Unification in Modal Theorem Proving

Modal formulas can be proved by translating them into a three-typed logic and then using unification and resolution, with axioms describing properties of the reachability relation among possible worlds. In this paper, we improve on the algorithms in [1], showing that strong skolemisation and occurrence checks are not needed for proving theorems of Q, T, Q4, and S4. We also extend the \u27path logic\u27 approach to S5, give the appropriate unification algorithm, and prove its correctness

Subsumption in Modal Logic

Subsumption has long been known as a technique to detect redundant clauses in the search space of automated deduction systems for classical first order logic. In recent years several automated deduction methods for non-classical modal logics have been developed. This thesis explores, how subsumption can be made to work in the context of these modal logic deduction methods. Many modern modal logic deduction methods follow an indirect approach. They translate the modal sentences into some other target language, and then determine whether there exists a proof in that language, rather than doing deduction in the modal language itself. Consequently, subsumption then needs to focus on the target language, in which the actual proof is done. World Path Logic (WPL) is introduced as a possible target language. Deduction in WPL works very much like in ordinary logic, the only significant difference is the need for a special purpose unification, which unifies world paths under an equational theory (E-unification). Relating WPL to a well understood first order logic of restricted quantification, the properties of WPL, that make deduction work, are examined. The obtained theoretical results are the basis for the following treatment of subsumption in WPL. Subsumption is analyzed treating a clause as a scheme standing for the set of its ground instances. Although the notion of ground instances in WPL is different from ordinary logic, it turns out that - just like in ordinary logic - a clause Cl subsumes another clause C2, if there exists a substitution 6 such that C10 £ C2. Once the special purpose unification has been implemented into a theorem prover to allow for deduction in WPL, existing subsumption tests then work without any further changes

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

Proof Complexity of Modal Resolution Systems

In this thesis we initiate the study of the proof complexity of modal resolution systems. To our knowledge there is no previous work on the proof complexity of such systems. This is in sharp contrast to the situation for propositional logic where resolution is the most studied proof system, in part due to its close links with satisfiability solving. We focus primarily on the proof complexity of two recently proposed modal resolution systems of Nalon, Hustadt and Dixon, one of which forms the basis of an existing modal theorem prover. We begin by showing that not only are these two proof systems equivalent in terms of their proof complexity, they are also equivalent to a number of natural refinements. We further compare the proof complexity of these systems with an older, more complicated modal resolution system of Enjalbert and Farinas del Cerro, showing that this older system p-simulates the more streamlined calculi. We then investigate lower bound techniques for modal resolution. Here we see that whilst some propositional lower bound techniques (i.e. feasible interpolation) can be lifted to the modal setting with only minor modifications, other propositional techniques (i.e. size-width) fail completely. We further develop a new lower bound technique for modal resolution using Prover-Delayer games. This technique can be used to establish "genuine" modal lower bounds (i.e lower bounds on the number of modal inferences) for the size of tree-like modal resolution proofs. We apply this technique to a new family of modal formulas, called the modal pigeonhole principle to demonstrate that these formulas require exponential size modal resolution proofs. Finally we compare the proof complexity of tree-like modal resolution systems with that of modal Frege systems, using our modal pigeonhole principle to obtain a "genuinely" modal separation between them

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logics

In this thesis we develop efficient methods for automated proof search within an important class of mathematical logics. The logics considered are the varying, cumulative and constant domain versions of the first-order modal logics K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of these non-classical logics is commonplace within Computing Science and Artificial Intelligence in applications in which efficient machine assisted proof search is essential. Traditional techniques for the design of efficient proof methods for classical logic prove to be of limited use in this context due to their dependence on properties of classical logic not shared by most of the logics under consideration. One major contribution of this thesis is to reformulate and abstract some of these classical techniques to facilitate their application to a wider class of mathematical logics. We begin with Bibel's Connection Calculus: a matrix proof method for classical logic comparable in efficiency with most machine orientated proof methods for that logic. We reformulate this method to support its decomposition into a collection of individual techniques for improving the efficiency of proof search within a standard cut-free sequent calculus for classical logic. Each technique is presented as a means of alleviating a particular form of redundancy manifest within sequent-based proof search. One important result that arises from this anaylsis is an appreciation of the role of unification as a tool for removing certain proof-theoretic complexities of specific sequent rules; in the case of classical logic: the interaction of the quantifier rules. All of the non-classical logics under consideration admit complete sequent calculi. We anaylse the search spaces induced by these sequent proof systems and apply the techniques identified previously to remove specific redundancies found therein. Significantly, our proof-theoretic analysis of the role of unification renders it useful even within the propositional fragments of modal and intuitionistic logic