Stephan Schmitt

MetaPRL – A Modular Logical Environment

MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCF-style tactic-based proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an open-source license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.

JProver: Integrating Connection-Based Theorem Proving into Interactive Proof Assistants

JProver is a first-order intuitionistic theorem prover that creates sequent-style proof objects and can serve as a proof engine in interactive proof assistants with expressive constructive logics. This paper gives a brief overview of JProvers proof technique, the generation of proof objects, and its integration into the Nuprl proof development system.

On Transforming Intuitionistic Matrix Proofs into Standard-Sequent Proofs

We present a procedure transforming intuitionistic matrix proofs into proofs within the intuitionistic standard sequent calculus. The transformation is based on L. Wallens proof justifying his matrix characterization for the validity of intuitionistic formulae. Since this proof makes use of Fittings non-standard sequent calculus our procedure consists of two steps. First a non-standard sequent proof will be extracted from a given matrix proof. Secondly we transform each non-standard proof into a standard proof in a structure preserving way. To simplify the latter step we introduce an extended standard calculus which is shown to be sound and complete.

Converting Non-Classical Matrix Proofs into Sequent-Style Systems

We present a uniform algorithm fot transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fittings prefixed tableaus we first develop a procedure for extracting a prefixed sequent proof from a given matrix proof. By considering the additional restrictions on the order of rule applications we then extend this procedure into an algorithm which generates a conventional sequent proof.

A Uniform Procedure for Converting Matrix Proofs into Sequent-Style Systems

We present a uniform algorithm for transforming machine-found matrix proofs in classical, constructive, and modal logics into sequent proofs. It is based on unified representations of matrix characterizations, of sequent calculi, and of prefixed sequent systems for various logics. The peculiarities of an individual logic are described by certain parameters of these representations, which are summarized in tables to be consulted by the conversion algorithm.

Guiding Program Development Systems by a Connection Based Proof Strategy

We present an automated proof method for constructive logic based on Wallens matrix characterization for intuitionistic validity. The proof search strategy extends Bibels connection method for classical predicate logic. It generates a matrix proof which will then be transformed into a proof within a standard sequent calculus. Thus we can use an efficient proof method to guide the development of constructive proofs in interactive proof/program development systems.

On intuitionistic proof transformations, their complexity, and application to constructive program synthesis

We present a translation of intuitionistic sequent proofs from a multi-succedent calculus LJ mc into a single-succedent calculus LJ. The former gives a basis for automated proof search whereas the latter is better suited for proof presentation and program construction from proofs in a system for constructive program synthesis. Well-known translations from the literature have a severe drawback; they use cuts in order to establish the transformation with the undesired consequence that the resulting program term is not intuitive. We establish a transformation based on permutation of inferences and discuss the relevant properties with respect to proof complexity and program terms. As an important result we show that LJ cannot polynomially simulate LJ mc (both without the cut rule), even in the prepositional fragment.

Intuitionistic Proof Transformations and Their Application to Constructive Program Synthesis

We present a translation of intuitionistic sequent proofs from a multi-succedent calculus LTmc into a single-succedent calculus LT. The former gives a basis for automated proof search whereas the latter is better suited for proof presentation and program construction from proofs in a system for constructive program synthesis. Well-known translations from the literature have a severe drawback; they use cuts in order to establish the transformation with the undesired consequence that the resulting program term is not intuitive. We establish a transformation based on permutation of inferences and discuss the relevant properties with respect to proof complexity and program terms.

Deleting Redundancy in Proof Reconstruction

We present a framework for eliminating redundancies during the reconstruction of sequent proofs from matrix proofs. We show that search-free proof reconstruction requires knowledge from the proof search process. We relate different levels of proof knowledge to reconstruction knowledge and analyze which redundancies can be deleted by using such knowledge. Our framework is uniformly applicable to classical logic and all non-classical logics which have a matrix characterization of validity and enables us to build adequate conversion procedures for each logic.

Matrix-based Constructive Theorem Proving

Formal methods for program verification, optimization, and synthesis rely on complex mathematical proofs, which often involve reasoning about computations. Because of that there is no single automated proof procedure that can handle all the reasoning problems occurring during a program derivation or verification. Instead, one usually relies on proof assistants like NuPRL (Constable et al., 1986), Coq (Dowek and et. al, 1991), Alf (Altenkirch et al., 1994) etc., which are based on very expressive logical calculi and support interactive and tactic controlled proof and program development. Proof assistants, however, suffer from a very low degree of automation, since all their inferences must eventually be based on sequent or natural deduction rules. Even proof parts that rely entirely on predicate logic can seldomly be found automatically, as there are no complete proof search procedures embedded into these systems. It is therefore desirable to extend the reasoning power of proof assistants by integrating well-understood techniques from automated theorem proving.