Giselle Reis

Algorithmic introduction of quantified cuts

Abstract We describe a method for inverting Gentzens cut-elimination in classical first-order logic. Our algorithm is based on first computing a compressed representation of the terms present in the cut-free proof and then cut-formulas that realize such a compression. Finally, a proof using these cut-formulas is constructed. Concerning asymptotic complexity, this method allows an exponential compression of quantifier complexity (the number of quantifier-inferences) of proofs.

Introducing Quantified Cuts in Logic with Equality

Cut-introduction is a technique for structuring and compressing formal proofs. In this paper we generalize our cut-introduction method for the introduction of quantified lemmas of the form ∀ x.A (for quantifier-free A) to a method generating lemmas of the form ∀ x 1 … ∀ x n .A. Moreover, we extend the original method to predicate logic with equality. The new method was implemented and applied to the TSTP proof database. It is shown that the extension of the method to handle equality and quantifier-blocks leads to a substantial improvement of the old algorithm.

An extended framework for specifying and reasoning about proof systems

It has been shown that linear logic can be successfully used as a framework for both specifying proof systems for a number of logics, as well as proving fundamental properties about the specified systems. This paper shows how to extend the framework with subexponentials in order to declaratively encode a wider range of proof systems, including a number of non-trivial proof systems such as multi-conclusion intuitionistic logic, classical modal logic S4, intuitionistic Lax logic, and Negri’s labelled proof systems for different modal logics. Moreover, we propose methods for checking whether an encoded proof system has important properties, such as if it admits cut-elimination, the completeness of atomic identity rules, and the invertibility of its inference rules. Finally, we present a tool implementing some of these specification/verification methods.

System Description: GAPT 2.0

GAPT General Architecture for Proof Theory is a proof theory framework containing data structures, algorithms, parsers and other components common in proof theory and automated deduction. In contrast to automated and interactive theorem provers whose focus is the construction of proofs, GAPT concentrates on the transformation and further processing of proofs. In this paper, we describe the current 2.0 release of GAPT.

An Adequate Compositional Encoding of Bigraph Structure in Linear Logic with Subexponentials

In linear logic, formulas can be split into two sets: classical those that can be used as many times as necessary or linear those that are consumed and no longer available after being used. Subexponentials generalize this notion by allowing the formulas to be split into many sets, each of which can then be specified to be classical or linear. This flexibility increases its expressiveness: we already have adequate encodings of a number of other proof systems, and for computational models such as concurrent constraint programming, in linear logic with subexponentials [Figure not available: see fulltext.]. Bigraphs were proposed by Milner in 2001 as a model for ubiquitous computing, subsuming models of computation such as CCS and the

Towards CERes in intuitionistic logic

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Ceres in intuitionistic logic

-calculus and capable of modeling connectivity and locality at the same time. In this work we present an encoding of the bigraph structure in [Figure not available: see fulltext.], thus giving an indication of the expressive power of this logic, and at the same time providing a framework for reasoning and operating on bigraphs. Our encoding is adequate and therefore the operations of composition and juxtaposition can be performed on the logical level. Moreover, all the proof-theoretical tools of [Figure not available: see fulltext.] become available for querying and proving properties of bigraph structures.

Checking Proof Transformations with ASP.

The proof of many foundational results in structural proof theory, such as the admissibility of the cut rule and the completeness of the focusing discipline, rely on permutation lemmas. It is often a tedious and error prone task to prove such lemmas as they involve many cases. This paper describes the tool Quati which is an automated tool capable of proving a wide range of inference rule permutations for a great number of proof systems. Given a proof system specification in the form of a theory in linear logic with subexponentials, Quati outputs in Open image in new window the permutation transformations for which it was able to prove correctness and also the possible derivations for which it was not able to do so. As illustrated in this paper, Quati’s output is very similar to proof derivation figures one would normally find in a proof theory book.