Arto Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, January 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are somewhat unsatisfactory as they afford very few models. A classic example of this phenomenon is present in the long history of the quest for equational axiomatizations of the algebra of regular languages. Salomaa gave two complete axiomatizations of the algebra of regular languages in [28]. However, one of them contains an infinitary rule, and, as argued by Kozen in [22] the other is not sound in most common interpretations of regular expressions (such as binary relations) because it uses a version of the unique fixed point rule. Implicational axiomatizations for the equational ....

....that the equality t(y) x.t(x, y) 1 is provable from the axioms in Ax. Proof. The proof follows the lines of similar arguments in, e.g. 6] and uses the Conway equations (1) 2) and their vector forms, axioms S1 and S2 from Table 2 and equation (5) Related arguments may be found in, e.g. [10, 26, 28, 29]. # From now on, we shall assume that terms in normal form are accessible. Moreover, we equip the transition system ts(t(x, y) with the initial state 1. Probabilistic bisimulations between two such transition systems will relate their initial states. 6 Completeness In Sect. 4, we established ....

A. Salomaa, Two complete axiom systems for the algebra of regular events, J. Assoc. Comput. Mach., 13 (1966), pp. 158--169.

.... of this coalgebraic treatment is coinduction, a proof technique for demonstrating the equivalence of regular expressions [14] Other methods for proving the equality of regular expressions have previously been established for instance, reasoning by using a sound and complete axiomatization [5,15], or by minimization of automata representing the Email: hubes cs.cornell.edu Email: riccardo cs.cornell.edu c #2003 Published by Elsevier Science B. V. expressions [3] However, the coinduction proof technique can give relatively short proofs, and is fairly simple to apply. Recently, ....

Salomaa, A., Two complete axiom systems for the algebra of regular events, Journal of the ACM 13 (1966), pp. 158--169. 16

.... consisting of well defined operations according to a calculus, an order relation and neutral element(s) These requirements are fulfilled by Finite State Automata (FSA) and Regular Expressions (RegEx) having equivalent recognition and generation capabilities, and building an event algebra [SAL1, SAL2]. 2.1 Finite State Modeling of GUI Deterministic finite state automata (FSA) also called finite state, sequential machines have been successfully used for many decades to model sequential systems, e.g. logic design of both combinatorial and sequential circuits [NAIT, DAVI, KISH] protocol ....

....and specification of sequential systems for good reasons. First, they have excellent recognition capabilities to effectively distinguish between correct and faulty events situations. Moreover, efficient algorithms exist for converting FSA into equivalent regular expressions (RegEx) and v.v. [GLUS, SAL1, SAL2]. RegEx, on the other hand, are traditional means to generate legal and illegal situations and events systematically. A FSM can be represented by a set of inputs, a set of outputs, a set of states, an output function that maps pairs of inputs and states to outputs, a next state ....

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A. Salomaa, "Two Complete Axiom Systems for the Algebra of Regular Events ", J. ACM 13, pp. 158-169, 1966

....the bulk of his treatment is infinitary. Redko, 1964] proved that there is no finite equational axiomatization. Schematic equational axiomatizations for the algebra of regular sets, necessarily representing infinitely many equations, have been given by [Krob, 1991] and [Bloom and Esik, 1993] [Salomaa, 1966] gave two finitary complete axiomatizations that are sound for the regular sets but not sound in general over other standard interpretations, including relational interpretations. The axiomatization given above is a finitary universal Horn axiomatization that is sound and complete for the ....

Arto Salomaa. Two complete axiom systems for the algebra of regular events. J. Assoc. Comput. Mach., 13(1):158--169, January 1966.

....timed automata, and demonstrates that bisimulation equivalence of timed automata are as mathematically tractable as those of standard process algebras. A classic result in automata theory is complete axiomatisation of language equivalence for nite state automata in terms of regular expressions [Sal66]. For timed automata it has been shown that the problem of deciding timed language equivalence is 1 hard [AD94] However, the more discriminating relation, timed bisimulation, is decidable [Cer92] The most recent development in algebraic characterisations for timed automata are presented in ....

A. Salomaa. Two Complete Axiom Systems for the Algebra of Regular Events. Journal of the ACM, 13:158-169. 1966.

.... , on languages together with the constants 0 and 1, the corresponding variety (that we denote L below) has no finite axiomatization. Infinite sets of equational axioms were obtained in [15, 5] but these are not minimal. For the variety L, positive answers to Problems 2 and 3 are given in [20, 3, 15, 14, 8, 18]. Due to Theorem 4.4, there is an O(n log n) algorithm for deciding the equational theory of Lg. By modifying the proof of Theorem 3.1 in [10] it is shown in [6] that deciding the inequational theory of the variety Lg is Pi 2 complete in the polynomial hierarchy. The same fact holds for the ....

Arto Salomaa. Two complete axiom systems for the algebra of regular events. J. ACM, 13:158-- 169, 1966.

....axiom system for the theory of regular expressions. An alternative equational axiomatization for regular expressions, developed within the framework of iteration theories [17] may be found in [16] Finite implicational proof systems for regular expressions have been developed by, e.g. Salomaa [61, 62] and Kozen [45] The interested reader is invited to consult [46, Sect. 15] for a thorough discussion of implicational proof systems for regular languages. Modifications of these proof systems to yield complete axiom systems based on conditional equations for the process semantics considered in ....

