In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easier-to-measure quantities x 1 ; : : : ; xn which are related to y by a known relation y = f(x 1 ; : : : ; xn ). Measurements are never 100% accurate; hence, the measured values e x i are different from x i , and the resulting estimate e y = f(ex 1 ; : : : ; e xn ) is different from the desired value y = f(x 1 ; : : : ; xn ). How different? Traditional engineering to error estimation in data processing assumes that we know the probabilities of different measurement error \Deltax i = e x i \Gamma x i . In many practical situations, we only know the upper bound \Delta i for this error; hence, after the measurement, the only information that we have about x i is that it belongs to the interval x i = [ex i \Gamma \Delta i ; e x i + \Delta i ]. In this case, it is important to find the range y of all possible values of y = f(x 1 ; : : : ; xn ) when x i 2 x i . We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems.

In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

this paper is to present full proofs of these results, in two parts. In this first part we deal only with theories of operations and numbers which may contain the operator. Then, in Part II, we shall consider the e#ect of adding class axioms. Essential use will be made in this part of proof-theoretic results by Jager [15] on certain formal theories of ordinals over PA.

Abstract not found

Contents 1 Introduction 1-1 2 Some Axiom Systems 2-1 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 2-1 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.1 CZF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2 2.3.2 Constructive Zermelo Fraenkel, CZF . . . . . . . . . . 2-3 2.3.3 Basic Constructive Set Theory, BCST . . . . . . . . . 2-3 3 Elementary Mathematics in Constructive Set Theory 3-1 3.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1 3.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.1 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . 3-2 3.2.2 Cartesian Products of Classes . . . . . . . . . . . . . . 3-2 3.2.3 Relations and Functions between Classes . . . . . . . . 3-3 3.3 The class of Natural

ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The El-operator is omitted. For example the \Pi-type is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...

policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomono-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.

This paper explores the application of rewriting logic to the executable formal modeling of real-time and hybrid systems. We give general techniques by which such systems can be specified as ordinary rewrite theories, and show that a wide range of real-time and hybrid system models, including object-oriented systems, timed automata [4], hybrid automata [2], timed and phase transition systems [28], and timed extensions of Petri nets [1,37], can indeed be expressed in rewriting logic quite naturally and directly. Since rewriting logic is executable and is supported by several language implementations, our approach complements property-oriented methods and tools less well suited for execution purposes. The relationships with the timed rewriting logic approach of Kosiuczenko and Wirsing [24,25] are also studied. 1 Introduction This paper explores the application of rewriting logic to the executable formal modeling of real-time and hybrid systems. The general conceptual advantage of using...

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truth-functional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.