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24
A completeness theorem for Kleene algebras and the algebra of regular events
 Information and Computation
, 1994
"... We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1 ..."
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Cited by 185 (22 self)
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We givea nitary axiomatization of the algebra of regular events involving only equations and equational implications. Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras. 1
A menagerie of nonfinitely based process semantics over BPA*—from ready simulation to completed traces
 Mathematical Structures in Computer Science
, 1998
"... Fokkink and Zantema ((1994) Computer Journal 37:259–267) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA ∗). In the light of this positive result on the mathematical tractability of ..."
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Cited by 24 (19 self)
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Fokkink and Zantema ((1994) Computer Journal 37:259–267) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA ∗). In the light of this positive result on the mathematical tractability of bisimulation equivalence over BPA ∗ , a natural question to ask is whether any other (pre)congruence relation in van Glabbeek’s linear time/branching time spectrum is finitely (in)equationally axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence, none of the preorders and equivalences in van Glabbeek’s linear time/branching time spectrum, whose discriminating power lies in between that of ready simulation and that of completed traces, has a finite equational axiomatization. This we achieve by exhibiting a family of (in)equivalences that holds in ready simulation semantics, the finest semantics that we consider, whose instances cannot all be proven by means of any finite set of (in)equations
Equational axioms for probabilistic bisimilarity
 IN PROCEEDINGS OF 9TH AMAST, LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finitestate agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571595). The axiomatization is obtained by extending ..."
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Cited by 18 (0 self)
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This paper gives an equational axiomatization of probabilistic bisimulation equivalence for a class of finitestate agents previously studied by Stark and Smolka ((2000) Proof, Language, and Interaction: Essays in Honour of Robin Milner, pp. 571595). The axiomatization is obtained by extending the general axioms of iteration theories (or iteration algebras), which characterize the equational properties of the fixed point operator on (#)continuous or monotonic functions, with three axiom schemas that express laws that are specific to probabilistic bisimilarity.
Bisimulation is not Finitely (First Order) Equationally Axiomatisable
 in Proceedings 9 th Annual Symposium on Logic in Computer Science
, 1994
"... This paper considers the existence of finite equational axiomatisations of bisimulation over a calculus of finite state processes. To express even simple properties such as ¯XE = ¯XE[E=X] equationally it is necessary to use some notation for substitutions. Accordingly the calculus is embedded in a s ..."
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Cited by 17 (0 self)
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This paper considers the existence of finite equational axiomatisations of bisimulation over a calculus of finite state processes. To express even simple properties such as ¯XE = ¯XE[E=X] equationally it is necessary to use some notation for substitutions. Accordingly the calculus is embedded in a simply typed lambda calculus, allowing axioms such as the above to be written as equations of higher type rather than as equation schemes. Notions of higher order transition system and bisimulation are then defined and using them the nonexistence of finite axiomatisations containing at most first order variables is shown. The same technique is then applied to calculi of star expressions containing a zero process  in contrast to the positive result given in [FZ93] for BPA ? , which differs only in that it does not contain a zero. 1 Introduction In this paper we consider the existence of finite equational axiomatisations for bisimulation over finite state processes. Such questions of axio...
Rational Series over Dioids and Discrete Event Systems
 In Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete Event Systems, number 199 in Lect. Notes. in Control and Inf. Sci, Sophia Antipolis
, 1994
"... this paper is obviously too short for such a program, we have chosen to propose an introductive guided tour. A more detailed exposition will be found in our references and in a more complete paper to appear elsewhere. 1 Rational Series in a Single Indeterminate ..."
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Cited by 16 (6 self)
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this paper is obviously too short for such a program, we have chosen to propose an introductive guided tour. A more detailed exposition will be found in our references and in a more complete paper to appear elsewhere. 1 Rational Series in a Single Indeterminate
Process Algebra with Recursive Operations
"... ing from just the two atomic actions in I def = fthrow; tailg, FIR b 1 yields I ((throw tail) throw head) = head: First, observe I (throw tail) = . Then, using (4), it easily follows that I ((throw tail) throw head) = head: This expresses that head eventually comes up, and thus ex ..."
