Automatic Proof Generation in Kleene Algebra
نویسنده
چکیده
In this paper, we develop the basic theory of disimulations, a type of relation between two automata which witnesses equivalence. We show that many standard constructions in the theory of automata such as determinization, minimization, inaccessible state removal, et al., are instances of disimilar automata. Then, using disimulations, we define an “algebraic” proof system for the equational theory of Kleene algebra in which a proof essentially consists of a sequence of matrices encoding automata and disimulations between them. We show that this proof system is complete for the equational theory of Kleene algebra, and that proofs in this system can be constructed by a PSPACE transducer.
منابع مشابه
Automatic Proof Generation in Kleene Algebra with Tests
Kleene algebra (KA) is the algebra of regular events. Familiar examples of Kleene algebras include regular sets, relational algebras, and trace algebras. A Kleene algebra with tests (KAT) is a Kleene algebra with an embedded Boolean subalgebra. The addition of tests allows one to encode while programs as KAT terms, thus the equational theory of KAT can express (propositional) program equivalenc...
متن کاملMyhill-Nerode Relations on Automatic Systems and the Completeness of Kleene Algebra
It is well known that finite square matrices over a Kleene algebra again form a Kleene algebra. This is also true for infinite matrices under suitable restrictions. One can use this fact to solve certain infinite systems of inequalities over a Kleene algebra. Automatic systems are a special class of infinite systems that can be viewed as infinite-state automata. Automatic systems can be collaps...
متن کاملOn the Coalgebraic Theory of Kleene Algebra with Tests
We develop a coalgebraic theory of Kleene algebra with tests (KAT) along the lines of Rutten (1998) for Kleene algebra (KA) and Chen and Pucella (Electron Notes Theor Comput Sci 82(1), 2003) for a limited version of KAT, resolving some technical issues raised by Chen and Pucella. Our treatment includes a simple definition of the Brzozowski derivative for KAT expressions and an automata-theoreti...
متن کاملKleene Algebra Completeness
This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an ε-free automaton to prove an equivalent to Kleene’s representation theorem (implementing Glushkov’s instead of Thom...
متن کاملA coalgebraic approach to Kleene algebra with tests
Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene algebra with tests does not lend itsel...
متن کامل