Kleene modules and linear languages

Leiß, Hans

doi:10.1016/j.jlap.2005.04.004

Cited by 9 publications

(12 citation statements)

References 9 publications

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“…A Kleene module [20] is a structure (K, L, :) where K is a Kleene algebra, L a join-semilattice with least element ⊥ and : a mapping of type L × K → L where P : (p + q) = P : p P : q, P : (p · q) = P : p : q,…”

Section: Hoare Logicmentioning

confidence: 99%

“…To obtain a predicate transformer semantics for forward reasoningà la Gordon [8] we have formalised the action of programs as SKAT terms which act on a Boolean algebra of predicates or assertions via a scalar product in a Kleene module [10,20]. We have instantiated this abstract algebra of assertions to the standard powerset algebra over program states realised as maps from variables to values.…”

Section: Introductionmentioning

confidence: 99%

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Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

Armstrong

Struth

Weber

2013

Interactive Theorem Proving

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Abstract. Schematic Kleene algebra with tests (SKAT) supports the equational verification of flowchart scheme equivalence and captures simple while-programs with assignment statements. We formalise SKAT in Isabelle/HOL, using the quotient type package to reason equationally in this algebra. We apply this formalisation to a complex flowchart transformation proof from the literature. We extend SKAT with assertion statements and derive the inference rules of Hoare logic. We apply this extension in simple program verification examples and the derivation of additional Hoare-style rules. This shows that algebra can provide an abstract semantic layer from which different program analysis and verification tasks can be implemented in a simple lightweight way.

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Section: Hoare Logicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

Armstrong

Struth

Weber

2013

Interactive Theorem Proving

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“…Definition 3 (Leiß 2006) An idempotent semimodule ðA; :Þ over a Kleene algebra K is called a Kleene module 2 if it also satisfies the action induction rule…”

Section: Proposition 1 Let S Be An Ordered *-Semiring Then S Is a Klmentioning

confidence: 99%

“…Proposition 2 (Leiß 2006) Let ðK; A; :Þ be a Kleene module. For any r 2 K and a 2 A; the element r Ã : a is the least solution of the equation x ¼ r : x þ a:…”

Section: Inductive Semimodulesmentioning

confidence: 99%

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Inductive semimodules and the vector modules over them

Feng

Jun

2008

Soft Comput

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The notion of an inductive semimodule over an ordered *-semiring is introduced and some related properties are investigated. Inductive semimodules are extensions of several important algebraic structures such as Kleene modules, Kleene algebras and inductive *-semirings. We prove that an inductive semimodule over an ordered *-semiring K is a Kleene module if and only if K is a Kleene algebra. Moreover, we establish that the vector module of an inductive semimodule over an ordered Conway semiring is again an inductive semimodule over the matrix semiring. Consequently, in an inductive semimodule over an ordered Conway semiring, least solutions to linear inequation systems can be denoted by linear expressions, avoiding the least fixed point operator. In addition, we also introduce a related notion called weak inductive semimodules, and propose several open problems on them.

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Modal Semirings Revisited

Desharnais¹,

Struth

Lecture Notes in Computer Science

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Kleene modules and linear languages

Cited by 9 publications

References 9 publications

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

Inductive semimodules and the vector modules over them

Modal Semirings Revisited

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