FC-normal and extended stratified logic program

This paper investigates the consistency property of FC-normal logic program and presents an equivalent deciding condition whether a logic program P is an FC-normal program. The deciding condition describes the characterizations of FC-normal program. By the Petri-net presentation of a logic program, the characterizations of stratification of FC-normal program are investigated. The stratification of FC-normal program motivates us to introduce a new kind of stratification, extended stratification, over logic program. It is shown that an extended (locally) stratified logic program is an FC-normal program. Thus, an extended (locally) stratified logic program has at least one stable model. Finally, we have presented algorithms about computation of consistency property and a few equivalent deciding methods of the finite FC-normal program.     

Disjunctive LP+integrity constraints= stable model semantics

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.     

Consistency Property of Finite FC-Normal Logic Programs

Marek＇s forward-chaining construction is one of the important techniques for investigating the non-monotonic reasoning. By introduction of consistency property over a logic program, they proposed a class of logic programs, FC-normal programs, each of which has at least one stable model. However, it is not clear how to choose one appropriate consistency property for deciding whether or not a logic program is FC-normal. In this paper, we firstly discover that, for any finite logic programⅡ, there exists the least consistency property LCon（Ⅱ） overⅡ, which just depends onⅡitself, such that, Ⅱ is FC-normal if and only ifⅡ is FC-normal with respect to （w.r.t.） LCon（Ⅱ）. Actually, in order to determine the FC-normality of a logic program, it is sufficient to check the monotonic closed sets in LCon（Ⅱ） for all non-monotonic rules, that is LFC（Ⅱ）. Secondly, we present an algorithm for computing LFC（Ⅱ）. Finally, we reveal that the brave reasoning task and cautious reasoning task for FC-normal logic programs are of the same difficulty as that of normal logic programs.     

Normalising the associative law: An experiment with Martin-Löf's type theory

Martin-Löf's type theory contains a logic, a specification language and a programming language, so it is a tool with different uses. Although it is traditionally used as anintegrated programming logic, it may well be used as anexternal logic, which is necessary if one wants to use the formalism of type theory to verify the correctness of an external program. Different tools, such as well founded recursion, measure functions, or the separation of correctness into termination and partial correctness, may be used to obtain a correct type theory program. Type theory is viewed as anopen system with respect toinductively defined types and predicates, which makes it easy to represent an external program as agraph. Formal proofs have been edited using Larry Paulson's ISABELLE.     

The Specification Logic νZ

This paper introduces a wide-spectrum specification logic νZ. The minimal core logic is extended to a more expressive specification logic which includes a schema calculus similar (but not equivalent) to Z, some new additional schema operators and extensions to a programming and program development logic.     

The expressiveness of locally stratified programs

This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence, to obtain all hyperarithmetic sets requires something new, in this case selecting one predicate from the model. We find that the expressive power of programs does not increase when one considers the programs which have a unique stable model or a total well-founded model. This shows that all these classes of structures (perfect models of logically stratified logic programs, well-founded models which turn out to be total, and stable models of programs possessing a unique stable model) are all closely connected with Kleene's hyperarithmetical hierarchy. Thus, for general logic programming, negation with respect to two-valued logic is related to the hyperarithmetic hierarchy in the same way as Horn logic is to the class of recursively enumerable sets. In particular, a set is definable in the well-founded semantics by a programP whose well-founded partial model is total iff it is hyperarithmetic.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by NSF Grant IRI-9012902 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by NSF Grant IRI-8905166 and partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Strong feedback cyclization for systems over rings

For nonnecessarily reachable systems over a commutative ring R, we define a strong form of feedback cyclization (FC). With this natural generalization of the FC property we obtain a feedback reduced form for systems over strong FC rings (i.e. rings for which every system verifies the FC property). In the particular case of reachable single input systems, this gives the usual control canonical form of Bumby et al. [Remarks on the pole-shifting problem over rings, J. Pure Appl. Algebra 20 (1981) 113–127]. Also it is proved that a commutative ring with 1 in its stable range has the strong FC property if and only if it has the UCU property, which is the natural parallel form of the result given by Brewer et al. [Pole assignability in polynomial rings, power series rings and Prüfer domains, J. Algebra 106 (1987) 265–286] for the reachable case. Many classes of rings which were known to be FC rings are in fact strong FC rings, but there are FC rings which are not strong FC rings.     

On Local Stratifiability of Logic Prograns and Databases

In this paper,we deal with the problem of verifying local stratifiability of logic programs and databases presented by Przymusinski.Necessary and sufficient condition for the local stratifiability of logic programs are presented and algorithms of performing the verification are developed.Finally,we prove that a database D B containing clauses with disjunctive consequents can be easily converted into a logic program P such that D B is locally statified iff P is locally stratified.     

