35 research outputs found
Inducing syntactic cut-elimination for indexed nested sequents
The key to the proof-theoretic study of a logic is a proof calculus with a
subformula property. Many different proof formalisms have been introduced (e.g.
sequent, nested sequent, labelled sequent formalisms) in order to provide such
calculi for the many logics of interest. The nested sequent formalism was
recently generalised to indexed nested sequents in order to yield proof calculi
with the subformula property for extensions of the modal logic K by
(Lemmon-Scott) Geach axioms. The proofs of completeness and cut-elimination
therein were semantic and intricate. Here we show that derivations in the
labelled sequent formalism whose sequents are `almost treelike' correspond
exactly to indexed nested sequents. This correspondence is exploited to induce
syntactic proofs for indexed nested sequent calculi making use of the elegant
proofs that exist for the labelled sequent calculi. A larger goal of this work
is to demonstrate how specialising existing proof-theoretic transformations
alleviate the need for independent proofs in each formalism. Such coercion can
also be used to induce new cutfree calculi. We employ this to present the first
indexed nested sequent calculi for intermediate logics.Comment: This is an extended version of the conference paper [20
From Display to Labelled Proofs for Tense Logics
We introduce an effective translation from proofs in the display calculus to proofs in the labelled calculus in the context of tense logics. We identify the labelled calculus proofs in the image of this translation as those built from labelled sequents whose underlying directed graph possesses certain properties. For the basic normal tense logic Kt, the image is shown to be the set of all proofs in the labelled calculus G3Kt
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 Sandqvistâs 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
Logics Based on Linear Orders of Contaminating Values
A wide family of many-valued logicsâfor instance, those based on the weak Kleene algebraâincludes a non-classical truth-value that is âcontaminatingâ in the sense that whenever the value is assigned to a formula Ïâ , any complex formula in which Ï appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper
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