Final Report Summary - STRUCTPROOFS (Structural Analysis of Mathematical Proofs)
Project context and objectives
The STRUCTPROOFS project has successfully contributed to a further unification of different traditions and methodologies in logic: classical proof theory on the one hand, and computational proof theory and its relation to the theory of programming languages, on the other hand. The theoretical results include a deeper understanding of the structure of formal proofs and arguments, of their computational treatment and their relation to neighbouring areas, such as the theory of formal languages. The practical applications of such results lie in the design of programming languages and in the verification of software and hardware. These areas are crucial for our society, which increasingly depends on the well-functioning of computer systems, and also in safety-critical areas such as control systems for planes, cars, etc.
The concrete mathematical problems that have been treated in the course of this project include the relationship between a proof and the Herbrand-disjunctions obtainable from it by cut-elimination. On the one hand, it turned out that, in an assymptotic setting, a surprisingly large number of Herbrand-disjunctions can be obtained. On the other hand, these disjunctions still share common structural features. Through the work on characterising these properties, a strong relationship between proof theory and formal language theory was uncovered which has opened many new avenues of investigation, such as algorithmic cut-introduction, which was partly started in this project.
As envisaged from the beginning, the techniques and methods developed in the host group were useful for carrying out this analysis of proofs. Some of the project objectives, like fast cut-elimination, could be treated effectively, while for others, such as the complexity of interpolants, only partial results could be obtained. In any case, the foundational results in structural proof theory that have been obtained are important and a useful contribution to the further mathematical understanding of logical reasoning.
Project results
A series of workshops was organised to further increase the collaboration of different communities (such as the Austrian and French) in this area. This project has significantly contributed to strengthening the relationship between these two groups and has led to a joint FWF/ANR-project which has been under way since January 2011 and will last three years. The results from STRUCTPROOFS have been disseminated via presentations at various national and international conferences and workshops as well as through publication in conference proceedings and journals.
The STRUCTPROOFS project has successfully contributed to a further unification of different traditions and methodologies in logic: classical proof theory on the one hand, and computational proof theory and its relation to the theory of programming languages, on the other hand. The theoretical results include a deeper understanding of the structure of formal proofs and arguments, of their computational treatment and their relation to neighbouring areas, such as the theory of formal languages. The practical applications of such results lie in the design of programming languages and in the verification of software and hardware. These areas are crucial for our society, which increasingly depends on the well-functioning of computer systems, and also in safety-critical areas such as control systems for planes, cars, etc.
The concrete mathematical problems that have been treated in the course of this project include the relationship between a proof and the Herbrand-disjunctions obtainable from it by cut-elimination. On the one hand, it turned out that, in an assymptotic setting, a surprisingly large number of Herbrand-disjunctions can be obtained. On the other hand, these disjunctions still share common structural features. Through the work on characterising these properties, a strong relationship between proof theory and formal language theory was uncovered which has opened many new avenues of investigation, such as algorithmic cut-introduction, which was partly started in this project.
As envisaged from the beginning, the techniques and methods developed in the host group were useful for carrying out this analysis of proofs. Some of the project objectives, like fast cut-elimination, could be treated effectively, while for others, such as the complexity of interpolants, only partial results could be obtained. In any case, the foundational results in structural proof theory that have been obtained are important and a useful contribution to the further mathematical understanding of logical reasoning.
Project results
A series of workshops was organised to further increase the collaboration of different communities (such as the Austrian and French) in this area. This project has significantly contributed to strengthening the relationship between these two groups and has led to a joint FWF/ANR-project which has been under way since January 2011 and will last three years. The results from STRUCTPROOFS have been disseminated via presentations at various national and international conferences and workshops as well as through publication in conference proceedings and journals.