On Stratified Belief Base Compilation

In this paper, we investigate the extent to which knowledge compilation can be used to circumvent the complexity of skeptical inference from a stratified belief base (SBB). We first analyze the compilability of skeptical inference from an SBB, under various requirements concerning both the selection policy under consideration, the possibility to make the stratification vary at the on-line query answering stage and the expected complexity of inference from the compiled form. Not surprisingly, the results are mainly negative. However, since they concern the worst case situation only, they do not prevent a compilation-based approach from being practically useful for some families of instances. While many approaches to compile an SBB can be designed, we are primarily interested in those which take advantage of existing knowledge compilation techniques for classical inference. Specifically, we present a general framework for compiling SBBs into so-called C-normal SBBs, where C is any tractable class for clausal entailment which is the target class of a compilation function. Another major advantage of the proposed approach lies in the flexibility of the C-normal belief bases obtained, which means that changing the stratification does not require to re-compile the SBB. For several families of compiled SBBs and several selection policies, the complexity of skeptical inference is identified. Some tractable restrictions are exhibited for each policy. Finally, some empirical results are presented.     

On the autoepistemic reconstruction of logic programming

Current semantics of logic programs normally ignore thesyntactical aspects of the programs. As a result, only the meanings ofsome well-behaved programs can be captured by these semantics. In this paper however, we propose a new semantics of logic programs that can reflectsome of the syntactical behaviours of the programs. The central notion of the semantics is the concept of aneutral clause p ← A which does not affect the behaviour of p in a program. The logic that underlies the semantics is based on anintensional extension of Levesque’s autoepistemicpredicate logic. It differs from existing autoepistemic logics in that it isquantificational andconstructive. We will also compare and contrast our semantics with some well-known semantics. In particular, we will show how to capture the undefined value of a logic program without resorting to a three-valued nonmonotonic formalism. This is achieved by translating an incoherent AE logic program to a program with multiple AE extensions whose intersection can then be used to characterize the undefined value of a logic program.     

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce aλ-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs ofΠ⁰₂formulas, i.e., those corresponding to program specifications.     

The kernel strategy and its use for the study of combinatory logic

Barendregt defines combinatory logic as an equational system satisfying the combinatorsS andK with ((Sx)y)z=(xz)(yz) and (Kx)y=x; the set consisting ofS andK provides abasis for all of combinatory logic. Rather than studying all of the logic, logicians often focus onfragments of the logic, subsets whose basis is obtained by replacingS orK or both by other combinators. In this article, we present a powerful new strategy, called thekernel strategy, for studying fragments in the context of questions concerned with fixed point properties. Interest in such properties rests in part with their relation to normal forms and paradoxes. We show how the kernel strategy was used to answer a number of open questions, offering abundant evidence that the availability of the kernel strategy marks a singular advance for automated reasoning. In all of our experiments with this strategy applied by an automated reasoning program, the rate of success has been impressively high and the CPU time to obtain the desired information startlingly small. For each fragment we study, we use the kernel strategy to attempt to determine whether the strong or the weak fixed point property holds. WhereA is a given fragment with basisB, the strong fixed point property holds forA if and only if there exists a combinatory such that, for all combinatorsx,yx=x(yx), wherey is expressed purely in terms of elements ofB. The weak fixed point property holds forA if and only if for all combinatorsx there exists a combinatory such thaty=xy, wherey is expressed purely in terms of the elements ofB and the combinatorx. Because the use of the kernel strategy is so effective in addressing questions focusing on either fixed point property, its formulation marks an important advance for combinatory logic. Perhaps of especial interest to logicians is an infinite class of infinite sets of tightly coupled fixed point combinators (presented here), whose unexpected discovery resulted directly from the application of the strategy by an automated reasoning program. We also offer various open questions for possible research and focus on an automated reasoning program and input files that may prove useful for such research.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Automata theory based on quantum logic: some characterizations

Automata theory based on quantum logic (abbr. l-valued automata theory) may be viewed as a logical approach of quantum computation. In this paper, we characterize some fundamental properties of l-valued automata theory, and discover that some properties of the truth-value lattices of the underlying logic are equivalent to certain properties of automata. More specifically (i) the transition relations of l-valued automata are extended to describe the transitions enabled by strings of input symbols, and particularly, these extensions depend on the distributivity of the truth-value lattices (Proposition 3.1); (ii) some properties of the l-valued successor and source operators and l-valued subautomata are demonstrated to be equivalent to a property of the truth-value lattices which is exactly equivalent to the distributive law (Proposition 4.3 and Corollary 4.4). This is a new characterization of Boolean algebras in the framework of l-valued automata theory; (iii) we verify that the intersection of two l-valued subautomata is still an l-valued subautomaton if and only if the multiplication (&) is distributive over the union in the truth-value lattices (Proposition 4.5), which is strictly weaker than the usual distributivity; (iv) we show that some topological characterizations in terms of the l-valued successor and source operators also rely on the distributivity of truth-value lattices (Theorem 5.6). Finally, we address some related topics for further study.     

