We present a proof method for intuitionistic logic based on Wallen’s matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof search the computed first-order and intuitionistic substitutions are used to simultaneously construct a sequent proof which is more human oriented than the matrix proof. This allows to use our method within interactive proof environments. Furthermore we can consider local substitutions instead of global ones and treat substitutions occurring in different branches of the sequent proof independently. This reduces the number of extra copies of formulae to be considered.

We present an automated proof method for constructive logic based on Wallen’s matrix characterization for intuitionistic validity. The proof search strategy extends Bibel’s 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.

Abstract: We present a uniform procedure for proof search in classical logic, intuitionistic logic, various modal logics, and fragments of linear logic. It is based on matrix characterizations of validity in these logics and extends Bibel’s connection method, originally developed for classical logic, accordingly. Besides combining a variety of different logics it can also be used to guide the development of proofs in interactive proof assistants and shows how to integrate automated and interactive theorem proving. 1

Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting’s 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. Our algorithm is based on unified representations of matrix characterizations for various logics as well as of prefixed and usual sequent calculi. The peculiarities of a logic are encoded by certain parameters which are summarized in tables to be consulted by the algorithm. 1

We present an approach to a coherent program synthesis system which integrates a variety of interactively controlled and automated techniques from theorem proving and algorithm design at different levels of abstraction. Besides providing an overall view we summarize the individual research results achieved in the course of this development.

Abstract. We present a matrix characterization of logical validity in the multiplicative fragment of linear logic. On this basis we develop a matrix-based proof search procedure for this fragment and a procedure which translates the machine-found proofs back into the usual sequent calculus for linear logic. Both procedures are straightforward extensions of methods which originally were developed for a uniform treatment of classical, intuitionistic and modal logics. They can be extended to further fragments of linear logic once a matrix characterization has been found. 1

Linear logic (LL) is the logical foundation of some type-theoretic languages and also of environments for specification and theorem proving. In this paper, we analyse the relationships between the proof net notion of LL and the connection notion used for efficient proof-search in different logics. Aiming at using proof nets as a tool for automated deduction in linear logic, we define a connection-based characterization of provability in Multiplicative Linear Logic (MLL). We show that an algorithm for proof net construction can be seen as a proof-search connection method. This central result is illustrated with a specific algorithm that is able to construct, for a provable MLL sequent, a set of connections, a proof net and a sequent proof. From these results we expect to extend to other LL fragments, we analyse what happens with the additive connectives of LL by tackling the additive fragment in a similar way.

Most automated theorem provers suffer from the problem that the resulting proofs are difficult to understand even for experienced mathematicians. An effective communication between the system and its users, however, is crucial for many applications, such as in a mathematical assistant system. Therefore, efforts have been made to transform machine generated proofs (e.g. resolution proofs) into natural deduction (ND) proofs. The state-of-the-art procedure of proof transformation follows basically its completeness proof: the premises and the conclusion are decomposed into unit literals, then the theorem is derived by multiple levels of proofs by contradiction. Indeterminism is introduced by heuristics that aim at the production of more elegant results. This indeterministic character entails not only a complex search, but also leads to unpredictable results. In this paper we first study resolution proofs in terms of meaningful operations employed by human mathematicians, and thereby esta...

The observation that humans often appear to use a more sophisticated logic than the classical one has sometimes served as an argument against the fundamental role of Automated Deduction in Intellectics. In response to this challenge, a variety of non-classical logics have been developed especially over the last decade, in order to provide adequate formalisms that cover different aspects of commonsense reasoning. We argue that both classical logic and non-classical derivatives, though different in the details, are based on common grounds. This insight gives rise to the question whether and how calculi originally designed for classical deduction can be adjusted to the various kinds of non-classical logic. We illustrate such adjusting modifications in terms of the connection method. We elaborate two fundamental concepts, namely the one of a connection, which represents the application of the modus ponens rule, and the one of a path, which represents the current assumptions made in the co...

Since almost 30 years software production has to face two major problems: the cost of non-standard software, caused by long development times and the constant need for maintenance, and a lack of confidence in the reliability of software. Recent accidents like the crash of KAL’s 747 in August 1997 or the