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Durham University

Computer Science


Publication details for Professor Iain Stewart

Gate, J. & Stewart. I.A. (2010), Frameworks for logically classifying polynomial-time optimisation problems, in Ablayev F. & Mayr E.F. eds, Lecture Notes in Computer Science 6072: Proceedings of 5th International Computer Science Symposium in Russia. Kazan, Russia, Springer-Verlag, Berlin, 120-131.

Author(s) from Durham


We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems.