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13
Parikh’s theorem in commutative Kleene algebra
 In Logic in Computer Science
, 1999
"... Parikh’s Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene ..."
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Parikh’s Theorem says that the commutative image of every context free language is the commutative image of some regular set. Pilling has shown that this theorem is essentially a statement about least solutions of polynomial inequalities. We prove the following general theorem of commutative Kleene algebra, of which Parikh’s and Pilling’s theorems are special cases: Every finite system of polynomial inequalities fi(x1�:::�xn) xi, 1 i n, over a commutative Kleene algebra K has a unique least solution in K n; moreover, the components of the solution are given by polynomials in the coefficients of the fi. We also give a closedform solution in terms of the Jacobian matrix of the system. 1
KATML: An interactive theorem prover for Kleene Algebra with Tests
 University of Manchester
, 2003
"... Abstract. We describe an implementation of an interactive theorem prover for Kleene algebra with tests (KAT). The system is designed to reflect the natural style of reasoning with KAT that one finds in the literature. We illustrate its use with some examples. 1 ..."
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Abstract. We describe an implementation of an interactive theorem prover for Kleene algebra with tests (KAT). The system is designed to reflect the natural style of reasoning with KAT that one finds in the literature. We illustrate its use with some examples. 1
On the elimination of hypotheses in Kleene algebra with tests
, 2002
"... The validity problem for certain universal Horn formulas of Kleene algebra with tests (KAT) can be efficiently reduced to the equational theory. This reduction is known as elimination of hypotheses. Hypotheses are used to describe the interaction of atomic programs and tests and are an essential com ..."
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The validity problem for certain universal Horn formulas of Kleene algebra with tests (KAT) can be efficiently reduced to the equational theory. This reduction is known as elimination of hypotheses. Hypotheses are used to describe the interaction of atomic programs and tests and are an essential component of practical program verification with KAT. The ability to eliminate hypotheses of a certain form means that the Horn theory with premises of that form remains decidable in PSPACE. It was known (Cohen 1994, Kozen and Smith 1996, Kozen 1997) how to eliminate hypotheses of the form q =0. In this paper we show how to eliminate hypotheses of the form cp = c for atomic p. Hypotheses of this form are useful in eliminating redundant code and arise quite often in the verification of compiler optimizations (Kozen and Patron 2000). 1
Kleene Algebra with Equations
"... Abstract. We identify sufficient conditions for the construction of free language models for systems of Kleene algebra with additional equations. The construction applies to a broad class of extensions of KA and provides a uniform approach to deductive completeness and coalgebraic decision procedur ..."
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Abstract. We identify sufficient conditions for the construction of free language models for systems of Kleene algebra with additional equations. The construction applies to a broad class of extensions of KA and provides a uniform approach to deductive completeness and coalgebraic decision procedures. 1
Adaptive Logics using the Minimal Abnormality strategy are
"... Abstract. In this paper complexity results for adaptive logics using the Minimal Abnormality strategy are presented. It is proven here that the consequence set of some premise sets is Π 1 1complete. So, the complexity results in (Horsten and Welch, 2007) are mistaken for the adaptive logics using M ..."
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Abstract. In this paper complexity results for adaptive logics using the Minimal Abnormality strategy are presented. It is proven here that the consequence set of some premise sets is Π 1 1complete. So, the complexity results in (Horsten and Welch, 2007) are mistaken for the adaptive logics using Minimal Abnormality strategy.
KAT and PHL in Coq
"... In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL encoding in K ..."
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In this article we describe an implementation of Kleene algebra with tests (KAT) in the Coq theorem prover. KAT is an equational system that has been successfully applied in program verification and, in particular, it subsumes the propositional Hoare logic (PHL). We also present an PHL encoding in KAT, by deriving its deduction rules as theorems of KAT. Some examples of simple program's formal correctness are given. This work is part of a study of the feasibility of using KAT in the automatic production of certificates in the context of (sourcelevel) ProofCarryingCode (PCC).
On Action Logic
"... Pratt [22] defines action algebras as Kleene algebras with residuals and action logic as the equational theory of action algebras. In opposition to Kleene algebras, action algebras form a (finitely based) variety. Jipsen [9] proposes a Gentzenstyle sequent system for action logic but leaves it as a ..."
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Pratt [22] defines action algebras as Kleene algebras with residuals and action logic as the equational theory of action algebras. In opposition to Kleene algebras, action algebras form a (finitely based) variety. Jipsen [9] proposes a Gentzenstyle sequent system for action logic but leaves it as an open question if this system admits cutelimination and if action logic is decidable. We show that Jipsen’s system does not admit cutelimination. We prove that the equational theory of *continuous action algebras and the simple Horn theory of *continuous Kleene algebras are not recursively enumerable and they possess FMP, but action logic does not possess FMP.
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, 2003
"... We show that the universal Horn theory of relational Kleene algebras is Π 1 1complete. 1 ..."
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We show that the universal Horn theory of relational Kleene algebras is Π 1 1complete. 1
Complexity of Kleene Algebra with Tests
, 2004
"... In this lecture we show that the equational theory of KAT is PSPACEcomplete. Thus KAT, while considerably more expressive than KA without tests, is no more difficult to decide. ..."
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In this lecture we show that the equational theory of KAT is PSPACEcomplete. Thus KAT, while considerably more expressive than KA without tests, is no more difficult to decide.
Adaptive Logics using the Minimal Abnormality strategy
"... Abstract. In this paper complexity results for adaptive logics using the Minimal Abnormality strategy are presented. It is proven here that the consequence set of some premise sets is Π 1 1complete. So, the complexity results in (Horsten and Welch, 2007) are mistaken for the adaptive logics using M ..."
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Abstract. In this paper complexity results for adaptive logics using the Minimal Abnormality strategy are presented. It is proven here that the consequence set of some premise sets is Π 1 1complete. So, the complexity results in (Horsten and Welch, 2007) are mistaken for the adaptive logics using Minimal Abnormality strategy.