Syntactic characterization in Lisp of the polynomial complexity classes and hierarchy

The definition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses Delta^p_n are presented. They are obtained by progressive restrictions of recursion in Lisp, and may be regarded as predicative according to a foundational point raised by Leivant

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Resource Bounded Unprovability of Computational Lower Bounds

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P not= NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This result gives a unified view to the existing two major negative results on proving P not= NP, Natural Proofs and relativizable proofs, through the two manners of characterization of PTM-omega-consistency. We also show that the PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory S, where S is a consistent PT-extension of T.Comment: 78 page