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Dynamic Logic
 Handbook of Philosophical Logic
, 1984
"... ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possibl ..."
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Cited by 825 (8 self)
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ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N. This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment. This is a rather unconventional program, since it is not effective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program x := 0 ; while x < y do x := x + 1: (1) In DL, such programs are firstclass objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. To discuss the effect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which
Action Logic and Pure Induction
 Logics in AI: European Workshop JELIA '90, LNCS 478
, 1991
"... In FloydHoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as onthefly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively ex ..."
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Cited by 51 (6 self)
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In FloydHoare logic, programs are dynamic while assertions are static (hold at states). In action logic the two notions become one, with programs viewed as onthefly assertions whose truth is evaluated along intervals instead of at states. Action logic is an equational theory ACT conservatively extending the equational theory REG of regular expressions with operations preimplication a!b (had a then b) and postimplication b/a (b ifever a). Unlike REG, ACT is finitely based, makes a reflexive transitive closure, and has an equivalent Hilbert system. The crucial axiom is that of pure induction, (a!a) = a!a. This work was supported by the National Science Foundation under grant number CCR8814921. 1 Introduction Many logics of action have been proposed, most of them in the past two decades. Here we define action logic, ACT, a new yet simple juxtaposition of old ideas, and show off some of its attractive aspects. The language of action logic is that of equational regular expressio...
Abstract State Machines: Verification Problems and Complexity
, 2000
"... Abstract state machines (ASMs) bilden das formale Fundament einer erfolgreichen ..."
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Abstract state machines (ASMs) bilden das formale Fundament einer erfolgreichen
Expressibility And Parallel Complexity
 SIAM J. of Comput
, 1989
"... It is shown that the time needed by a concurrentread, concurrentwrite parallel random access machine (CRAM) to check if an input has a certain property is the same as the minimal depth of a firstorder inductive definition of the property. This in turn is equal to the number of "iterations" of a ..."
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It is shown that the time needed by a concurrentread, concurrentwrite parallel random access machine (CRAM) to check if an input has a certain property is the same as the minimal depth of a firstorder inductive definition of the property. This in turn is equal to the number of "iterations" of a firstorder sentence needed to express the property. The second contribution of this paper is the introduction of a purely syntactic uniformity notion for circuits. It is shown that an equivalent definition for the uniform circuit classes AC i ; i 1 is given by firstorder sentences "iterated" log i n times. Similarly, uniform AC 0 is defined to be the firstorder expressible properties (which in turn is equal to constant time on a CRAM by our main theorem). A corollary of our main result is a new characterization of the PolynomialTime Hierarchy (PH): PH is equal to the set of languages accepted by a CRAM using exponentially many processors and constant time. Key words. C...
Alternating Turing Machines and the Analytical Hierarchy
"... We use notions originating in Computational Complexity to provide insight into the analogies between Computational Complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a mac ..."
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We use notions originating in Computational Complexity to provide insight into the analogies between Computational Complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is inductive (Π1 1). Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyperarithmetical (∆1 1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy. The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the analogy between the polynomialtime and the arithmetical hierarchies, as well as between their respective limits, namely the classes of the polynomialspace and Π1 1 languages.