E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. SpringerVerlag, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....been presented which, we hope, solves some of the problems encountered in AI, and related to the diculty of formalizing the idea of inertia into logical descriptions of dynamic systems. Earlier, and independently, Lukaszewicz and Madali nska Bugaj discovered that the use of Dijkstra s calculus ([7, 8]) of predicate transformers copes with the Frame Problem, cf. 19, 20] Whereas their approach is more general, the advantage of the framework presented here relates to its logical nature. As a result, we are able to encode our system in a general purpose theorem prover Isabelle, cf. 4] Thus, ....

Dijkstra, E. W. and C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.

....been presented which, we hope, solves some of the problems encountered in AI, and related to the di#culty of formalizing the idea of inertia into logical descriptions of dynamic systems. Earlier, and independently, L ukaszewicz and Madali#ska Bugaj discovered that the use of Dijkstra s calculus ([8, 9]) of predicate transformers copes with the Frame Problem, cf. 20, 21] Their approach is more general ours is based on the weakest liberal preconditions, whereas they, after Dijkstra, consider strongest postconditions as well. Nevertheless, our framework has the advantage of being logical in ....

Dijkstra, E. W. and C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.

....and only if, both and its converse relation, are simulations. We say that M is simulated by (bisimilar to) N if there exists a simulation (bisimulation) relation from M to N . Notation. In the rest of this paper, proofs are presented in a format popularized by Dijkstra and Scholten in [DS90] Here, individual steps of a proof are linked by a transitive connective such as or ) along with a hint for why the connection holds. The notation (Qx : r(x) p(x) is used to represent an operation where x is the dummy variable, r(x) is the range of x, and p(x) is the term being operated ....

E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics. Springer Verlag, 1990.

....and Tweedledee Argument in [BCG82] But, tit for tat is invalid in the case of non well founded game relations. One of our concerns is to identify precisely where the property of wellfoundedness is vital to the development of the theory. Throughout this paper, we use the Dijkstra Scholten [DS90] notation for predicates and predicate transformers. In particular, we use square brackets to indicate that a predicate is true at all positions. For a given relation R , dom:R and rng:R are predicates characterising the domain and range of R , respectively. Formally, for all positions s , ....

.... g) f ( f g) g ( f g) and together cover all positions: f g ( f g) Lemma 25 If f and g are conjugate, monotonic predicate transformers, f f f g] We remind the reader that continued equivalences should be read associatively, and not conjunctionally [DS90]. 10 f f = f by de nition of and ( least and greatest ) f ) f] g f ( f = f [ g f] g f ( g f g : 3.4 Application to Win Lose Equations Since Some:M Some:M are conjugate, monotonic predicate transformers, 20) and lemma 24 suggest that the winning positions ....

E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, Berlin, 1990.

....ranking based on voting procedures is the subject of social choice theory, e.g. see [6] and references given there. Notations. We replace the definitions and arguments by case distinctions in natural language of [1] by more concise formulas in predicate logic, in a version as suggested by [2]. So, we use for logical equivalence, for conjunction (and) for disjunction (or) for implication, for negation. Universal and existential quantification of a predicate R(x) with x ranging over a set X is denoted by (# x X : R(x) and (# x X : R(x) If f is a numerical ....

....over a set X is denoted by (# x X : R(x) and (# x X : R(x) If f is a numerical function on X, a similar notation is used for the minimum (MIN x X : f(x) and the sum ( X : f(x) We write #V for the number of elements of a set V . At some points, we also use the format of [2] to give predicate calculations with hints between braces to justify the steps. 2 Policies and utility functions This section contains the main definitions. Given finite sets States and Actions, we define policies as functions that yield total preorders on Actions, with deterministic policies as ....

Dijkstra, E.W., Scholten, C.S.: Predicate calculus and program semantics. Springer V. 1990.

....equational style and compared. 4 Quantification and the predicate calculus The treatment of quantification in our course unifies what, until now, has been a rather chaotic topic. Quantification in mathematics assumes many forms, for example: E a=i = 1 2 3 (x) l x 3 = b[x] O = b[1] 0 A b[2]=0 A b[3] 0 (Bx) l x 3 A b[x] O = b[1] 0 V b[2] 0V b[3] 0 There appears to be no consistency of concept or notation here. Compounding the problem is that students are not taught rules for manipulating specific quantifiers much less general 11 Table 4: Axioms for Quantification (23) Empty ....

