Monotonicity in Logic and Complexity

Project Information

MiLC

Grant agreement ID: 753431

DOI

10.3030/753431

Project closed

EC signature date 17 March 2017

Start date 1 September 2017

End date 31 August 2019

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 200 194,80

EU contribution

€ 200 194,80

Coordinated by

KOBENHAVNS UNIVERSITET
Denmark

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

Reporting period: 2017-09-01 to 2019-08-31

Summary of the context and overall objectives of the project

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 from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

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