Results 1 
9 of
9
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 ..."
Abstract

Cited by 820 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 50 (6 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract state machines (ASMs) bilden das formale Fundament einer erfolgreichen
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 ..."
Abstract
 Add to MetaCart
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.
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& ..."
Abstract
 Add to MetaCart
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...
Abstract ARTICLE IN PRESS Annals of Pure and Applied Logic ( ) – Computational inductive de nability
"... It is shown that over any countable rstorder structure, IND programs with dictionaries accept ..."
Abstract
 Add to MetaCart
It is shown that over any countable rstorder structure, IND programs with dictionaries accept
unknown title
"... CHRIS FREILING This is not about the symbolic manipulation of functions so popular these days. Rather it is about the more abstract, but infinitely less practical, problem of the primitive. Simply stated: ..."
Abstract
 Add to MetaCart
CHRIS FREILING This is not about the symbolic manipulation of functions so popular these days. Rather it is about the more abstract, but infinitely less practical, problem of the primitive. Simply stated:
How To Compute Antiderivatives
, 1995
"... oped by Lebesgue (see appendix for details). By proving that at least one of these techniques would always succeed, the process could be continued until the definite integral over all possible intervals was obtained. At this point, the antiderivative F (x) = R x 0 f(x) dx (up to a constant) beco ..."
Abstract
 Add to MetaCart
oped by Lebesgue (see appendix for details). By proving that at least one of these techniques would always succeed, the process could be continued until the definite integral over all possible intervals was obtained. At this point, the antiderivative F (x) = R x 0 f(x) dx (up to a constant) becomes apparent. The trouble with Denjoy's procedure is that it needs to be continued transfinitely and, in fact, may require arbitrarily large countable ordinals to complete. He called his process "totalization". The question was immediately raised (for example in Lusin's thesis) as to whether such use of transfinite numbers was really necessary. Could perhaps a di#erent approach avoid these countable ordinals (or at least arbitrarily large ones) and still recover the primitive? Received March 15, 1995. Research supported by the National Science Foundation. The author would like to thank
Negation and Inductive Norms
, 2003
"... In 1982, N. Immerman proved that (positive) least fixed point logic was closed under negation. He used a construction similar to that of Moschovakis [34]: if a logic admits an "inductive norm" that partitions a relation into blocks labelled by integers, then an appropriate "stage c ..."
Abstract
 Add to MetaCart
In 1982, N. Immerman proved that (positive) least fixed point logic was closed under negation. He used a construction similar to that of Moschovakis [34]: if a logic admits an "inductive norm" that partitions a relation into blocks labelled by integers, then an appropriate "stage comparison relation" might be used to construct a negation of that relation within that logic. In this paper, we generalize this construction to many fragments of positive least fixed point logic, and in particular we will find that if such a fragment is closed under (on a class of finite structures), and admits "stage comparison relations" (on M), then it is closed under negation (on M).