142 research outputs found
Towards the Integration of an Intuitionistic First-Order Prover into Coq
An efficient intuitionistic first-order prover integrated into Coq is useful
to replay proofs found by external automated theorem provers. We propose a
two-phase approach: An intuitionistic prover generates a certificate based on
the matrix characterization of intuitionistic first-order logic; the
certificate is then translated into a sequent-style proof.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
leanCoP: lean connection-based theorem proving
AbstractThe Prolog programimplements a theorem prover for classical first-order (clausal) logic which is based on the connection calculus. It is sound and complete (provided that an arbitrarily large I is iteratively given), and demonstrates a comparatively strong performance
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
The Pocket Reasoner -- Automatic Reasoning on Small Devices
Automated reasoning in classical first-order logic is a core research field in Artificial Intelligence. Most of the fully automated reasoning tools are large and complex systems implementing proof search methods that have significant memory requirements. This paper presents an automated reasoning tool implemented on an iPod Nano. It is based on leanCoP, a very compact Prolog implementation of the connection calculus, which operates on the structure of the given formula without generating new subformula instances. Hence, the memory requirements are significantly lower, allowing leanCoP to run on devices with only little (random-access) memory. The paper presents details of the proof search calculus, its implementation, and a practical evaluation of the presented reasoning tool
A tableau-like proof procedure for normal modal logics
AbstractIn this paper a new proof procedure for some propositional and first-order normal modal logics is given. It combines a tableau-like approach and a resolution-like inference. Completeness and decidability for some propositional logics are proved. An extension for the first-order case is presented
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
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