Herbrand Model Research Articles

An Extension Principle for Fuzzy Logics

AbstractLet S be a set, P(S) the class of all subsets of S and F(S) the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P(S) into a fuzzy closure operator J* defined in F(S). This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory.Mathematics Subject Classification: 03B52.

Rewrite-based Equational Theorem Proving with Selection and Simplification

We present various refutationally complete calculi for first-order clauses with equality that allow for arbitrary selection of negative atoms in clauses. Refutational completeness is established via the use of well-founded orderings on clauses for defining a Herbrand model for a consistent set of clauses. We also formulate an abstract notion of redundancy and show that the deletion of redundant clauses during the theorem proving process preserves refutational completeness. It is often possible to compute the closure of non-trivial sets of clauses under application of non-redundant inferences. The refutation of goals for such complete sets of clauses is simpler than for arbitrary sets of clauses, in particular it suffices to consider only those proofs that have support from the goals, without compromising refutational completeness. For certain classes of formulas the search space can be restricted even further, as we demonstrate for so-called quasi-Horn clauses. The results in this paper contain as special cases or generalize many known results about Knuth–Bendix-like completion procedures (for equations, Horn clauses, and Horn clauses with Boolean conditions), completion of first-order clauses by clausal rewriting, and inductive theorem proving for Horn clauses.

A LOGICAL FRAMEWORK FOR CONCEPT LEARNING WITH BACKGROUND KNOWLEDGE

A central process in many kinds of learning is the process of generalization or concept learning from a set of training instances (a set of examples and counterexamples) in the presence of some Background Knowledge. Given a set of examples and counterexamples of a concept, the learner induces a general concept that describes all of the positive examples and none of the counterexamples, and is consistent with the Background Knowledge. We present here a logical framework to induce concept descriptions from a given set of examples and counterexamples in the presence of Background Knowledge described by a set of Horn clauses. In particular, we provide: 1) a definition of what is meant by “a learnable concept” from examples and counterexamples in the presence of Background Knowledge described by a set H of Horn clauses”. Broadly speaking, in our framework, a learnable concept C is an atom which is true (valid) in the least Herbrand model of H but false in some other models of H and, 2) a methodology to induce general concept descriptions from concept examples and counterexamples in Horn theories. We give an automatic method, based on a well-founded ordering on the elements of H, which is the basis for checking validity in the least Herbrand model of H. This work generalizes, unifies, and provides a logical framework of previous studies on the concept-learning-from-examples paradigm.

Generalized constraint propagation over the CLP scheme

Constraint logic programming is often described as logic programming with unification replaced by constraint solving over a computation domain. There is another, very different, CLP paradigm based on constraint satisfaction, where program-defined goals can be treated as constraints and handled using propagation. This paper proposes a generalization of propagation that enables it to be applied on arbitrary computation domains revealing that the two paradigms of CLP are orthogonal and can be freely combined. The main idea behind generalized propagation is to use whatever constraints are available over the computation domain to express restrictions on problem variables. Generalized propagation on a goal G requires that the system extracts a constraint approximating all the answers to G. This paper introduces a generic algorithm for generalized propagation called topological branch and bound, which avoids enumerating all the answers to G. Generalized propagation over the Herbrand universe has been implemented in a system called Propia, and we describe its behavior in some applications.

Finite representation of infinite query answers

We define here a formal notion of finite representation of infinite query answers in logic programs. We apply this notion to Datalog nS programs may be infinite and consequently queries may have infinite answers. We present a method to finitely represent infinite least Herbrand models of Datalog nS program (and its underlying computational engine) can be forgotten. Given a query to be evaluated, it is easy to obtain from the relational specification finitely many answer substitutions that represent infinitely many answer substitutions to the query. The method involved is a combination of a simple, unificationless, computational mechanism (graph traversal, congruence closure, or term rewriting) and standard relational query evaluation methods. Second, a relational specification is effectively computable and its computation is no harder, in the sense of the complexity class, than answering yes-no queries. Our method is applicable to every range-restricted Datalog nS program. We also show that for some very simple non-Datalog nS logic programs, finite representations of query answers do not exist.

Basic transformation operations which preserve computed answer substitutions of logic programs

Some transformation operations for logic programs, basic for partial deduction, program specialization, and transformation, and for program synthesis from specifications, are studied with respect to the minimal S-model semantics defined in [31, 15–17]. Such a semantics is, in our opinion, more interesting than the usual least Herbrand model one since it captures the program's behavior with respect to computed answers. The S-semantics is also the strongest semantics which is maintained by unrestricted unfolding [31]. For such operations, we single out general applicability conditions, and prove that they guarantee that the minimal S-model semantics of a program is not modified by the transformation. Some sufficient conditions, which are very common in practice and easy to verify, since they are mostly syntactical, are also supplied with simple exemplifications.

Unfold⧸fold transformation of general logic programs for the well-founded semantics

This paper proposes a framework for unfold⧸fold transformation of general logic programs. The framework is an extension of that of Tamaki and Sato [25] (resp., Seki [21]) defined for definite programs (resp., stratified programs) which was shown to preserve the least Herbrand model semantics (resp., the perfect model semantics). Unlike the previous work, the framework given in this paper imposes no syntactic restrictions on negations occurring in programs, and it is shown that a transformed program has the same well-founded semantics as that of an initial program. We also discuss a relationship between unfold⧸fold transformation and partial deduction for the well-founded semantics.

