Frameworks for Logically Classifying Polynomial-Time Optimisation Problems

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 NPcomplete. 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.

• Type: Conference
• Year: 2010
• Where: CSR
• Authors: James Gate, Iain A. Stewart
• Keywords: CSR 2010, Existential Second-order Logic, Logical Framework, Polynomially-bounded P-optimisation Problems, Theoretical Computer Science