Section: Scientific Foundations

Deep Inference and Categorical Axiomatizations

Deep inference [44] , [45] is a novel methodology for presenting deductive systems. Unlike traditional formalisms like the sequent calculus, it allows rewriting of formulas deep inside arbitrary contexts. The new freedom for designing inference rules creates a richer proof theory. For example, for systems using deep inference, we have a greater variety of normal forms for proofs than in sequent calculus or natural deduction systems. Another advantage of deep inference systems is the close relationship to categorical proof theory. Due to the deep inference design one can directly read off the morphism from the derivations. There is no need for a counter-intuitive translation.

One reason for using categories in proof theory is to give a precise algebraic meaning to the identity of proofs: two proofs are the same if and only if they give rise to the same morphism in the category. Finding the right axioms for the identity of proofs for classical propositional logic has for long been thought to be impossible, due to “Joyal's Paradox”. For the same reasons, it was believed for a long time that it it not possible to have proof nets for classical logic. Nonetheless, Lutz Straßburger and François Lamarche provided proof nets for classical logic in [4] , and analyzed the category theory behind them in [48] . In [10] and [63] , one can find a deeper analysis of the category theoretical axioms for proof identification in classical logic. Particular focus is on the so-called medial rule which plays a central role in the deep inference deductive system for classical logic.

The following research problems are investigated by members of the Parsifal team:

• Find deep inference system for richer logics. This is necessary for making the proof theoretic results of deep inference accessible to applications as they are described in the previous sections of this report.

• Investigate the possibility of focusing proofs in deep inference. As described before, focusing is a way to reduce the non-determinism in proof search. However, it is well investigated only for the sequent calculus. In order to apply deep inference in proof search, we need to develop a theory of focusing for deep inference.

• Use the results on deep inference to find new axiomatic description of categories of proofs for various logics. So far, this is well understood only for linear and intuitionistic logics. Already for classical logic there is no common accepted notion of proof category. How logics like LINC can be given a categorical axiomatisation is completely open.