derivatives of the maximum function

Directional differentiability of the function (x) = sup{f(x, u), u U} is proved for a class of smooth functions f. The result is applied to study the directional differentiability of the function (x) = sup{f(x, y), y F(x)}, where F is a multivalued mapping.Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 171–173, March–April, 1992.     

The complexity of homomorphisms and renamings for minimal unsatisfiable formulas

We investigate the complexity of deciding whether for minimal unsatisfiable formulas F and H there exists a variable renaming, a literal renaming or a homomorphism such that (F) = H. A variable renaming is a permutation of variables. A literal renaming is a permutation of variables which additionally replaces some of the variables by its complements. A homomorphism can be considered as a literal renaming which can map different literals to one literal.