....and the predicate calculus The treatment of quantification in our course unifies what, until now, has been a rather chaotic topic. Quantification in mathematics assumes many forms, for example: E a=i = 1 2 3 (x) l x 3 = b[x] O = b[1] 0 A b[2] 0 A b[3] 0 (Bx) l x 3 A b[x] O = b[1] 0 V b[2]=0V b[3] 0 There appears to be no consistency of concept or notation here. Compounding the problem is that students are not taught rules for manipulating specific quantifiers much less general 11 Table 4: Axioms for Quantification (23) Empty range: x [fals: P) the identity of ) 24) ....

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Dijkstra, E.W., and C. Scholten. Predicate Calculus and Program Semantics. Springer- Verlag, New York 1990.

....F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrences of P in R ) and if Q , then S . Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. [3, 6]) so named because a formalization of (1) is the central inference rule. However, di#ering formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in ....

....at all occurrences of P in R ) and if Q , then S . Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. 3, 6] so named because a formalization of (1) is the central inference rule. However, di#ering formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in terms of substitution that does avoid such capture [6] This article explores the relation between these three formalizations. In Sect. 2, ....

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Dijkstra, E.W., and C.S. Scholten. Predicate Calculus and Program Semantics. Springer Verlag, NY, 1990.

....formalized using 2 as 2(fx Qg = fx Rg) j 2(Q j R) but it cannot be formalized as a formula without something like 2 . The use of the everywhere operator 2P was introduced to researchers in the formal development of programs by Dijkstra (using the notation [P ] in the early 1980 s (see e.g. [5, 4]) It can be used to formalize the Hoare triple fPg S fQg , with the meaning execution of statement S begun in a state in which P is true is guaranteed to terminate in a state in which Q is true . 10] Using weakest precondition predicate transformer wp [3] we define: fPg S fQg : 2(P ) wp(S; ....

....Joe Halpern) The resulting axiomatization is still unsatisfactory to us, because of the need to reformulate ffi of a conjectured theorem 3ffi in disjunctive normal form. This reformulation is not in keeping with our usual way of proving theorems (using a calculational approach, where suitable [7, 8, 5]) For example, to prove 3ffi 3fl , we would be forced to prove that one of ffi and fl were satisfiable, rather than simply performing syntactic manipulations to obtain 3ffi 3fl , as is our preference. Inference rule Textual Substitution provides an alternative that is more in tune with the way ....

Dijkstra, E.W., and C.S. Scholten. Predicate Calculus and Program Semantics. Springer Verlag, New York, 1990.

....herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of these organizations or the U.S. Government. Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. [3, 6]) so named because a formalization of (1) is the central inference rule. However, differing formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in ....

....or implied, of these organizations or the U.S. Government. Since then, 1) has become a cornerstone of calculational formulations of logic (see e.g. 3, 6] so named because a formalization of (1) is the central inference rule. However, differing formalizations of (1) have been used. One [3] is in terms of function application, another [7] is in terms of substitution that doesn t avoid capture of free occurrences of variables, and a third is in terms of substitution that does avoid such capture [6] This article explores the relation between these three formalizations. In Sect. 2, ....

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Dijkstra, E.W., and C.S. Scholten. Predicate Calculus and Program Semantics. Springer Verlag, NY, 1990.

....law has been pervasive among users of mathematics. It allows them to infer new truths from old ones in a simple and efficient way. Why not make such a calculational style available to users of logic A first cut at an equational logic for computer scientists appeared in the monograph [4]. Three years later, a freshman sophomore level text incorporating the approach appeared [5] The new approach offers hope for a new view of logic and an entirely different method of teaching logic and proof. One of the principles guiding research by those studying the formal development of ....

Dijkstra, E.W., and C.S. Scholten. Predicate Calculus and Program Semantics. Springer Verlag, New York 1990.

....of the predicate calculus, but conventional logics, like natural deduction, required too much formal detail. The equational approach was used informally in the late 1970 s, and the first text on the formal development of programs [4] used a rudimentary equational logic. Dijkstra and Scholten [2] developed the equational logic on which ours is based. The unification of quantification was also developed in the early 1980 s; the first text we know of that talks about this topic is by Backhouse [1] ....

Dijkstra, E.W., and C. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, New York 1990.

....in this context. We surmise that by the very nature of the [a] modality (related to weakest preconditions) the framework so far seems to fit for prediction but is not very suitable for postdiction or explanation of scenarios ( 11] Perhaps extending it with the notion of strongest postconditions ([2], 9] 8] would be helpful here. 7 Related Work We were much inspired by work by ( 9] 8] In this work the authors also attempted to employ proven verification and correctness methods and logics from imperative programming for reasoning about action and change in AI. In particular ....

