Frameworks for Logically Classifying Polynomial-Time Optimisation Problems

  • James Gate
  • Iain A. Stewart
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6072)


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.


Feasible Solution Logical Framework Horn Clause Decision Version Relation Symbol 
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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • James Gate
    • 1
  • Iain A. Stewart
    • 1
  1. 1.School of Engineering and Computing SciencesDurham University, Science LabsDurhamU.K.

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