A Model-Theoretic Reconstruction of the Operational Semantics of Logic Programs

In this paper we define a new notion or truth on Herbrand interpretations extended with variables which allows us to capture, by means of suitable models, various observable properties, such as the ground success set, the set of atomic consequences, and the computed answer substitutions. The notion of truth extends the classical one to account for non-ground formulas in the interpretations. The various operational semantics are all models. An ordering on interpretations is defined to overcome the problem that the intersection of a set of models is not necessarily a model. The set of interpretations with this partial order is shown to be a complete lattice, and the greatest lower bound of any set of models is shown to be a model. Thus there exists a least model, which is the least Herbrand model and therefore the ground success set semantics. Richer operational semantics are non-least models, which can, however, be effectively defined by fixpoint constructions. The model corresponding to the computed answer substitutions operational semantics is the most primitive one (the others can easily be obtained from it).

Logic Programs for Primitive Recursive Sets

Meyer and Ritchie have previously given a description of primitive recursive functions by loop-programs. In this paper a class of logic programs is described which computes the primitive recursive sets on Herbrand universes. Furthermore, an internal description of primitive recursive functions and sets on Herbrand universes is given.

Relating minimal models and pre-requisite-free normal defaults

Given any disjunctive logic program p, we show that the minimal Herbrand models of P are in a precise one-one correspondence with the default logic theory Δ P obtained by adding to P, the default logic schema :¬A A .

Analysis of the behavior of logic-based computation for deductive databases and default reasoning

One advantage of logic programming is the ability to use formal logic to analyze program behavior. Formal logic provides a basis for demonstrating that a program will behave as the user intends. This paper attempts to present to the non-logician known program behavior results for logic programs in the context of deductive databases. Previous results are extended through an investigation of a class of programs more general than logic programs. Specific attention is given to computational approaches which involve the solution of a finite sequence of progressively larger integer linear programming problems (ILP's). Constraints for such problems are constructed by a form of logical unification. Results indicate that under certain conditions the solution to the last ILP yields a Herbrand model for the original set of statements in logic.

Towards a unified theory of intensional logic programming

Intensional Logic Programming is a new form of logic programming based on intensional logic and possible worlds semantics. Intensional logic allows us to use logic programming to specify nonterminating computations and to capture the dynamic aspects of certain problems in a natural and problem-oriented style. The meanings of formulas of an intensional first-order language are given according to intensional interpretations and to elements of a set of possible worlds. Neighborhood semantics is employed as an abstract formulation of the denotations of intensional operators. Then we investigate general properties of intensional operators such as universality, monotonicity, finitariness and conjunctivity. These properties are used as constraints on intensional logic programming systems. The model-theoretic and fixpoint semantics of intensional logic programs are developed in terms of least (minimum) intensional Herbrand models. We show in particular that our results apply to a number of intensional logic programming languages such as Chronolog proposed by Wadge and Templog by Abadi and Manna. We consider some elementary extensions to the theory and show that intensional logic program clauses can be used to define new intensional operators. Intensional logic programs with intensional operator definitions are regarded as metatheories.

A domain-theoretic approach to functional and logic programming

AbstractThe integration of functional and logic programming languages has been a topic of great interest in the last decade. Many proposals have been made, yet none is completely satisfactory especially in the context of higher order functions and lazy evaluation. This paper addresses these shortcomings via a new approach: domain theory as a common basis for functional and logic programming. Our integrated language remains essentially within the functional paradigm. The logic programming capability is provided by set abstraction (via Zermelo-Frankel set notation), using the Herbrand universe as a set abstraction generator, but for efficiency reasons our proposed evaluation procedure treats this generator's enumeration parameter as a logical variable. The language is defined in terms of (computable) domain-theoretic constructions and primitives, using the lower (or angelic) powerdomain to model the set abstraction facility. The result is a simple, elegant and purely declarative language that successfully combines the most important features of both pure functional programming and pure Horn logic programming. Referential transparency with respect to the underlying mathematical model is maintained throughout. An implicitly correct operational semantics is obtained by direct execution of the denotational semantic definition, modified suitably to permit logical variables whenever the Herbrand universe is being generated within a set abstraction. Completeness of the operational semantics requires a form of parallel evaluation, rather than the more familiar left-most rule.

