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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - MiLC (Monotonicity in Logic and Complexity)

Teaser

Summary

This project studied logical characterisations of monotone complexity classes. Monotone computation is all around us, and more or less corresponds to \'computing without flipping bits\'. Such models are used for a wide variety of mathematical and computational problems, such as sorting a list or detecting cliques in graphs. Naturally it is important to understand the resource usage, or complexity, of programs in these models. To this end, we developed novel approaches using proof theory and recursion theory, to establish comprehensive logical frameworks for reasoning about monotone computation and complexity.

Work performed

Togther with Isabel Oitvaem, I have developed monotone functional models of computation inspired by the previous work of Grigni & Sipser and Lautemann, Schwentick & Stewart. We have furthermore given recursion theoretic characterisations in the form of function algebras that smoothly scale to other classes by a simple uniformity constraint.

I have developed a full monotone version of Cook\'s correspondence, for monotone polynomial time, unifying monotone circuit complexity, monotone proof complexity and monotone models of computation via a logical correspondence. As a result I was able to arrive at alternative proofs and generalised results in proof complexity. Some of these ideas and methods were also applied in joint work with Buss and Knop for the proof complexity of branching programs and logspace computation.

Final results

Beyond the current state of the art, I expect that the substructural direction will develop over the next few years via PhD students and collaborators. This remains the missing piece for unifying different communities in implicit complexity, namely linear logic and proof complexity. Eventually I hope these developments will be implemented via proof assistants such as Coq, whereby monotone programs with controled resources may be synthesised for general use.