Computational Natural Deduction

The formalization of the notion of a logically sound argument as a natural deduction proof offers the prospect of a computer program capable of constructing such arguments for conclusions of interest. We present a constructive definition for a new subclass of natural deduction proofs, called atomic normal form (ANF) proofs. A natural deduction proof is readily understood as an argument leading from a set of premisses, by way of simple principles of reasoning, to the conclusion of interest. ANF extends this explanative power of natural deduction. The very detailed steps of the argument are replaced by derived rules of inference, each of which is justified by a particular input formula. ANF constitutes a proof theoretically well motivated normal form for natural deduction. Computational techniques developed for resolution refutation based systems are directly applicable to the task of constructing ANF proofs. We analyse a range of languages in this framework, extending from the simple Horn language to the full classical calculus. This analysis is applied to provide a natural deduction based account for existing logic programming languages, and to extend current logic programming implementation techniques towards more expressive languages. We consider the visualization of proofs, failure demonstrations, search spaces and the proof search process. Such visualization can be used for the purposes of explanation and to gain an understanding of the proof search process. We propose introspection based architecture for problem solvers based on natural deduction. The architecture offers a logic based meta language to overcome the combinatorial and other practical problems faced by the problem solver

Propositional Logics Complexity and the Sub-Formula Property

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle for a deductive system for a logic L states that whenever F1,...,Fk proves A, there is a proof in which each formula occurrence is either a sub-formula of A or of some of Fi. In this work we extend Statman result and show that any propositional (possibly modal) structural logic satisfying a particular formulation of the sub-formula principle is in PSPACE. If the logic includes the minimal purely implicational logic then it is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our technique can be used to prove that any finitely many-valued logic has the set of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192

G\"odel's Notre Dame Course

This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics