Deduction Plans: A Basis for Intelligent Backtracking (original) (raw)

Intelligent Backtracking in Plan-Based Deduction

IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000

This paper develops a method of mechanical deduction based on a graphical representation of the structure of proofs. Attempts to find a refutation(s) are recorded in the form of plans, corresponding to portions of an AND/OR graph search space and representing a purely deductive structure of derivation. This method can be applied to any initial base (set of nonnecessarily Horn clauses). Unlike the exhaustive (blind) backtracking which treats all the goals deduced in the course of a proof as equally probable sources of failure, his approach detects the exact source of failure. Only a small fragment of the solution space is kept on disk as a collection of pairs, each of which consists of a plan and a graph of constraints. The search strategy and the method of nonredundant processing of individual pairs which leads to a solution (if it exists) is presented. This approach is compared-on a special case-with a blind backtracking algorithm for which an exponential improvement is demonstrated. Some important implementation problems are discussed, and toplevel design of a mechanical deduction system implementing our algorithm is presented. It is proven that the algorithm is complete in the following sense: if for a given base a resolution refutation exists, then this refutation is found by the algorithm.

Structuring deduction by using abstractions

During the last decade a variety of industrial strength formal methods has emerged and has been applied to industrial test cases to demonstrate their adequacy and scalability. Formal techniques require a sucient tool support especially when dealing with proof obligations. The size and the complexity of the arising problems demand for techniques to structure the deduction. In this paper we present techniques to realize a general divide-and-conquer approach in the framework of proof planning. In order to tackle di erent subgoals by di erent proof methods we propose the use of the color-calculus as an underlying constraint mechanism to resolve possible threats.

Natural deduction via graphs: formal definition and computation rules

Mathematical Structures in Computer Science, 2007

We introduce the formalism of deduction graphs as a generalization of both Gentzen-Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that subproofs can be shared, like in flag deductions (and unlike natural deduction), but also that the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. In the paper we give a precise definition of deduction graphs and we give examples to illustrate them. Furthermore we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure and we therefore also propose a translation to a context calculus with lets, that faithfully captures the structure of deduction graphs. The proof nets of linear logic also present a graph-like presentation of natural deduction. We point out some similarities of the two formalisms.

Special relations in automated deduction

Journal of the ACM, 1986

Two deduction rules are introduced to give streamlined treatment to relations of special importance in an automated theorem-proving system. These rules, the relation replacement and relation matching rules, generalize to an arbitrary binary relation the paramodulation and E-resolution rules, respectively, for equality, and may operate within a nonclausal or clausal system. The new rules depend on an extension of the notion of polarity to apply to subterms as well as to subsentences, with respect to a given binary relation. The rules allow us to eliminate troublesome axioms, such as transitivity and monotonicity, from the system; proofs are shorter and more comprehensible, and the search space is correspondingly deflated. if X=y and if X=y then f(x, z) = f(y, z) then f(z, x) = f(z, y), and for a binary predicate symbol p(x, y), we must introduce two predicate-substitutivity axioms, if X =y and if X = y then if p(xz) then p(y, z) then if p(z,x) then p(z,y).

A General Criterion for Avoiding Infinite Unfolding During Partial Deduction of Logic Programs

Well-founded orderings are a commonly used tool for proving the termination of programs. We introduce related concepts specialised to SLD-trees. Based on these concepts, we formulate formal and practical criteria for controlling the unfolding during the construction of SLD-trees that form the basis of a partial deduction. We provide algorithms that allow to use these criteria in a constructive way. In contrast to the many ad hoc techniques proposed in the literature, our technique provides both a formal and practically applicable framework. the order in which answers are produced, as e.g. Ref. 29).

Conjunctive partial deduction: foundations, control, algorithms, and experiments

The Journal of Logic Programming, 1999

Partial deduction in the Lloyd-Shepherdson framework cannot achieve certain optimisations which are possible by unfold/fold transformations. We introduce conjunctive partial deduction, an extension of partial deduction accommodating such optimisations, e.g., tupling and deforestation.

Parameterized abstractions used for proof-planning

In order to cope with large case studies arising from the application of formal methods in an industrial setting, this paper presents new techniques to support hierarchical proof planning. Following the paradigm of di erence reduction, proofs are obtained by removing syntactical di erences between parts of the formula to be proven step by step. To guide this manipulation we introduce dynamic abstractions of terms. These abstractions are parameterized by the individual goals of the manipulation and are especially designed to ease the proof search based on heuristics. The hierarchical approach and thus the decomposition of the original goal into several subgoals enables the use of di erent abstractions or di erent parameters of an abstraction within the proof search. In this paper we will present one of these dynamic abstractions together with heuristics to guide the proof search in the abstract space.