## Deep Inference and the Calculus of Structures Implementations

Ozan Kahramanogullari, Pierre-Etienne Moreau and Antoine Reilles are implementing calculus-of-structures deductive systems in Maude and in Tom. Ozan managed to achieve efficiency without sacrificing proof theoretic cleanliness, and he is obtaining results of independent theoretical interest. There are two slides presentations:

Max Schäfer has built a graphical proof editor in Java, called GraPE, for the Maude modules written by Ozan Kahramanogullari; this means that one can interactively build and find proofs in several deep-inference systems.

The following papers are available.

The calculus of structures is a recently developed proof theoretical formalism that extends one-sided sequent calculus, with the gain of interesting proof theoretical properties. In contrast to sequent calculus, it does not rely on the notion of main connective and, like in term rewriting, permits the application of inference rules anywhere deep inside a formula. In this paper, exploiting this resemblance, we present a procedure turning derivations in the calculus of structures in four steps into rewritings in a term rewriting system modulo equality.

Pdf    13 June 2004
Technical Report WV-04-03, Technische Universität Dresden

System BV is an extension of multiplicative linear logic with a noncommutative self-dual operator. We first map derivations of system BV of the calculus of structures to rewritings in a term rewriting system modulo equality, and then express this rewriting system as a Maude system module. This results in an automated proof search implementation for this system, and provides a recipe for implementing existing calculus of structures systems for other logics. Our result is interesting from the view of applications, specially, where sequentiality is essential, e.g., planning and natural language processing. In particular, we argue that we can express plans as logical formulae by using the sequential operator of BV and reason on them in a purely logical way.

Pdf    13 June 2004
Proceedings of ESSLLI 2004-Student Session

System BV is an extension of multiplicative linear logic with a non-commutative self-dual operator. In this paper, we present systems equivalent to system BV where equalities for unit are oriented from left to right and new structural rules are introduced to preserve completeness. While the first system allows units to appear in the structures, the second system makes it possible to completely remove the units from the language of BV by proving the normal forms of the structures that are provable in BV. The resulting systems provide better performance in automated proof search by disabling redundant applications of inference rules due to the unit. As evidence, we provide a comparison of the performance of these systems in a Maude implementation.

Pdf    4 August 2004
ISCIS 2004, LNCS 3280, pp. 986–995

Deep inference is a proof theoretical methodology that generalizes the traditional notion of inference in the sequent calculus: In contrast to the sequent calculus, the deductive systems with deep inference do not rely on the notion of main connective, and permit the application of the inference rules at any depth inside logical expressions, in a way which resembles the application of term rewriting rules. Deep inference provides a richer combinatoric analysis of proofs for different logics. In particular, construction of exponentially shorter proofs becomes possible. In this paper, aiming at the development of computation as proof search tools, we propose the Maude language as a means for designing and implementing different deep inference deductive systems and proof strategies that work on these systems. We demonstrate these ideas on classical logic and argue that these ideas can be analogously carried to other deductive systems for other logics.

Pdf    16 September 2007
Proceedings of RULE '07, to appear on Electronic Notes in Theoretical Computer Science

The calculus of structures is a proof theoretical formalism which generalizes sequent calculus with the feature of deep inference: in contrast to sequent calculus, the calculus of structures does not rely on the notion of main connective and, like in term rewriting, it permits the application of inference rules at any depth inside a formula. Tom is a pattern matching processor that integrates term rewriting facilities into imperative languages. In this paper, relying on the correspondence between the systems in the calculus of structures and term rewriting systems, we present an implementation of system BV of the calculus of structures in Java by exploiting the term rewriting features of Tom. This way, by means of the expressive power due to Java, it becomes possible to implement different search strategies. Since the systems in the calculus of structures follow a common scheme, we argue that our implementation can be generalized to other systems in the calculus of structures for classical logic, modal logics, and different fragments of linear logic.

Pdf    23 April 2005
Proceedings of Structures and Deduction '05, pp. 158–172

• Canonical Abstract Syntax Trees

This paper presents GOM, a language for describing abstract syntax trees and generating a Java implementation for those trees. GOM includes features allowing to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a rewrite system. This explicitly guarantees that the client program only manipulates normal forms for this rewrite system, a feature which is only implicitly used in many implementations.

Pdf    28 March 2006
Proceedings of 6th International Workshop on Rewriting Logic and Its Applications, Electronic Notes in Theoretical Computer Science 176, 2007, pp. 165–179

18.9.2007    Alessio Guglielmi    email