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204
Computational Complexity and Feasibility of Data Processing and Interval Computations, With Extension to Cases When We Have Partial Information about Probabilities
, 2003
"... In many reallife 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 easiertomeasure quantities x 1 ; : : : ; xn which are related to y by a known relation y = f(x 1 ; : : : ; xn ). Measurements a ..."
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Cited by 184 (118 self)
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In many reallife 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 easiertomeasure 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.
Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... 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 b ..."
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Cited by 113 (22 self)
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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 BrouwerHeytingKolmogorov 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.
Systems of explicit mathematics with nonconstructive µoperator. Part I
, 1993
"... 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 prooftheoret ..."
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Cited by 48 (16 self)
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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 prooftheoretic results by Jager [15] on certain formal theories of ordinals over PA.
Notes on Constructive Set Theory
, 1997
"... Contents 1 Introduction 11 2 Some Axiom Systems 21 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 CZF 0 . . . . ..."
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Cited by 44 (9 self)
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Contents 1 Introduction 11 2 Some Axiom Systems 21 2.1 Classical Set Theory . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Intuitionistic Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3 Constructive Set Theory . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 CZF 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Constructive Zermelo Fraenkel, CZF . . . . . . . . . . 23 2.3.3 Basic Constructive Set Theory, BCST . . . . . . . . . 23 3 Elementary Mathematics in Constructive Set Theory 31 3.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.2 Cartesian Products of Classes . . . . . . . . . . . . . . 32 3.2.3 Relations and Functions between Classes . . . . . . . . 33 3.3 The class of Natural
The Realizability Approach to Computable Analysis and Topology
, 2000
"... policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government. ..."
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Cited by 41 (19 self)
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policies, either expressed or implied, of the NSF, NAFSA, or the U.S. government.
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... 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 ..."
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Cited by 40 (4 self)
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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 Eloperator is omitted. For example the \Pitype 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...
Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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Cited by 38 (20 self)
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin 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.
Specification of RealTime and Hybrid Systems in Rewriting Logic
, 1999
"... This paper explores the application of rewriting logic to the executable formal modeling of realtime 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 realtime and hybrid system models, including object ..."
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Cited by 33 (17 self)
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This paper explores the application of rewriting logic to the executable formal modeling of realtime 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 realtime and hybrid system models, including objectoriented 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 propertyoriented 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 realtime and hybrid systems. The general conceptual advantage of using...