Language, Life, Limits

In the context of second-order polynomial-time computability, we prove that there is no general function space construction. We proceed to identify restrictions on the domain or the codomain that do provide a function space with polynomial-time function evaluation containing all polynomial-time computable functions of that type. As side results we show that a polynomial-time counterpart to admissibility of a representation is not a suitable criterion for natural representations, and that the Weihrauch degrees embed into the polynomial-time Weihrauch degrees

A Modal µ-Calculus and a Proof System for Value Passing Processes

A first-order modal ¯-calculus is introduced as a convenient logic for reasoning about processes with value passing. For this logic we present a proof system for model checking sequential processes defined in the value passing CCS. Soundness of the proof system is established. The use of the system is demonstrated on two small but instructive examples. 1 Introduction The propositional modal ¯-calculus is a particularly expressive logic for reasoning about branching-time properties of communicating systems. Many other logics, like dynamic logic and CTL, have uniform encodings in this logic [11]. Over the last decade, many proof systems for checking validity of formulae of this logic w.r.t. particular states (or sets of states) of particular models have been proposed. Since the semantics of the logic is given w.r.t. labelled transition systems (LTS), some of these proof systems [1,4,6,7,15] refer directly to LTS, while other, compositional approaches [2,8] refer to descriptions in some ..

Parallel computable higher type functionals (Extended Abstract)

) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursion-theoretic, proof-theoretic and machine-theoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifier-free, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit..

Interpretation of stream programs: characterizing type 2 polynomial time complexity

We study polynomial time complexity of type 2 functionals. For that purpose, we introduce a first order functional stream language. We give criteria, named well-founded, on such programs relying on second order interpretation that characterize two variants of type 2 polynomial complexity including the Basic Feasible Functions (BFF). These characterizations provide a new insight on the complexity of stream programs. Finally, we adapt these results to functions over the reals, a particular case of type 2 functions, and we provide a characterization of polynomial time complexity in Recursive Analysis

Until-since temporal logic based on parallel time with common past. Deciding algorithms

We present a framework for constructing algorithms recognizing admissible inference rules (consecutions) in temporal logics with Until and Since based on Kripke/Hintikka structures modeling parallel time with common past. Logics with various branching factor after common past are considered. The offered technique looks rather flexible, for instance, with similar approach we showed [33] that temporal logic based on sheafs of integer numbers with common origin is decidable by admissibility. In this paper we extend obtained algorithms to logics . We prove that any logic is decidable w.r.t. admissible consecutions (inference rules), as a consequence, we solve satisfiability problem and show that any itself is decidable

Branching time logics BTL, U,S , N,N −1(Z)α with operations until and since based on bundles of integer numbers, logical consecutions, deciding algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers with a single node (glued zeros). For , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in . As a consequence, it implies that itself is decidable and solves the satisfiability problem