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13
Evolving Algebras: An Attempt To Discover Semantics
, 1993
"... Machine (a virtual machine model which underlies most of the current Prolog implementations and incorporates crucial optimization techniques) starting from a more abstract EA for Prolog developed by Borger in [Bo1Bo3]. Q: How do you tailor an EA machine to the abstraction level of an algorithm wh ..."
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Cited by 74 (12 self)
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Machine (a virtual machine model which underlies most of the current Prolog implementations and incorporates crucial optimization techniques) starting from a more abstract EA for Prolog developed by Borger in [Bo1Bo3]. Q: How do you tailor an EA machine to the abstraction level of an algorithm whose individual steps are complicated algorithms all by themselves? For example, the algorithm may be written in a high level language that allows, say, multiplying integer matrices in one step. A: You model the given algorithm modulo those algorithms needed to perform single steps. In your case, matrix multiplication will be built in as an operation. Q: Coming back to Turing, there could be a good reason for him to speak about computable functions rather than algorithms. We don't really know what algorithms are. A: I agree. Notice, however, that there are different notions of algorithm. On the one hand, an algorithm is an intuitive idea which you have in your head before writing code. Th...
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...
On Kleene Algebras and Closed Semirings
 of Lect. Notes in Comput. Sci
, 1990
"... Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and ..."
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Cited by 42 (7 self)
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Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures [2, 14, 15, 5, 6, 1, 10, 7]. In this paper we establish some new relationships among these structures. Our main results are: ffl There is a Kleene algebra in the sense of [6] that is not *continuous. ffl The categories of *continuous Kleene algebras [5, 6], closed semirings [1, 10] and Salgebras [2] are strongly related by adjunctions. ffl The axioms of Kleene algebra in the sense of [6] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103]. ffl Righthanded Kleene algebras are not necessarily lefthanded Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15]. In Rov...
Process Logic: Expressiveness, Decidability, Completeness
, 1982
"... this paper have natural algebraic and topological interpretations: Let L be the Boolean algebra of formulas of PL modulo the PL axioms of Section 4, and let rim= {nXlXe Z}, fL=/fXlXe m } ..."
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Cited by 40 (1 self)
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this paper have natural algebraic and topological interpretations: Let L be the Boolean algebra of formulas of PL modulo the PL axioms of Section 4, and let rim= {nXlXe Z}, fL=/fXlXe m }
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
Kleene algebra with tests: Completeness and decidability
 In Proc. of 10th International Workshop on Computer Science Logic (CSL’96
, 1996
"... Abstract. Kleene algebras with tests provide a rigorous framework for equational speci cation and veri cation. They have been used successfully in basic safety analysis, sourcetosource program transformation, and concurrency control. We prove the completeness of the equational theory of Kleene alg ..."
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Cited by 22 (11 self)
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Abstract. Kleene algebras with tests provide a rigorous framework for equational speci cation and veri cation. They have been used successfully in basic safety analysis, sourcetosource program transformation, and concurrency control. We prove the completeness of the equational theory of Kleene algebra with tests and *continuous Kleene algebra with tests over languagetheoretic and relational models. We also show decidability. Cohen's reduction of Kleene algebra with hypotheses of the form r = 0 to Kleene algebra without hypotheses is simpli ed and extended to handle Kleene algebras with tests. 1
Dynamic Algebras: Examples, Constructions, Applications
 Studia Logica
, 1991
"... Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an Rmodule with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this con ..."
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Cited by 17 (1 self)
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Dynamic algebras combine the classes of Boolean (B 0 0) and regular (R [ ; ) algebras into a single finitely axiomatized variety (B R 3) resembling an Rmodule with "scalar" multiplication 3. The basic result is that is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper or...
Linear logic for generalized quantum mechanics
 In Proc. Workshop on Physics and Computation (PhysComp'92
, 1993
"... Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of l ..."
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Cited by 16 (2 self)
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Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computeraided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1
On Models for Propositional Dynamic Logic
 Theoretical Computer Science
, 1991
"... In this paper we study some foundational aspects of the theory of PDL. We prove a claim made by Parikh [12], namely, the existence of a Kripke model U that is universal in the sense that every other Kripke model M can be isomorphically embedded in it. Using this model we give different and particul ..."
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Cited by 1 (0 self)
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In this paper we study some foundational aspects of the theory of PDL. We prove a claim made by Parikh [12], namely, the existence of a Kripke model U that is universal in the sense that every other Kripke model M can be isomorphically embedded in it. Using this model we give different and particularly easy proofs of the Completeness Theorem for the Segerberg axiomatization of PDL and the Small Model Theorem. We also give an infinitary axiomatization for PDL and prove it complete using a syntax model A, by a technique that is wellknown from Modal Logic. We prove that U and A are isomorphic. Lastly, we briefly turn to Dynamic Algebras and show that the characteristic algebras of U is initial in the class of ?continuous Dynamic Algebras. 1 Introduction Logics of Programs are formal systems for reasoning about the behavior of computer programs. In these formal systems, computer programs are viewed as a means to enable certain logical formulae. The formulae may be propositional or fir...
On Axiomatizations for Propositional Logics of Programs
, 1988
"... this paper, we focus attention on a propositional program logic, namely Propositional Dynamic Logic or PDL in short ..."
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Cited by 1 (1 self)
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this paper, we focus attention on a propositional program logic, namely Propositional Dynamic Logic or PDL in short