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Calculational semantics: Deriving programming theories from equations by functional predicate calculus

Author:

Raymond T. BouteAuthors Info & Claims

ACM Transactions on Programming Languages and Systems (TOPLAS), Volume 28, Issue 4

Pages 747 - 793

https://doi.org/10.1145/1146809.1146814

Published: 01 July 2006 Publication History

Abstract

The objects of programming semantics, namely, programs and languages, are inherently formal, but the derivation of semantic theories is all too often informal, deprived of the benefits of formal calculation “guided by the shape of the formulas.” Therefore, the main goal of this article is to provide for the study of semantics an approach with the same convenience and power of discovery that calculus has given for many years to applied mathematics, physics, and engineering. The approach uses functional predicate calculus and concrete generic functionals; in fact, a small part suffices. Application to a semantic theory proceeds by describing program behavior in the simplest possible way, namely by program equations, and discovering the axioms of the theory as theorems by calculation. This is shown in outline for a few theories, and in detail for axiomatic semantics, fulfilling a second goal of this article. Indeed, a chafing problem with classical axiomatic semantics is that some axioms are unintuitive at first, and that justifications via denotational semantics are too elaborate to be satisfactory. Derivation provides more transparency. Calculation of formulas for ante- and postconditions is shown in general, and for the major language constructs in particular. A basic problem reported in the literature, whereby relations are inadequate for handling nondeterminacy and termination, is solved here through appropriately defined program equations. Several variants and an example in mathematical analysis are also presented. One conclusion is that formal calculation with quantifiers is one of the most important elements for unifying continuous and discrete mathematics in general, and traditional engineering with computing science, in particular.

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Cited By

Woodcock J(2021)Hoare and He’s Unifying Theories of ProgrammingTheories of Programming10.1145/3477355.3477369(285-316)Online publication date: 4-Oct-2021
https://dl.acm.org/doi/10.1145/3477355.3477369
Boute R(2018)Pointfree expression and calculationFormal Methods in System Design10.1007/s10703-010-0100-237:2-3(95-140)Online publication date: 28-Dec-2018
https://dl.acm.org/doi/10.1007/s10703-010-0100-2
Kryvolap ANikitchenko MSchreiner W(2013)Extending Floyd-Hoare Logic for Partial Pre- and PostconditionsInformation and Communication Technologies in Education, Research, and Industrial Applications10.1007/978-3-319-03998-5_18(355-378)Online publication date: 2013
https://doi.org/10.1007/978-3-319-03998-5_18
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Reviews

Reviewer: Wolfgang Schreiner

Scholars in many disciplines envy physics and its mathematical formalism, where natural processes can be described by precise equations and equational reasoning can derive consequences and yield insights that would be hard to obtain by intuition alone. This is also true for computer scientists who aim to elaborate formal calculi for describing the behavior of programs in order to get a better understanding of their properties. In pursuit of this goal, the paper presents a set of calculational semantics that allow equational reasoning on programs. The core problem is that programs process discrete objects (computer stores) in a way that can only be adequately described by a logic with quantifiers. To support equational reasoning on such formulas, the author introduces a functional predicate calculus, where logical formulas are represented by Boolean functions and logical equivalence becomes function equality. It is shown how program behavior can be described by equations and how the laws of various other approaches to program semantics can be derived by formal calculations. This approach has various elder relatives. For instance, the late Dijkstra, a pioneer in the field of programming theory, together with Scholten, propagated an equational style of program reasoning, and Hoare and Jifeng introduced an algebra of programming based on a view of programs as relations between states. While the calculational semantics presented in this paper does not enter new territory, it nevertheless provides a succinct working language and a framework for linking other theories. Online Computing Reviews Service

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cover image ACM Transactions on Programming Languages and Systems

ACM Transactions on Programming Languages and Systems Volume 28, Issue 4

July 2006

217 pages

ISSN:0164-0925

EISSN:1558-4593

DOI:10.1145/1146809

Issue’s Table of Contents

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 July 2006

Published in TOPLAS Volume 28, Issue 4

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Cited By

Woodcock J(2021)Hoare and He’s Unifying Theories of ProgrammingTheories of Programming10.1145/3477355.3477369(285-316)Online publication date: 4-Oct-2021
https://dl.acm.org/doi/10.1145/3477355.3477369
Boute R(2018)Pointfree expression and calculationFormal Methods in System Design10.1007/s10703-010-0100-237:2-3(95-140)Online publication date: 28-Dec-2018
https://dl.acm.org/doi/10.1007/s10703-010-0100-2
Kryvolap ANikitchenko MSchreiner W(2013)Extending Floyd-Hoare Logic for Partial Pre- and PostconditionsInformation and Communication Technologies in Education, Research, and Industrial Applications10.1007/978-3-319-03998-5_18(355-378)Online publication date: 2013
https://doi.org/10.1007/978-3-319-03998-5_18
Schreiner W(2012)Computer-Assisted Program Reasoning Based on a Relational Semantics of ProgramsElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.79.879(124-142)Online publication date: 21-Feb-2012
https://doi.org/10.4204/EPTCS.79.8
Erascu MJebelean T(2012)Soundness of a Logic-Based Verification Method for Imperative LoopsProceedings of the 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing10.1109/SYNASC.2012.63(127-134)Online publication date: 26-Sep-2012
https://dl.acm.org/doi/10.1109/SYNASC.2012.63
Erascu MJebelean T(2010)A Purely Logical Approach to the Termination of Imperative LoopsProceedings of the 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing10.1109/SYNASC.2010.64(142-149)Online publication date: 23-Sep-2010
https://dl.acm.org/doi/10.1109/SYNASC.2010.64
Erascu MJebelean T(2009)A Calculus for Imperative ProgramsProceedings of the 2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing10.1109/SYNASC.2009.42(77-84)Online publication date: 26-Sep-2009
https://dl.acm.org/doi/10.1109/SYNASC.2009.42
Boute R(2009)Making Temporal Logic CalculationalProceedings of the 2nd World Congress on Formal Methods10.1007/978-3-642-05089-3_25(387-402)Online publication date: 4-Nov-2009
https://dl.acm.org/doi/10.1007/978-3-642-05089-3_25
Boute R(2008)Simple Gedanken Experiments in Leveraging Applications of Formal MethodsLeveraging Applications of Formal Methods, Verification and Validation10.1007/978-3-540-88479-8_60(847-861)Online publication date: 2008
https://doi.org/10.1007/978-3-540-88479-8_60
Boute R(2006)Using domain-independent problems for introducing formal methodsProceedings of the 14th international conference on Formal Methods10.1007/11813040_22(316-331)Online publication date: 21-Aug-2006
https://dl.acm.org/doi/10.1007/11813040_22

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