A method for simultaneous search for refutations and models by equational constraint solving

A method is proposed to systematize the simultaneous search for a refutation and Herbrand models of a given conjecture. It is based on an extension of resolution using equational problems and the inference system included in the method is proved to be sound and refutationally complete. For some classes of formulas the method is indeed a decision procedure. In particular it is a decision procedure for the Bernays-Schönfinkel class (a class for which no resolution term ordering strategy is known to be a decision procedure). Some examples of model construction - including one for which other resolution based decision procedures fail to detect satisfiability - are developed in detail. The method is also useful in cases in which model construction is not required. The search space in resolution based deductions can be greatly decreased. This is shown in solving a question-answering problem, considered to be hard. Models are built by constructing relations on Herbrand universe. The relationship between these models and finite ones is established. The class of these constructible relations is precisely characterized. Some of the rules introduced, in order to extend resolution, are essentially new. It is proved that they are necessary for enlarging the class of models the method is able to build. A brief comparison with existing methods which bear similarity with ours, either in the use of constraints or in the search of a model, shows the originality of our proposal. Some hints about directions for extending the class of formulas for which models can be constructed are given.

Unfold/fold transformations of logic programs

Unfold/fold transformations have been used in logic programming for some years to transform programs into more efficient ones. We describe recent work on the extent to which these transformations produce programs which are equivalent to the original one. Various notions of equivalence are considered: same success set; finite failure set; least Herbrand model; completion. This is used to illustrate the rather unsatisfactory relationship between logic programming and logic shown by the wide variety of different declarative semantics proposed for logic programs.

Complexity of resolution proofs and function introduction

The length of resolution proofs is investigated, relative to the model-theoretic measure of Herband complexity. A concept of resolution deduction (R-deduction) is introduced which is somewhat more general than the classical concepts. It is shown that proof complexity is exponential in terms of Herband complexity and that this bound is tight. The concept of R-deduction is extended to FR-deduction, where, besides resolution, a function introduction rule (based on a re-Skolemization technique) is allowed. As an example, consider the clause P(x, y) ∨ Q(x, y): conclude P(x,f(x)) ∨ Q(a, y), where a, f(x) are new function symbols extending Herbrand's universe. (We use the derivability of (∀x)(∃y) P(x, y) ∨ (∃x)(∀) Q(x, y) from (∀x)(∀y)( P(x, y) ∨ Q(x, y)) and Skolemization). By modification of a construction of Statman it is shown that FR-deductions may be nonelementarily shorter than R-deductions (thus giving a nonelementary speedup with respect to Herbrand complexity and search space). Finally it is proved that FR-deduction has weaker effects only (maximally double exponential speedup) if clausal logic is restricted to Horn logic.

Short note: procedural semantics and negative information of elementary formal system

We discuss procedural semantics and inference of negated ground atoms in elementary formal system (EFS). EFS is now a logic programming with associative unification. There are two problems on the SLD-resolution when we infer negated atoms. One is existence of infinitely many unifiers for two atoms, even maximally general. This prevents us finding a proper completed definitions of EFS's corresponding to the negation as failure rule. The other problem is existence of infinite derivations. They make it difficult to reject atoms not in the least Herbrand model. In the note, we give solutions for these problems. When we use the SLD-resolution to accept formal languages defined by EFS's, we assume that every refutation begins from a ground goal. Under the assumption we solve the first problem by introducing the variable-bounded EFS, which is powerful enough to define languages. The solution for the second problem is to bound the length of every SLD-derivation. We present it as an algorithm to decide whether a ground atom is in the least Herbrand model or not. We introduce the weakly reducing EFS as a class of EFS's where our algorithm is a complete realization of the closed world assumption.

How complicated is the set of stable models of a recursive logic program?

Gelfond and Lifschitz (1988) proposed the notion of a stable model of a logic program. We establish that the set of all stable models in a Herbrand universe of a recursive logic program is, up to recursive renaming, the set of all infinite paths of a recursive, countably branching tree, and conversely. As a consequence, the problem, given a recursive logic program, of determining whether it has at least one stable model, is Σ 1 1-complete. Due to the equivalences established in the authors' previous nonmonotone rule systems papers (Marek, Nerode and Remmel (1990)), this applies equally to truth maintenance systems and default logics.

Compositional model-theoretic semantics for logic programs

We present a compositional model-theoretic semantics for logic programs, where the composition of programs is modelled by the composition of the admissible Herbrand models of the programs. An Herbrand model is admissible if it is supported by the assumption of a set of hypotheses. On one hand, the hypotheses supporting a model correspond to an open interpretation of the program intended to capture possible compositions with other programs. On the other hand, admissible models provide a natural model-theory for a form of hypothetical reasoning, called abduction. The application of admissibel models to programs with negation is discussed.

Logic programming with functions and predicates: The language Babel

We investigate the experimental programming language BABEL, designed to achieve integration of functional programming (as embodied in HOPE, Standard ML, or MIRANDA) and logic programming (as embodied in PROLOG) in a simple, flexible, and mathematically well-founded way. The language relies on a constructor discipline, well suited to accommodate PROLOG terms and HOPE-like patterns. From the syntactical point of view, BABEL combines pure PROLOG with a first order functional notation. On the other side, the language uses narrowing as the basis of a lazy reduction semantics which embodies both rewriting and SLD resolution and supports computation with potentially infinite data structures. There is also a declarative semantics, based on Scott domains, which provides a notion of least Herbrand model for BABEL programs. We develop both semantics and prove the existence of least Herbrand models, as well as a soundness result for the reduction semantics w.r.t. the declarative one. We also sketch a completeness result for the reduction semantics and illustrate the features of the language through some programming examples.