this paper let \Delta-formulas and \Delta-terms be those defined as \Delta R with the construct the-least omitted. (They define predicates and operations also known as basic [Gan74] or rudimentary [Jen72] ones.) Note, that provablytotal \Sigma -definable operations in KP 0 coincide with those definable by \Delta-terms [Saz85, Saz85a, Saz87]. Therefore, we may use the name KP 0 also for the subtheory of KPR 0 , based on this \Delta-language, which does not involve both the term construct the-least and the corresponding axiom. Analogously, \Delta R (\Delta

This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of \Pi

In this paper we shall investigate fragments of Kripke--Platek set theory with Infinity which arise from the full theory by restricting Foundation to \Pi n Foundation, where n 2. The strength of such fragments will be characterized in terms of the smallest ordinal ff such that L ff is a model of every \Pi 2 sentence which is provable in the theory. 1 Introduction Kripke-Platek set theory plus Infinity (hereinafter called KP!) is a truly remarkable subsystem of ZF. Though considerably weaker than ZF, a great deal of set theory requires only the axioms of this subsystem (cf.[Ba]). KP! consists of the axioms Extensionality, Pair, Union, (Set)Foundation, Infinity, along with the schemas of \Delta 0 --Collection, \Delta 0 --Separation, and Foundation for Definable Classes. So KP! arises from ZF by completely omitting Power Set and restricting Separation and Collection to \Delta 0 --formulas. These alterations are suggested by the informal notion of "predicative". KP! is an impredicat...

We show that the addition of name induction to the theory EETJ + (LEM -I N ) of explicit elementary types with join yields a theory proof-theoretically equivalent to ID_1.

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the non-constructive µ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semi-formal systems and asymmetrical interpretations in standard structures for explicit mathematics.

. The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. MSC:03F15,03F35 Keywords: Anti-foundation axiom, Kripke-Plate set theory, subsystems of second order arithmeic 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [6], [2]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [4]). Instead of the Foundation Axiom these set theories adopt the so-called AntiFoundation Axiom, AFA, which gives rise to a rich universe of ...

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Abstract. Many situations can be modeled as solutions of systems of simultaneous equations. If the functions of these equations monotonically increase in all bound variables, then the existence of extremal fixed point solutions for the equations is guaranteed. Among all solutions, these fixed points uniformly take least or greatest values for all bound variables. Hence, we call them homogeneous fixed points. However, there are systems of equations whose functions monotonically increase in some variables and decrease in others. The existence of solutions of such equations cannot be guaranteed using classical fixed point theory. In this paper, we define general conditions to guarantee the existence and computability of fixed point solutions of such equations. In contrast to homogeneous fixed points, these fixed points take least values for some variables and greatest values for others. Hence, we call them heterogeneous fixed points. We illustrate heterogeneous fixed point theory through points-to analysis. 1

We present a formal theory on the semantics of logic programs and abductive logic programs with first order integrity constraints. The theory provides an elegant, uniform formalisation for the three most widely accepted families of semantics: completion semantics, stable semantics and wellfounded semantics. The theory is based on the notion of a justification, which is a mathematical object describing, given an interpretation, how the truth value of a literal can be justified on the basis of the program. We identify the three different notions of justifications underlying the three types of semantics. In addition, we defend an alternative declarative reading of logic programming, different from the current predominant view of logic programming as a form of defeasible logic. Logic programs are interpreted as sets of definitions of predicates. The framework is suited to evaluate the extent to which this intuition is supported by the three classes of semantics. 1 Introduction. At this mo...

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