The weak pigeonhole principle for function classes in S^1_2

It is well known that S^1_2 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective (dual) weak pigeonhole principle in S^1_2 for provably weaker function classes.Comment: 11 page

Some applications of logic to feasibility in higher types

In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook result on mashine representation of basic feasible functionals. Our proofs are based on essential applications of logic. We introduce a weak fragment of second order arithmetic with second order variables ranging over functions from N into N which suitably characterizes basic feasible functionals, and show that it is a useful tool for investigating the properties of basic feasible functionals. In particular, we provide an example how one can extract feasible "programs" from mathematical proofs which use non-feasible functionals (like second order polynomials)

On Rules and Parameter Free Systems in Bounded Arithmetic

We present model–theoretic techniques to obtain conservation results for first order bounded arithmetic theories, based on a hierarchical version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865

Formalizing Termination Proofs under Polynomial Quasi-interpretations

Usual termination proofs for a functional program require to check all the possible reduction paths. Due to an exponential gap between the height and size of such the reduction tree, no naive formalization of termination proofs yields a connection to the polynomial complexity of the given program. We solve this problem employing the notion of minimal function graph, a set of pairs of a term and its normal form, which is defined as the least fixed point of a monotone operator. We show that termination proofs for programs reducing under lexicographic path orders (LPOs for short) and polynomially quasi-interpretable can be optimally performed in a weak fragment of Peano arithmetic. This yields an alternative proof of the fact that every function computed by an LPO-terminating, polynomially quasi-interpretable program is computable in polynomial space. The formalization is indeed optimal since every polynomial-space computable function can be computed by such a program. The crucial observation is that inductive definitions of minimal function graphs under LPO-terminating programs can be approximated with transfinite induction along LPOs.Comment: In Proceedings FICS 2015, arXiv:1509.0282