An applicative theory for FPH

In this paper we introduce an applicative theory which characterizes the polynomial hierarchy of time.Comment: In Proceedings CL&C 2010, arXiv:1101.520

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

On Sharing, Memoization, and Polynomial Time (Long Version)

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity

A Logical Account of PSPACE

International audienceThe Soft Type Assignment system has been introduced as a language for Polytime Computations. In this work we extend the language by boolean additives constants obtaining a system named BSTA which we will show to be correct and complete for the complexity class of problem decidable in polynomial space

On sharing, memoization, and polynomial time

International audienceWe study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity

Dagstuhl News January - December 2001

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

A hierarchy of ramified theories below primitive recursive arithmetic

The arithmetical theory EA(I;O) developed by Çagman, Ostrin and Wainer ([18] and [48]) provides a formal setting for the variable separation of Bellantoni-Cook predicative recursion [6]. As such, EA(I;O) separates variables into outputs, which are quantified over, and inputs, for which induction applies. Inputs remain free throughout giving inductions in EA(I;O) a pointwise character termed predicative induction. The result of this restriction is that the provably recursive functions are the elementary functions. An infinitary analysis brings out a connection to the Slow-Growing Hierarchy yielding є0 as the appropriate proof-theoretic ordinal in a pointwise sense. Chapters 1 and 2 are devoted to an exposition of these results. In Chapter 3 a new principle of 1-closure is introduced in constructing a conservative extension of EA(I;O) named EA1. This principle collapses the variable separation in EA(I;O) and allows quantification over inputs by acting as an internalised ω-rule. EA1 then provides a natural setting to address the problem of input substitution in ramified theories. Chapters 4 and 5 introduce a hierarchy of theories based upon alternate additions of the predicative induction and ∑1-closure principles. For 0 < k є N, the provably recursive functions of the theories EAk are shown to be the Grzegorczyk classes Ek+2. Upper bounds are obtained via embeddings into appropriately layered infinitary systems with carefully controlled bounding functions for existential quantifiers. The theory EA-ω, defined by closure under finite applications of these two principles, is shown to be equivalent to primitive recursive arithmetic. The hierarchy generated may be considered as an implicit ramification of the sub-system of Peano Arithmetic which restricts induction to ∑1-formulae.EThOS - Electronic Theses Online ServiceGBUnited Kingdo