John S. Schlipf

The expressive powers of the logic programming semantics

Abstract We study the expressive of two semantics far deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older semantics, the two-valued and three-valued program completion semantics. We identify the expressive power of the stable semantics and, in fairly general circumstances, that of the well-founded semantics. In particular, over infinite Herbrand universes, the four semantics all have the same expressive power. We discuss a feature of certain logic programming semantics, which we call the Principle of Stratification, a feature allowing a program to be built easily in modules. The three-valued program completion and well-founded semantics satisfy this principle. Over infinite Herbrand models, we consider a notion of translatability between the three-valued program completion and well-founded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show the well-founded semantics to be more expressive than the three-valued program completion. The proof is a corollary of our result that over non-Herbrand infinite models, the well-founded semantics is more expressive than the three-valued program completion semantics.

Formalizing a Logic for Logic Programming

Much research in the last few years has centered upon an idea called negation as failure. Thebasic idea of negation as failure is that, if an atomic “fact” (atomic sentence) is true, it must bedemonstrably true – so if we cannot demonstrate that the atomic sentence is true, we should inferit to be false.Negation as failure clearly is not logically sound. In classical logic, to infer a sentence to befalse, we must demonstrate that the atomic sentence is false – not merely that it is undemonstrable.Neverthelessithasnaturalappeal. Mostnoticeably, aspointedoutbyMcCarthy, Reiter, andothers,it is invoked in a great many “commonsense” human inferences .Example 1.1 (Reiter) If a patient goes to a physician with a problem that could be caused byeither pneumonia or a sprained ankle, the physician assumes – in the absence of evidence to thecontrary – that the patient has only one of these problems.Example 1.2 If company records do not show that Mr. Jones was ever a vice president, thenmanagement infers that he never was a vice president.A major research topic concerning negation as failure is the attempt to find the “correct”semantics. We shall use the term semantics to mean what is usually called declarative semantics.The declarative semantics of a logic program tells what is to be inferred – but does not prescribehow. Our point of view here is that goal of the study of semantics for negation as failure is tofind a reasonable and fairly elegant mathematical formalism that captures as much as possible ofordinary human negationasfailure type reasoning. There is of course a danger: human negation-asfailure type reasoning often appears to ad hoc, so it seems entirely possible that there is noentirely satisfactory way to capture it formally. But we believe the goal of finding one approachthat captures much, or most, of “common sense reasoning” needs to be pursued.Our starting point is not a search through ordinary human usage. Rather, we start by comparingvarious semantics people have already given for negation as failure. These semantics, we feel, allhave clear “common sense” justification. We shall identify certain features of these semantics asgoals of a semantics for a common sense negation as failure.

On finding solutions for extended Horn formulas

Abstract We present a simple quadratic-time algorithm for solving the satisfiability problem for a special class of boolean formulas. This class properly contains the class of extended Horn formulas and balanced formulas. Previous algorithms for these classes require testing membership in the classes. However, the problem of recognizing balanced formulas is complex, and the problem of recognizing extended Horn formulas is not known to be solvable in polynomial time. Our algorithm requires no such test for membership.

Decidability and definability with circumscription

Abstract We consider McCarthys notions of predicate circumscription and formula circumscription. We show that the decision problems “does θ have a countably infinite minimal model” and “does φ hold in every countably infinite minimal model of θ” are complete Σ 1 2 and complete π 1 2 over the integers, for both forms of circumscription. The set of structures definable (up to isomorphism) as first order definable subsets of countably infinite minimal models is the set of structures which are Δ 1 2 over the integers, for both forms of circumscription. Thus, restricted to countably infinite structures, predicate and formula circumscription define the same sets and have equally difficult decision problems. With general formula circumscription we can define several infinite cardinals, so the decidability problems are dependent upon the axioms of set theory.

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 Kleenes 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.

Computing Well-founded Semantics Faster

We address methods of speeding up the calculation of the well-founded semantics for normal propositional logic programs. We first consider two algorithms already reported in the literature and show that these, plus a variation upon them, have much improved worst-case behavior for special cases of input. Then we propose a general algorithm to speed up the calculation for logic programs with at most two positive subgoals per clause, intended to improve the worst case performance of the computation. For a logic program P in atoms A1, the speed up over the straight Van Gelder alternating fixed point algorithm (assuming worst-case behavior for both algorithms) is approximately (¦P¦/¦A¦)(1/3). For ¦P¦≥¦A¦4, the algorithm runs in time linear in ¦P¦.

SBSAT: a State-Based, BDD-Based Satisfiability Solver

We present a new approach to SAT solvers, supporting efficient implementation of highly sophisticated search heuristics over a range of propositional inputs, including CNF formulas, but particularly sets of arbitrary boolean constraints, represented as BDDs. The approach preprocesses the BDDs into state machines to allow for fast inferences based upon any single input constraint. It also simplifies the set of constraints, using a tool set similar to standard BDD engines. And it memoizes search information during an extensive preprocessing phase, allowing for a new form of lookahead, called local-function-complete lookahead. This approach has been incorporated, along with existing tools such as lemmas, to build a SAT tool we call SBSAT.

Complexity and undecidability results for logic programming

This paper surveys complexity, degree of uncomputability, and expressive power results for logic programming. Some major decision problem complexity results and other results for logic programming are also covered. It also proves several new results filling in previous gaps in the literature. The paper considers seven logic programming semantics: the van Emden-Kowalski semantics for definite (Horn) logic programs; the perfect model semantics for stratified and for locally stratified logic programs; and the two- and three-valued program completion semantics, the well-founded semantics, and the stable semantics, all for normal logic programs, under skeptical inference. The main results concern expressibility and query complexity/uncomputability in five contexts: for propositional logic programs, for first order logic programs with infinite Herbrand universes on their Herbrand universes (a closed domain assumption), for first order logic programs with infinite Herbrand universes on those universes extended with infinitely many new elements (an open domain assumption), and for logic programs without function or constant symbols evaluated over varying extensional databases (DATALOG-type results, data complexity results only) under both closed and open domain assumptions. Several of the open domain assumption results are new to this paper. Other results surveyed are (1) results about the family of all stable models of a program and (2) decision questions about when a logic program has nice properties with respect to a semantics (e.g., a unique stable model). One decision result, for well-founded semantics, is new to this paper.

An algorithm for the class of pure implicational formulas

Abstract Heusch introduced the notion of pure implicational formulas. He showed that the falsifiability problem for pure implicational formulas with k negations is solvable in time O (n k ) . Such falsifiability results are easily transformed to satisfiability results on CNF formulas. We show that the falsifiability problem for pure implicational formulas is solvable in time O (k k n 2 ) , which is polynomial for a fixed k. Thus this problem is fixed-parameter tractable.

The expressive powers of the logic programming semantics (extended abstract)

We compare the expressive powers of three semantics for deductive databases and logic programming: the 3-valued program completion semantics, the well-founded semantics, and the stable semantics, We identify the expressive power of the stable semantics, and in fairly general circumstances that of the well-founded semantics. Over infinite Herbrand models, where the three semantics have equivalent expressive power, we also consider a notion of uniform translatability between the 3-valued program completion and well-founded semantics. In this sense of uniform translatability we show the well-founded semantics to be more expressive.