知识库维护过程中检查其协调性的有效方法

本文首先描述了知识库维护过程中的协调性问题，然后给出了扩充逻辑程序设计的框架，在此框架下，每个逻辑程序等价于一个知识库．为了检查知识库的协调性，本文为知识库中的推理规则构造了正支持集和负支持集，并给出了一些定义；基于这些概念和定义，提出了知识库维护过程中检查知识库协调性的一种有效方法，并证明了相关的定理；基于此方法，实现了一个算法ＣＨＩＭＥ，并给出了用ＣＨＩＭＥ分析一些知识库的实验结果．本文还提到一些相关的工作，最后给出结论.     

On the computational cost of disjunctive logic programming: Propositional case

This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satisfied by all models according to a given semantics. We concentrate on finite propositional disjunctive programs with as well as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is ₂ ^p -complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is ₂ ^p -complete; analogous results are derived for the answer sets semantics of extended disjunctive logic programs. Besides, we generalize previously derived complexity results for the generalized closed world assumption and other more sophisticated variants of the closed world assumption. Furthermore, we use the close ties between the logic programming framework and other nonmonotonic formalisms to provide new complexity results for disjunctive default theories and disjunctive autoepistemic literal theories.Parts of the results in this paper appeared in form of an abstract in the Proceedings of the Twelfth ACM SIGACT SIGMOD-SIGART Symposium on Principles of Database Systems (PODS-93), pp. 158–167. Other parts appeared in shortened form in the Proceedings of the International Logic Programming Symposium, Vancouver, October 1993 (ILPS-93), pp. 266–278. MIT Press.     

Some new results on the algebraic characterizations of an equality constrained optimization problem equivalent to the transportation problem

An equality constrained optimization problem equivalent to the transportation problem with m sources and n destinations is described. The optimality condition and some algebraic characterizations of the problem are investigated using its Hessian matrix. In addition, several algebraic characterizations of an equivalent case of the transportation problem are given using the spectral decomposition and generalized inverses of its coefficient matrix. It is shown that the transportation problem and its equivalent case have common algebraic characterizations.     

N-Prolog and equivalence of logic programs

The aim of this work is to develop a declarative semantics for N-Prolog with negation as failure. N-Prolog is an extension of Prolog proposed by Gabbay and Reyle (1984, 1985), which allows for occurrences of nested implications in both goals and clauses. Our starting point is an operational semantics of the language defined by means of top-down derivation trees. Negation as finite failure can be naturally introduced in this context. A goal-G may be inferred from a database if every top-down derivation of G from the database finitely fails, i.e., contains a failure node at finite height.Our purpose is to give a logical interpretation of the underlying operational semantics. In the present work (Part 1) we take into consideration only the basic problems of determining such an interpretation, so that our analysis will concentrate on the propositional case. Nevertheless we give an intuitive account of how to extend our results to a first order language. A full treatment of N-Prolog with quantifiers will be deferred to the second part of this work.Our main contribution to the logical understanding of N-Prolog is the development of a notion of modal completion for programs, or databases. N-Prolog deductions turn out to be sound and complete with respect to such completions. More exactly, we introduce a natural modal three-valued logic PK and we prove that a goal is derivable from a propositional program if and only if it is implied by the completion of the program in the logic PK. This result holds for arbitrary programs. We assume no syntactic restriction, such as stratification (Apt et al. 1988; Bonner and McCarty 1990). In particular, we allow for arbitrary recursion through negation.Our semantical analysis heavily relies on a notion of intensional equivalence for programs and goals. This notion is naturally induced by the operational semantics, and is preserved under substitution of equivalent subexpressions. Basing on this substitution property we develop a theory of normal forms of programs and goals. Every program can be effectively transformed into an equivalent program in normal form. From the simple and uniform structure of programs in normal form one may directly define the completion.     

On Local StratifiabUity of Logic Programs and Databases

In this paper,we deal with the problem of verifying local stratifiability of logic programs anddatabases presented by Przymusinski.Necessary and sufficient conditions for the local stratifiability oflogic programs are presented and algorithms for performing the verification are developed.Finally,weprove that a database DB containing clauses with disjunctive consequents can be easily converted into alogic program P such that DB is locally stratified iff P is locally stratified.     

Local system equivalence

The important concept of the strict system equivalence of polynomial realisations has been extended to realisations over an arbitrary principal ideal domainR. We use the algebraic concept of the localisation of a ring at a prime ideal to study the local properties, as opposed to the global properties, of realisations over arbitrary principal ideal domains. This allows us to gain a new understanding of strict system equivalence. We show that twoR-realisations of aK-matrix, whereK is the field of fractions of the principal ideal domainR, are strictly system equivalent if and only if they are locally system equivalent at every prime ideal (p) ofR. 1980 Mathematics Subject Classification Number: 93B99