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27
A completeness theorem for Kleene algebras and the algebra of regular events
 Information and Computation
, 1994
"... We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1 ..."
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Cited by 185 (22 self)
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We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1
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 50 (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
, 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 40 (6 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: There is a Kleene algebra in the sense of [6] that is not *continuous. The categories of *continuous Kleene algebras [5, 6], closed semirings [1, 10] and Salgebras [2] are strongly related by adjunctions. 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]. Righthanded Kleene algebras are not necessarily lefthanded Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15].
Certification of compiler optimizations using Kleene algebra with tests
 STUCKEY (EDS.), PROC. RST INTERNAT. CONF. COMPUTATIONAL LOGIC (CL2000), LECTURE NOTES IN ARTI CIAL INTELLIGENCE
, 2000
"... We use Kleene algebra with tests to verify a wide assortment ofcommon compiler optimizations, including dead code elimination, common subexpression elimination, copy propagation, loop hoisting, induction variable elimination, instruction scheduling, algebraic simplification, loop unrolling, elimin ..."
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Cited by 32 (10 self)
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We use Kleene algebra with tests to verify a wide assortment ofcommon compiler optimizations, including dead code elimination, common subexpression elimination, copy propagation, loop hoisting, induction variable elimination, instruction scheduling, algebraic simplification, loop unrolling, elimination of redundant instructions, array bounds check elimination, and introduction of sentinels. In each of these cases, we give a formal equational proof of the correctness of the optimizing transformation.
Peirce Algebras
, 1992
"... We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming o ..."
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Cited by 25 (10 self)
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We present a twosorted algebra, called a Peirce algebra, of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a relationforming operator on sets (the Peirce product of Boolean modules) and a setforming operator on relations (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the socalled terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.
Cutfree Display Calculi for Relation Algebras
, 1997
"... . We extend Belnap's Display Logic to give a cutfree Gentzenstyle calculus for relation algebras. The calculus gives many axiomatic extensions of relation algebras by the addition of further structural rules. It also appears to be the first purely propositional Gentzenstyle calculus for relation ..."
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Cited by 21 (14 self)
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. We extend Belnap's Display Logic to give a cutfree Gentzenstyle calculus for relation algebras. The calculus gives many axiomatic extensions of relation algebras by the addition of further structural rules. It also appears to be the first purely propositional Gentzenstyle calculus for relation algebras. 1 Introduction Given a nonempty set U , the universal relation U \Theta U is the set of all ordered pairs (a; b) where a 2 U and b 2 U . Any subset of U \Theta U is a binary relation over U , and the set of all subsets of U \Theta U is the set of all binary relations over U . Thus any two binary relations R and S are each just a set of ordered pairs, and we can use the settheoretic operations of complement, intersection and union to build other relations. The identity relation is f(a; a) j a 2 Ug while the "relative" analogues of complement, intersection and union are converse (` R) = f(b; a) j (a; b) 2 Rg, composition (R ffi S) = f(a; b) j 9c; (a; c) 2 R and (c; b) 2 Sg and ...
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 21 (10 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
A RelationAlgebraic Approach to the Region Connection Calculus
 Fundamenta Informaticae
, 2001
"... We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads ..."
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Cited by 19 (0 self)
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We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the "part  of" relation, and that the table is closed under first order definable relations iff B is homogeneous. 1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient  or not present  to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often l...
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...
Typed Kleene algebra
, 1998
"... In previous work we havefound it necessary to argue that certain theorems of Kleene algebra hold even when the symbols are interpreted as nonsquare matrices. In this note we de ne and investigate typed Kleene algebra, a typed version of Kleene algebra in which objects have types s! t. Although nonsq ..."
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Cited by 16 (4 self)
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In previous work we havefound it necessary to argue that certain theorems of Kleene algebra hold even when the symbols are interpreted as nonsquare matrices. In this note we de ne and investigate typed Kleene algebra, a typed version of Kleene algebra in which objects have types s! t. Although nonsquare matrices are the principal motivation, there are many other useful interpretations: traces, binary relations, Kleene algebra with tests. We give a set of typing rules and show that every expression has a unique most general typing (mgt). Then we prove the following metatheorem that incorporates the abovementioned results for nonsquare matrices as special cases. Call an expression 1free if it contains only the Kleene algebra operators (binary) +, (unary) +, 0, and,but no occurrence of 1 or. Then every universal 1free formula that is a theorem of Kleene algebra is also a theorem of typed Kleene algebra under its most general typing. The metatheorem is false without the restriction to 1free formulas.