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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 system of polynomial inequalities f i (x 1 ; : : : ; xn ) x i , 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 f i . We also give a closedform solution in terms of the Jacobian matrix. 1 Introduction Parikh's theorem [8] says that every contextfree language is "letterequivalent" to a regular set; formally, the commutative image of any contextfree language is also the commutative image of some regular set. The commutative image of a string x over the alphabet fa 1 ; : : : ; a k g is ...
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
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).
speci cation techniques � F.3.2 [Logics and Meanings of Programs]: Semantics of Programming
"... We introduce Kleene algebra with tests, an equational system for manipulating programs. We give a purely equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by awhile program with at most one while lo ..."
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We introduce Kleene algebra with tests, an equational system for manipulating programs. We give a purely equational proof, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by awhile program with at most one while loop. The proof illustrates the use of Kleene algebra with tests and commutativity conditions in program equivalence proofs.