Representing scope in intuitionistic deductions

AbstractIntuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever — rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions

Representing Scope in Intuitionistic Deductions

Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semantics-based translation methods for modal deduction. This calculus is simple and is justified by direct proof-theoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoever—rules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how λ-terms are to be extracted from its deductions

Towards a Proof Theory of G\"odel Modal Logics

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments

Two Categories of Refutation Decision Procedures for Classical and Intuitionistic Propositional Logic

An automatic theorem prover is a computer program that proves theorems without the assistance of a human being. Theorem proving is an important basic tool in proving theorems in mathematics, establishing the correctness of computer programs, proving the correctness of communication protocols, and verifying integrated circuit designs. This dissertation introduces two new categories of theorem provers, one for classical propositional logic and another for intuitionistic propositional logic. For each logic a container property and generalized algorithm are introduced. Many methods have been developed over the years to prove theorems in propositional logic. This dissertation describes and presents example proofs for five of these methods: natural deduction, Kripke tableau, analytic tableau, matrix, and resolution. Each of these methods uses refutation to prove a theorem. In refutation, the proposed theorem is assumed to be false. The theorem prover is successful, only if the analysis of this assumption leads to a contradiction. Each of these methods, except resolution, share a common algorithm. To prove this, the container is introduced. A data structure used by a method is a container, if it meets a set of properties. A generalized algorithm that proves theorems is introduced. Since each step in this algorithm uses only operations that are provided by the container. The steps it performs can be translated to any method that can be described using a container. This allows the data structures representing a partial proof in one method, to be transformed into the data structures representing the “same ” proof in another method. This can be very beneficial in a situation where another method would be more efficient in advancing the proof. In addition to being able to switch between methods, an heuristic for one method can be examined to see if it can be applied to the other methods. This development is repeated for intuitionistic logic. Each of these methods, except resolution, is modified to prove theorems in intuitionistic logic. An intuitionistic container is presented. Each one of the intuitionistic methods is proven to have the properties of the intuitionistic container. Lastly, a generalized algorithm using the intuitionistic container is presented. This algorithm proves theorems in intuitionistic logic. Examples showing successful and unsuccessful proof attempts are presented

Frege systems for quantified Boolean logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 279611 and under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 61507. Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University Rome.Peer ReviewedPostprint (published version