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Cited by 10 (5 self)
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ing from just the two atomic actions in I def = fthrow; tailg, FIR b 1 yields I ((throw tail) throw head) = head: First, observe I (throw tail) = . Then, using (4), it easily follows that I ((throw tail) throw head) = head: This expresses that head eventually comes up, and thus excludes the infinite sequence of steps present in I ((throw tail) throw head). 7.2 Empty Process Let the symbol " denote the empty process, introduced as a unit for sequential composition by Koymans and Vrancken in [58] (see also [28, 74]). Obvious as " may be (being a unit for \Delta), its introduction is nontrivial because at the same time it must be a unit for k as well. In the design of BPA, PA, ACP and related axiom systems, it has proved useful to study versions of the theory, both with and without ". Just for this reason the star operation with its (original) defining equation as given by Kleene in [54] was introduced in process algebra. Taking y = " in x y, one obtains x ...
On Action Algebras
 Logic and Information Flow
, 1993
"... Action algebras have been proposed by Pratt [22] as an alternative to Kleene algebras [8, 9]. Their chief advantage over Kleene algebras is that they form a finitelybased equational variety, so the essential properties of (iteration) are captured purely equationally. However, unlike Kleene algeb ..."
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Cited by 10 (1 self)
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Action algebras have been proposed by Pratt [22] as an alternative to Kleene algebras [8, 9]. Their chief advantage over Kleene algebras is that they form a finitelybased equational variety, so the essential properties of (iteration) are captured purely equationally. However, unlike Kleene algebras, they are not closed under the formation of matrices, which renders them inapplicable in certain constructions in automata theory and the design and analysis of algorithms. In this paper we consider a class of action algebras called action lattices. An action lattice is simply an action algebra that forms a lattice under its natural order. Action lattices combine the best features of Kleene algebras and action algebras: like action algebras, they form a finitelybased equational variety; like Kleene algebras, they are closed under the formation of matrices. Moreover, they form the largest subvariety of action algebras for which this is true. All common examples of Kleene algebras appeari...
Rational and recognisable power series
 DRAFT OF A CHAPTER FOR THE HANDBOOK OF WEIGHTED AUTOMATA
"... ..."
On the Complexity of Reasoning in Kleene Algebra
 Information and Computation
, 1997
"... We study the complexity of reasoning in Kleene algebra and *continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexi ..."
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Cited by 9 (4 self)
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We study the complexity of reasoning in Kleene algebra and *continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for * continuous Kleene algebra, ffl if E contains only commutativity assumptions pq = qp, the problem is \Pi 0 1 complete; ffl if E contains only monoid equations, the problem is \Pi 0 2 complete; ffl for arbitrary equations E, the problem is \Pi 1 1  complete. The last problem is the universal Horn theory of the *continuous Kleene algebras. This resolves an open question of Kozen (1994). 1 Introduction Kleene algebra (KA) is fundamental and ubiquitous in computer science. Since its invention by Kleene in 1956, it has arisen in various forms in program logic and semantics [17, 28], relational algebra [27, 32], aut...
On a Question of A. Salomaa: The Equational Theory of Regular Expressions over a Singleton Alphabet is not Finitely Based
 Comput. Sci
, 1996
"... Salomaa ((1969) Theory of Automata, page 143) asked whether the equational theory of regular expressions over a singleton alphabet has a finite equational base. In this paper, we provide a negative answer to this long standing question. The proof of our main result rests upon a modeltheoretic argume ..."
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Cited by 8 (0 self)
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Salomaa ((1969) Theory of Automata, page 143) asked whether the equational theory of regular expressions over a singleton alphabet has a finite equational base. In this paper, we provide a negative answer to this long standing question. The proof of our main result rests upon a modeltheoretic argument. For every finite collection of equations, that are sound in the algebra of regular expressions over a singleton alphabet, we build a model in which some valid regular equation fails. The construction of the model mimics the one used by Conway ((1971) Regular Algebra and Finite Machines, page 105) in his proof of a result, originally due to Redko, to the effect that infinitely many equations are needed to axiomatize equality of regular expressions. Our analysis of the model, however, needs to be more refined than the one provided by Conway ibidem.