Using Extended Tactics to Do Proof Transformations

In this thesis we develop a comprehensive human-oriented theorem proving system that integrates several different proof systems. The main theorem proving environment centers around a natural Gentzen first-order logic system. This allows construction of natural proofs, encourages user involvement in the search for proofs, and facilitates understanding of the resulting proofs. We integrate more abstract automatically generated proofs such as resolution refutations by transforming them to proofs in the Gentzen system. Expansion trees are another proof system used as an intermediate stage in transformations between the abstract and natural systems. They are a compact representation useful for transformations and other computations. We develop a programming language approach to theorem proving based on tactics and tacticals. Our extended tactics provide a method for doing proof transformations, as well as facilitate interactive theorem proving, allowing full integration of interactive and automatic theorem proving. In the system, we explicitly represent proofs in each proof system and view expansion tree proofs as types for Gentzen proof terms. This explicit proof representation allows proofs to be manipulated as meaningful data objects and used in various computations. For example, the proof terms in the natural Gentzen system can be used to obtain natural language explanations of proofs. We foresee several applications for this kind of theorem proving system, such as use as a logic tutor, a tool for doing mathematics, or an enhanced reasoner and explanation facility for existing A1 systems

Computational Aspects of Proofs in Modal Logic

Various modal logics seem well suited for developing models of knowledge, belief, time, change, causality, and other intensional concepts. Most such systems are related to the classical Lewis systems, and thereby have a substantial body of conventional proof theoretical results. However, most of the applied literature examines modal logics from a semantical point of view, rather than through proof theory. It appears arguments for validity are more clearly stated in terms of a semantical explanation, rather than a classical proof-theoretic one. We feel this is due to the inability of classical proof theories to adequately represent intensional aspects of modal semantics. This thesis develops proof theoretical methods which explicitly represent the underlying semantics of the modal formula in the proof. We initially develop a Gentzen style proof system which contains semantic information in the sequents. This system is, in turn, used to develop natural deduction proofs. Another semantic style proof representation, the modal expansion tree is developed. This structure can be used to derive either Gentzen style or Natural Deduction proofs. We then explore ways of automatically generating MET proofs, and prove sound and complete heuristics for that procedure. These results can be extended to most propositional system using a Kripke style semantics and a fist order theory of the possible worlds relation. Examples are presented for standard T, S4, and S5 systems, systems of knowledge and belief, and common knowledge. A computer program which implements the theory is briefly examined in the appendix

Resolution Proof Technique in Linear Temporal Logic.

This dissertation presents a resolution proof technique for Propositional Linear Temporal Logic of discrete time with the Until operator. The presented proof technique stems from the resolution method developed by L. Farinas del Cerro and A. Cavalli. However, their method is incomplete, and their completeness proof, as originally reported, is incorrect. Unlike Farinas\u27s method, our proof technique incorporated the Until operator, which is a very powerful and useful in describing complex temporal relationships which are common in many areas of computer science. Our technique is also proved complete. The presented resolution method for linear temporal logic is similar to classical resolutions: the main goal is to show unsatisfiability of a set of temporal clauses by locating, either directly or indirectly, a state which contains unsatisfiability. Subsequent resolvents of a refutation are obtained by resolving out complementary literals referring to the same instant of time. In order to increase the efficiency of the resolution proof technique, we have developed a refinement of the presented basic method. This refinement is similar to the well-known Ordered Linear (OL) strategy for classical resolution. We also present the lifting of the basic resolution method to predicate linear temporal logic. Unlike First Order Logic, clauses of predicate linear temporal logic may contain variables which are quantified existentially, because skolemization is not valid here. We use standard unification with substitution on universally quantified variables. However, if a term substituted in place of a variable contains any flexible symbols, a constant or a function is flexible if it has different translation in different states, then all occurrences of the substituted variable must refer to the same instant of time (state). Otherwise, the unification may lead to incorrect results. Resolution in predicate linear temporal logic, though very useful from a practical standpoint, is incomplete, since no predicate temporal logic with arithmetic model of time is complete

Combinatorial Flows and Their Normalisation

This paper introduces combinatorial flows that generalize combinatorial proofs such that they also include cut and substitution as methods of proof compression. We show a normalization procedure for combinatorial flows, and how syntactic proofs are translated into combinatorial flows and vice versa

Proof Checking and Logic Programming

International audienceIn a world where trusting software systems is increasingly important, formal methods and formal proof can help provide trustable foundations. Proof checking can help to reduce the size of the trusted base since we do not need to trust an entire theorem prover if we can check the proofs they produce by a trusted (and smaller) checker. Many approaches to building proof checkers require embedding within them a full programming language. In most many modern proof checkers and theorem provers, that programming language is a functional programming language, often a variant of ML. In fact, parts of ML (e.g., strong typing , abstract datatypes, and higher-order programming) were designed to make ML into a trustworthy " metalanguage " for checking proofs. While there is considerable overlap in the foundations of logic programming and proof checking (both benefit from unification, backtracking search, efficient term structures, etc), the discipline of logic programming has, in fact, played a minor role in the history of proof checking. I will argue that logic programming can have a major role in the future of this important topic

Hybrid speciation in Heliconius butterflies? A review and critique of the evidence

The evidence supporting the recent hypothesis of a homoploid hybrid origin for the butterfly species Heliconius heurippa is evaluated. Data from selective breeding experiments, mate-choice studies, and a wide variety of DNA markers are reviewed, and an alternative hypothesis for the origin of the species and its close relatives is proposed. A scenario of occasional red wing-pattern mutations in peripheral populations of Heliconius cydno with subsequent adaptive convergence towards sympatric mimicry rings involving H. melpomene and H. erato is offered as an alternative to the HHS hypothesis. Recent twists of this tale are addressed in a postscript

A proposal for broad spectrum proof certificates

International audienceRecent developments in the theory of focused proof systems provide flexible means for structuring proofs within the sequent calculus. This structuring is organized around the construction of ''macro'' level inference rules based on the ''micro'' inference rules which introduce single logical connectives. After presenting focused proof systems for first-order classical logics (one with and one without fixed points and equality) we illustrate several examples of proof certificates formats that are derived naturally from the structure of such focused proof systems. In principle, a proof certificate contains two parts: the first part describes how macro rules are defined in terms of micro rules and the second part describes a particular proof object using the macro rules. The first part, which is based on the vocabulary of focused proof systems, describes a collection of macro rules that can be used to directly present the structure of proof evidence captured by a particular class of computational logic systems. While such proof certificates can capture a wide variety of proof structures, a proof checker can remain simple since it must only understand the micro-rules and the discipline of focusing. Since proofs and proof certificates are often likely to be large, there must be some flexibility in allowing proof certificates to elide subproofs: as a result, proof checkers will necessarily be required to perform (bounded) proof search in order to reconstruct missing subproofs. Thus, proof checkers will need to do unification and restricted backtracking search

Proof Transformation with Built-in Equality Predicate

One of the main reasons why computer generated proofs are not widely accepted is often their complexity and incomprehensibility. Especially proofs of mathematical theorems with equations are normally presented in an inadequate and not intuitive way. This is even more of a problem for the presentation of inferences drawn by automated reasoning components in other AI systems. For first order logic, proof transformation procedures have been designed in order to structure proofs and state them in a formalism that is more familiar to human mathematicians. In this report we generalize these approaches, so that proofs involving equational reasoning can also be handled. To this end extended refutation graphs are introduced to represent combined resolution and paramodulation proofs. In the process of transforming these proofs into natural deduction proofs with equality, the inherent structure can also be extracted by exploiting topological properties of refutation graphs