....of actions and is in fact very close to the dynamic logic framework: formulas of the form [a] are actually the same as the wlp (weakest liberal precondition) of action a with respect to postcondition . In ( 9] 8] a central role is played by the following theorem from Dijkstra and Scholten ([2]) which says that a state cr a A wlp(S, fi) iff 11 there is a computation c under control of S starting in a state satisfying c and terminating in a state satisfying such that cr is the initial state of c. What all this amounts to is that when in [9] weakest (liberal) preconditions and the ....

E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.

....to show that the method we propose can be used within different fields of mathematics. The notion of derivation that we use generalises a linear proof format, often referred to as calculational derivations. This format is attributed to Wim Feijen and is described by Dijkstra and Scholten [3]. The calculation format has been advocated by Gries and Schneider as a vehicle for teaching logic as a tool rather than as an object of study [5, 6] Our derivation style also builds on the idea of contextual rewriting described by Staples and Robinson [10] This idea has been developed further ....

E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics. Springer--Verlag, 1990.

....logical redundancy and multiple exits we can systematically apply loop rationalization to understand, analyze and improve such delinquent program structures. 2. Strongest Postcondition Calculations Central to the process of loop rationalization is the use of Strongest Postcondition calculations [6,18], that provide explicit guidance for detecting loop exits termination and the subsequent transformations. Because this method, like the use of weakest precondition calculations [6] is based on formal calculations and formal semantics rather than pattern matching, it can be easily adapted to ....

....Calculations Central to the process of loop rationalization is the use of Strongest Postcondition calculations [6,18] that provide explicit guidance for detecting loop exits termination and the subsequent transformations. Because this method, like the use of weakest precondition calculations [6], is based on formal calculations and formal semantics rather than pattern matching, it can be easily adapted to reengineering of programs in any imperative programming language. Strongest postcondition calculations may be used to assist with the loop rationalization of loop bodies. In ....

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Dijkstra, E.W., Scholten, C.S.: Predicate Calculus and Program Semantics, SpringerVerlag 1989.

....software it points to a powerful measurement discipline for software quality The feature of formal methods is that they allow us to focus on the semantics of programs. A powerful theoretical tool that can be used to help capture the semantics of program structures is the strongest postcondition [7,8]. In developing a framework for assessing, improving and measuring software quality we may exploit the application of strongest postconditions. Their important advantage is that they allow the derivation of specifications from program components. Such specifications formally and accurately ....

Dijkstra, E.W., Scholten, C.S.: Predicate Calculus and Program Semantics, Springer-Verlag 1989.

....to Model Checking : by fixpoint evaluation [CE 81, EL 86] and using automata theory [VW 86] 2. 1 Notation Quantified expressions are written in the format (Qx : r : p) where Q is the quantifier (e.g. 9; 8; min; max ) x the bound variable, r the range, and p the expression being quantified [DS 90] When the range of x is clear from the context, it may be dropped, in which case the quantified expression has the form (Qx : p) Sets defined by set comprehension are written in the format fxjP (x)g, which represents the set of all elements x for which the predicate P holds. The powerset of a ....

....are written in the format fxjP (x)g, which represents the set of all elements x for which the predicate P holds. The powerset of a set S is denoted by P(S) For a binary relation R, we often write s R t for readability in place of (s; t) 2 R. 8 Proofs are often given in the calculational style [DS 90] A proof in this format is constructed out of the following basic unit : op ( hint justifying P op Q ) The binary operator op is a transitive relation on the expressions, which allows the chaining together (using transitivity) of a number of such proof units. Typical operators are ) ....

Dijkstra, E. W., Scholten. C. S. Predicate Calculus and Program Semantics, Springer-Verlag, 1990.

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E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. SpringerVerlag, 1990.

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Dijkstra E.W., Scholten C.S. Predicate Calculus and Program Semantics. Springer-Verlag (1990).

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E. W. Dijkstra, C. S. Scholten. Predicate Calculus and Program Semantics, Springer-Verlag, New York, 1990.

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Dijkstra, E. W. and Scholten, C. S. (1990) Predicate Calculus and Program Semantics. Springer Verlag.

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E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.

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E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, New York, 1990.

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E. W. Dijkstra and C. S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, 1990.

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E.W. Dijkstra and C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, Berlin, 1990.

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E.W. Dijkstra, C.S. Scholten. Predicate Calculus and Program Semantics. Springer-Verlag, Berlin, 1990.

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