Tautology testing with a generalized matrix reduction method

AbstractA formalization of the tautology problem in terms of matrices is given. From that a generalized matrix reduction method is derived. Its application to a couple of selected examples indicates a relatively efficient behaviour in testing the validity of a given formula in propositional logic—not only for machines but also for humans. A further result from that formalization is a reduction of the tautology problem to a part of Presburger arithmetic which involves formulas of the ∀∃∃⋯∃-type where all quantifiers have finite range

Let's plan it deductively!

AbstractThe paper describes a transition logic, TL, and a deductive formalism for it. It shows how various important aspects (such as ramification, qualification, specificity, simultaneity, indeterminism etc.) involved in planning (or in reasoning about action and causality for that matter) can be modelled in TL in a rather natural way. (The deductive formalism for) TL extends the linear connection method proposed earlier by the author by embedding the latter into classical logic, so that classical and resource-sensitive reasoning coexist within TL. The attraction of a logical and deductive approach to planning is emphasized and the state of automated deduction briefly described

Research Perspectives for Logic and Deduction

The article is meant to be kind of the author's manifesto for the role of logic and deduction within Intellectics. Based on a brief analysis of this role the paper presents a number of proposals for future scientic research along the various di-mensions in the space of logical explorations. These dimensions include the range of possible applications including modelling intelligent behavior, the grounding of logic in some semantic context, the choice of an appropriate logic from the great variety of alternatives, then the choice of an appropriate formal system for repre-senting the chosen logic, and nally the issue of developing the most ecient search strategies. Among the proposals is a conjecture concerning the treatment of cuts in proof search. Often a key advance is a matter of applying a small change to a single formula. Ray Kurzweil [Kur05, p.5]

Deductive synthesis of recursive plans in linear logic

Linear logic has previously been shown to be suitable for describing and deductively solving planning problems involving conjunction and disjunction. We introduce a recursively defined datatype and a corresponding induction rule, thereby allowing recursive plans to be synthesised. In order to make explicit the relationship between proofs and plans, we enhance the linear logic deduction rules to handle plans as a form of proof term

Middle-out reasoning for synthesis and induction

We develop two applications of middle-out reasoning in inductive proofs: Logic program synthesis and the selection of induction schemes. Middle-out reasoning as part of proof planning was first suggested by Bundy et al [Bundy et al 90a]. Middle-out reasoning uses variables to represent unknown terms and formulae. Unification instantiates the variables in the subsequent planning, while proof planning provides the necessary search control. Middle-out reasoning is used for synthesis by planning the verification of an unknown logic program: The program body is represented with a meta-variable. The planning results both in an instantiation of the program body and a plan for the verification of that program. If the plan executes successfully, the synthesized program is partially correct and complete. Middle-out reasoning is also used to select induction schemes. Finding an appropriate induction scheme during synthesis is difficult, because the recursion of the program, which is un..