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16
Fractal Symbolic Analysis
, 2001
"... Modern compilers perform wholesale restructuring of programs to improve their efficiency. Dependence analysis is the most widely used technique for proving the correctness of such transformations, but it suffers from the limitation that it considers only the memory locations read and written by a st ..."
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Modern compilers perform wholesale restructuring of programs to improve their efficiency. Dependence analysis is the most widely used technique for proving the correctness of such transformations, but it suffers from the limitation that it considers only the memory locations read and written by a statement, and does not assume any particular interpretation for the operations in that statement. Exploiting the semantics of these operations permits more transformations to be proved correct, and is critical for automatic restructuring of codes such as LU with partial pivoting.
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 algebras with tests and the static analysis of programs
, 2003
"... We propose a general framework for the static analysis of programs based on Kleene algebra with tests (KAT). We show how KAT can be used to statically verify compliance with safety policies specified by security automata. We prove soundness and completeness over relational interpretations. We illust ..."
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We propose a general framework for the static analysis of programs based on Kleene algebra with tests (KAT). We show how KAT can be used to statically verify compliance with safety policies specified by security automata. We prove soundness and completeness over relational interpretations. We illustrate the method on an example involving the correctness of a device driver. 1
Halting and Equivalence of Schemes over Recursive Theories
"... Let Σ be a fixed firstorder signature. In this note we consider the following decision problems. (i) Given a recursive ground theory T over Σ, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models of T? (ii) Given a recursive gr ..."
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Let Σ be a fixed firstorder signature. In this note we consider the following decision problems. (i) Given a recursive ground theory T over Σ, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models of T? (ii) Given a recursive ground theory T over Σ and two program schemes p and q over Σ, are p and q equivalent in all models of T? When T is empty, these two problems are the classical halting and equivalence problems for program schemes, respectively. We show that problem (i) is r.e.complete and problem (ii) is Π0 2complete. Both these problems remain hard for their respective complexity classes even if T is empty and Σ is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively complete deductive system for scheme equivalence. Key words: model theory, Kleene algebra, dynamic logic
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
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).
Halting and Equivalence of Program Schemes in Models of Arbitrary Theories
"... In this note we consider the following decision problems. Let Σ be a fixed firstorder signature. (i) Given a firstorder theory or ground theory T over Σ of Turing degree α, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models ..."
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In this note we consider the following decision problems. Let Σ be a fixed firstorder signature. (i) Given a firstorder theory or ground theory T over Σ of Turing degree α, a program scheme p over Σ, and input values specified by ground terms t1,...,tn, doesp halt on input t1,...,tn in all models of T? (ii) Given a firstorder theory or ground theory T over Σ of Turing degree α and two program schemes p and q over Σ, arep and q equivalent in all models of T? When T is empty, these two problems are the classical halting and equivalence problems for program schemes, respectively. We show that problem (i) is Σ α 1complete and problem (ii) is Π α 2complete. Both problems remain hard for their respective complexity classes even if Σ is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively complete deductive system for scheme equivalence over models of theories of any Turing degree.
KAT + B!
"... It is known that certain program transformations require a small amount of mutable state, a feature not explicitly provided by Kleene algebra with tests (KAT). In this paper we show how to axiomatically extend KAT with this extra feature in the form of mutable tests. The extension is conservative an ..."
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It is known that certain program transformations require a small amount of mutable state, a feature not explicitly provided by Kleene algebra with tests (KAT). In this paper we show how to axiomatically extend KAT with this extra feature in the form of mutable tests. The extension is conservative and is formulated as a general commutative coproduct construction. We give several results on deductive completeness and complexity of the system, as well as some examples of its use. Categories and Subject Descriptors F.3.3 [Logics and Meanings of Programs]: Studies of Program Constructs—Program and recursion schemes
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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