A syntactic soundness proof for free-variable tableaux with on-the-fly Skolemization

We prove the syntactic soundness of classical tableaux with free variables and on-the-fly Skolemization. Soundness proofs are usually built from semantic arguments, and this is to our knowledge, the first proof that appeals to syntactic means. We actually prove the soundness property with respect to cut-free sequent calculus. This requires great care because of the additional liberty in freshness checking allowed by the use of Skolem terms. In contrast to semantic soundness, we gain the possibility to state a cut elimination theorem for sequent calculus, under the proviso that completeness of the method holds. We believe that such techniques can be applied to tableaux in other logics as well

A Tableau Calculus for Pronoun Resolution

We present a tableau calculus for reasoning in fragments of natural language. We focus on the problem of pronoun resolution and the way in which it complicates automated theorem proving for natural language processing. A method for explicitly manipulating contextual information during deduction is proposed, where pronouns are resolved against this context during deduction. As a result, pronoun resolution and deduction can be interleaved in such a way that pronouns are only resolved if this is licensed by a deduction rule; this helps us to avoid the combinatorial complexity of total pronoun disambiguation.Comment: 16 page

HyperS tableaux - heuristic hyper tableaux

Several syntactic methods have been constructed to automate theorem proving in first-order logic. The positive (negative) hyper-resolution and the clause tableaux were combined in a single calculus called hyper tableaux in [1]. In this paper we propose a new calculus called hyperS tableaux which overcomes substantial drawbacks of hyper tableaux. Contrast to hyper tableaux, hyperS tableaux are entirely automated and heuristic. We prove the soundness and the completeness of hyperS tableaux. HyperS tableaux are applied in the theorem prover Sofia, which additionally provides useful tools for clause set generation (based on justificational tableaux) and for tableau simplification (based on redundancy), and advantageous heuristics as well. An additional feature is the support of the so-called parametrized theorems, which makes the prover able to give compound answers

Enumerating Query Plans via Conditional Tableau Interpolation

Database query optimization studies the problem of finding equivalent and efficient query execution plans for user queries under schema constraints. Logic-based approaches to query optimization leverage automated theorem proving and Craig interpolation to enumerate query plans that are correct and performance-optimal. In this thesis, we investigate and improve one of the state-of-the-art logic-based query optimizers – the Interpolation Test Bed (ITB). We begin by formally capturing the physical data independence framework and query optimization problem with first-order logic. Then, we give a gentle introduction to the classical results from logic that form the basis of logic-based query optimizers. We re-establish the correctness of ITB’s conditional tableau interpolation mechanism by reduction to free-variable tableau interpolation. To facilitate the reduction proof, we introduce interpolation rules for the free-variable tableau and prove the correctness of interpolation. Then we show the correctness of conditional tableau interpolation by reduction. We investigate a limitation of ITB’s forward chaining design, which causes missing optimal plans. To address this limitation, we propose a rewriting procedure inspired by Magic Set Transformation (MST), to extend the plan space for the current ITB system. We show that the propose rewriting procedure effectively generates the missing query plans, which are otherwise not found, while accommodating the existing forward chaining design

First-Order Models for Configuration Analysis

Our world teems with networked devices. Their configuration exerts an ever-expanding influence on our daily lives. Yet correctly configuring systems, networks, and access-control policies is notoriously difficult, even for trained professionals. Automated static analysis techniques provide a way to both verify a configuration\u27s correctness and explore its implications. One such approach is scenario-finding: showing concrete scenarios that illustrate potential (mis-)behavior. Scenarios even have a benefit to users without technical expertise, as concrete examples can both trigger and improve users\u27 intuition about their system. This thesis describes a concerted research effort toward improving scenario-finding tools for configuration analysis. We developed Margrave, a scenario-finding tool with special features designed for security policies and configurations. Margrave is not tied to any one specific policy language; rather, it provides an intermediate input language as expressive as first-order logic. This flexibility allows Margrave to reason about many different types of policy. We show Margrave in action on Cisco IOS, a common language for configuring firewalls, demonstrating that scenario-finding with Margrave is useful for debugging and validating real-world configurations. This thesis also presents a theorem showing that, for a restricted subclass of first-order logic, if a sentence is satisfiable then there must exist a satisfying scenario no larger than a computable bound. For such sentences scenario-finding is complete: one can be certain that no scenarios are missed by the analysis, provided that one checks up to the computed bound. We demonstrate that many common configurations fall into this subclass and give algorithmic tests for both sentence membership and counting. We have implemented both in Margrave. Aluminum is a tool that eliminates superfluous information in scenarios and allows users\u27 goals to guide which scenarios are displayed. We quantitatively show that our methods of scenario-reduction and exploration are effective and quite efficient in practice. Our work on Aluminum is making its way into other scenario-finding tools. Finally, we describe FlowLog, a language for network programming that we created with analysis in mind. We show that FlowLog can express many common network programs, yet demonstrate that automated analysis and bug-finding for FlowLog are both feasible as well as complete

A mechanization of sorted higher-order logic based on the resolution principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic SUM HOL suitable for automatic theorem proving applications. SUM HOL is based on a sorted Lambda-calculus SUM A->, which is obtained by extending Church\u27;s simply typed Lambda-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of SUM HOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. The main technical difficulty of unification in ! is that the analysis of general bindings is much more involved than in the unsorted case, since in the presence of term declarations well-sortedness is not a structural property. This difficulty is overcome by a structure theorem that links the structure of a formula to the structure of its sorting derivation. We develop two notions of set-theoretic semantics for SUM HOL. General SUM-models are a direct generalization of Henkin\u27;s general models to the sorted setting. Since no known machine-oriented calculus can adequately mechanize full extensionality, we generalize general SUM-models further to SUM-model structures, which allow full extensionality to fail. The notions of SUM-model structures and general SUM-models allow us to prove model existence theorems for them. These model-theoretic variants of Andrews unifying principle for type theory\u27; can be used as a powerful tool in completeness proofs of higher-order calculi. Finally, we use our pre-unification algorithms as a central inference procedure for a sorted higherorder resolution calculus in the spirit of Huet\u27;s Constrained Resolution. This calculus is proven sound and complete with respect to our semantics. It differs from Huet\u27;s calculus by allowing early unification strategies and using variable dependencies. For the completeness proof we make use of our model existence theorem, and prove a strong lifting lemma