We present an extension of Horn-clause logic which can hypothetically add and delete tuples from a database. Such logics have been discussed in the literature, but their complexities and expressibilities have remained an open question. This paper examines two such logics in the function-free, predicate case. It is shown, in particular, that augmenting Horn-clause logic with hypothetical addition increases its data-complexity from PTIME to PSPACE. When deletions are added as well, complexity increases again, to EXPTIME. We then augment the logic with negation-as-failure and develop the notion of stratified hypothetical rulebases. It is shown that negation does not increase complexity. To establish expressibility, we view the logic as a query language for relational databases. It is shown that any typed generic query that is computable in PSPACE can be expressed as a stratified rulebase of hypothetical additions. Similarly, any typed generic query that is computable in EXPTIME can be exp...

Abstract. An overview of a software system under development for “assembling ” legal documents is presented. The system applies Artificial Intelligence (AI) methods and is founded on a theory construction or abduction view of legal reasoning. The AI methods employed include an Assumption-based Truth Maintenance System (ATMS), a Natural Deduction theorem prover, and an implementation of Poole's approach to default reasoning. Legal Document Assembly There has been very little published work on the application of Artificial Intelligence (AI) technology to the problem of constructing legal documents. The bulk of the legal expert systems literature is concerned “only ” with the problem of supporting the task of deciding legal cases, i.e. of identifying and analyzing the legal issues raised by the facts of some case. With the exception of the German KOKON project [Kowalewski 86], where an attempt was made to represent legal rules using Horn clauses, I am aware of no previous attempt to develop an integrated approach to legal analysis and documentation using AI methods. The work described here is one attempt to fill this gap. The usual approach to “assembling ” legal documents is procedural. James Sprowl

This paper shows that classical logic is inappropriate for hypothetical reasoning and develops an alternative logic for this purpose. The paper focuses on a form of hypothetical reasoning which appears computationally tractable. Specifically, Horn-clause logic is augmented with rules, called embedded implications, which can hypothetically add atomic formulas to a rulebase. By introducing the notion of rulebase independence, we show that these rules can express hypothetical queries which classical logic cannot; and by adopting methods from modal logic, we show these rules to be intuitionistic. In particular, they form a subset of intuitionistic logic having semantic properties similar to those of Horn-clause logic. This report is an expanded version of a paper published in the Proceedings of the Seventh National Conference on Artificial Intelligence, St. Paul, Minnesota, August 21--26 1988, American Association for Artificial Intelligence (AAAI). 1 Introduction Several researchers...

We consider the computational complexity of evaluating nested counterfactuals over a propositional knowledge base. Counterfactual implication p ? q models a statement "if p, then q," where p is known or expected to be false, and is different from material implication p ) q. A nested counterfactual is a counterfactual statement where the conclusion q is a (possibly negated) counterfactual. Statements of the form p 1 ? (p 2 ? \Delta \Delta \Delta (p n ? q) \Delta \Delta \Delta) intuitively correspond to hypothetical queries involving a sequence of revisions. We show that evaluating such statements is \Pi P 2 -complete, and that this task becomes PSPACE-complete if negation is allowed in the nesting. We also consider nesting a counterfactual in the premise, i.e. (p ? q) ? r and show that evaluating such statements is most likely much harder than evaluating p ? (q ? r). 1 Introduction A counterfactual is a conditional statement "if p, then q," where the premise p is eit...

This paper develops a declarative language with intuitionistic semantics which expresses exactly the generic database queries. Syntactically, the language is an extension of Datalog (function-free Horn logic) which allows rules themselves to appear in the bodies of other rules. Such rules are called embedded implications. Several researchers have studied restricted versions of these rules, but in their full incarnation, universal quantifiers may appear in the premises, as in the rule A / 8 x [B(x) / C(x)]. This paper focuses on these embedded universal quantifiers. It is shown, for instance, that such quantifiers give the logic the ability to create new constant symbols hypothetically during inference. This, in turn, allows the logic to simulate unbounded counters and arbitrary Turing machines. In addition, when the logic is augmented with negation-as-failure, it becomes expressively complete, that is, it can express any database query which is typed and generic. Similar results exist...

This paper examines an extension of Horn logic in which rules can add entries to a database hypothetically. Several researchers have developed logical systems along these lines, but the complexity and expressibility of such logics is only now being explored. It has been shown, for instance, that the data-complexity of these logics is PSPACE-complete in the function-free, predicate case. This paper extends this line of research by developing syntactic restrictions with lower complexity. These restrictions are based on two ideas from Horn-clause logic: linear recursion and stratified negation. In particular, a notion of stratification is developed in which negation-as-failure alternates with linear recursion. The complexity of such rulebases depends on the number of layers of stratification. The result is a hierarchy of syntactic classes which corresponds exactly to the polynomial-time hierarchy of complexity classes. In particular, rulebases with k strata are data-complete for \Sigma P...

Intuitionistic logic programming is an extension of Horn-clause logic programming in which implications may appear "embedded" on the right-hand side of a rule. Thus, rules of the form A(x) / [B(x) / C(x)] are allowed. These rules are called embedded implications . In this paper, we develop a language in which negation-as-failure is combined with embedded implications in a principled way. Although this combination has been studied by other researchers, Gabbay has argued in [10] that the entire idea is logically incoherent since modus ponens would not be valid in such a system. We show how to solve this problem by drawing a distinction between rules and goals. To specify the semantics of rules and goals, we then develop an analogue of Przymusinski's perfect model semantics for stratified Horn-clause logic [20]. Several modifications are necessary to adapt this idea from classical logic to intuitionistic logic, but we eventually show how to define a preferred model of a stratified intui...

In this paper we present a modal extension of logic programming, which allows both multiple modalities and embedded implications. We show that this extension is well suited for structuring knowledge and, specifically, for defining module constructs within programs, for representing agents beliefs, and also for hypothetical reasoning. The language contains modalities [a i ] to represent agent beliefs, and a modality 2 which is a kind of common knowledge operator. It allows sequences of modalities to occur in front of clauses, goals and clause heads, and hypothetical implications to occur in goals and in clause bodies. A goal directed proof procedure of the language is presented, and several examples of its use for defining modules are given. In particular, the language is shown to capture different proposal for module definition and composition presented in the literature. The modal logic, of which our programming language is a clausal fragment, is introduced through its Kripke semantic...

. In conventional logic programming systems, control information is expressed by clause and goal order and by purely procedural constructs, e.g., the Prolog cut. This approach destroys the equivalence of declarative and procedural semantics in logic programs. In this paper, we argue that in order to comply with the logic programming paradigm, control information should also be expressed declaratively. A program should be divided into a logical theory that specifies the problem to be solved and control information that specifies the strategy of the deduction process. Control information is expressed through meta level control clauses. These control clauses are evaluated dynamically in order to select the subgoal that will be resolved next and to select the resolving clause. Program clauses have guards that allow clause determinism to be expressed. A major design goal for the presented work is to keep the declarative and the procedural semantics of logic programs equivalent. The emphasis...

This paper addresses a limitation of most deductive database systems: they