PhD Thesis Nondeterminism and language design in deep inference


This thesis studies the design of deep inference deductive systems. In the systems with deep inference, in contrast to traditional proof-theoretic systems, inference rules can be applied at any depth inside logical expressions. Deep applicability of inference rules provides a rich combinatorial analysis of proofs. Deep inference also makes it possible to design deductive systems that are tailored for computer science applications and otherwise provably not expressible. By applying the inference rules deeply, logical expressions can be manipulated starting from their sub-expressions. This way, we can simulate analytic proofs in traditional deductive formalisms. Furthermore, we can also construct much shorter analytic proofs than in these other formalisms. However, deep applicability of inference rules causes much greater nondeterminism in proof construction. This thesis attacks the problem of dealing with nondeterminism in proof search while preserving the shorter proofs that are available thanks to deep inference. By redesigning the deep inference deductive systems, some redundant applications of the inference rules are prevented. By introducing a new technique which reduces nondeterminism, it becomes possible to obtain a more immediate access to shorter proofs, without breaking certain proof theoretical properties such as cut-elimination. Di fferent implementations presented in this thesis allow to perform experiments on the techniques that we developed and observe the performance improvements. Within a computation-as-proof-search perspective, we use deep inference deductive systems to develop a common proof-theoretic language to the two fi elds of planning and concurrency.

Paper Details


O. Kahramanogullari