A. Salomaa, Two complete axiom systems for the algebra of regular events, J. Assoc.

....that is comparable to that of the standard notions from the theory of formal languages. For example, the complete axiomatization of bisimulation equivalence for the regular fragment of CCS [32] provided by Milner in his classic paper [31] parallels those obtained by Salomaa for regular languages [37, 38], and have contributed to the realization that the notion of process is at least as elegant and mathematically tractable as that of language. Despite these successes, process theory has traditionally lacked a systematic investigation of (equational) axiomatizations of process equivalences over ....

A. Salomaa, Two complete axiom systems for the algebra of regular events, J. Assoc.

....and complete axiomatization of path constraint implication. However, obtaining such an axiomatization appears to be highly nontrivial. Note that even an axiomatization of classical regular expression equivalence (in the absence of constraints) is far from obvious (see the set of axioms provided in [29]) 4.2 Word constraints We next consider two particular cases of the implication problem. We show that for word constraints, implication is decidable in ptime. We are then able to extend this result to implication of full path constraints by word constraints, with pspace complexity. Note that ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. J. ACM, 13(1):158--169, 1966.

....focussed on such questions, with the interest driven in part by analogies drawn between classes of concurrent system models and classes of generators for families of formal languages. In [114] Milner 4 exploits the relationship between regular (finite state) automata as discussed by Salomaa in [130] and regular behaviours to present the decidability and a complete axiomatisation of bisimulation equivalence for finite state behaviours, whilst in his textbook [115] he demonstrates that the halting problem for Turing machines can be encoded as a bisimulation question for the full CCS calculus ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13:158--169, 1966.

....e.g. they defined a unary version of Kleene s star in the presence of an empty word. The unary Kleene star has been studied extensively ever since. Redko ( Red64] see also [Con71] proved for the unary Kleene star that a complete finite axiomatisation for language equality does not exist. Salomaa [Sal66] presented a complete finite axiomatisation which incorporates one conditional axiom, namely (translated to our setting) x = y Delta x z and y does not have the empty word property = x = y z A process y has the empty word property if it incorporates the empty word ffl, or in other words ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, 1966.

....53 was introduced in [3] where it was called definiteness . Reference [3] was about applying regular algebra to path finding problems, and a fundamental fact exploited in that paper was that the property of being a regular algebra is preserved by matrix formation. Salomaa s axiomatisation [17] of regular algebra, however, involved the use of the so called empty word property , the formulation of which does not extend to matrices. As a replacement for Salomaa s rule the following rule was postulated in [3] as an axiom of regular algebra: R satisfies (54) j 8(S; T : T = S t T ffi R j ....

Arto Salomaa. Two complete axiom systems for the algebra of regular events. J. Assoc. Comp. Mach., 13(1):158--169, January 1966.

....equivalence, yet it breaks if it is defined as weak equality. Let us write 0 if Tree( Tree( 0 ) Axiomatizations of are given by Amadio Cardelli [AC91] and Ariola Klop [AK95] It is clear, however, that this kind of axiomatization has been known for a long time; see for example Salomaa [Sal66], Milner [Mil84] and Kozen [Koz94] All these axiomatizations are a variant of the inference system presented in Figure 1. 1 (In Rule Contract, recursive type is contractive in type variable ff if ff occurs in only under , if at all. 1.4. Recursive Subtyping Amadio and Cardelli [AC93] ....

....between bisimulation, final coalgebras, attendant coinduction principles and equality of infinite trees precise in a category theoretic setting. Amadio and Cardelli [AC91, AC93] have defined subtyping for recursive types and given an axiomatization inspired by work relying on the Contract Rule [Sal66, Mil84]. They also present an algorithm , which is the basis for efficient subtype checking and can be understood as an inference system based on a finitary coinduction principle (analogous to 20 author short title Figure 9, though without an operational interpretion) In contrast to our work the ....

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A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the Association for Computing Machinery (JACM), 13(1):158--169, 1966.

....are new. The relation with the more traditional methods for proving language equality, has already been discussed at the end of Section 10. Another well known way of proving equality of (regular expressions denoting) rational languages is to use a complete axiom system, such as given by Salomaa in [Sal66], and apply purely algebraic reasoning. The reader is invited to consult [Gin68, pp.68 69] which contains a minor variation on the example K = L, at the end 40 of Section 10, and convince himself of the greater complexity of that approach. The connection between finality and minimality, in ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, 1966.

....[Brz64] and Conway s book [Con71] contain, more generally, many of the ingredients that have been used in the present paper. A well known way of proving equality of regular expressions is to use a complete axiom system (of which the laws in Section 6 form a subset) such as given by Salomaa in [Sal66], and apply purely algebraic reasoning. The reader is invited to consult [Gin68, pp.68 69] from which the example E 1 = F 1 in Section 6 was taken, and convince himself of the greater complexity of that approach. The most common and practical way of proving equality of two expressions is firstly, ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, 1966.

....for this reason he did not define E as a unary operation. Four years later, Redko [69] proved that there does not exist a sound and complete finite equational axiomatisation for regular expressions. This proof was simplified and corrected by Pilling; see [35, Chapter 11] In 1966, Salomaa [70] presented a sound and complete finite axiomatisation for regular expressions, with as basic ingredient an implicational axiom dating back to Arden [9] namely (in process algebra notation) x = y Delta x) z = x = y z if y does not have the so called empty word property. According to ....

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, 1966.

....axiom system for the theory of regular expressions. An alternative equational axiomatization for regular expressions, developed within the framework of iteration theories [4] may be found in [3] Finite implicational proof systems for regular expressions have been developed by, e.g. Salomaa [16, 17] and Kozen [9] The interested reader is invited to consult [10, Sect. 15] for a thorough discussion of implicational proof systems for regular languages. The research reported in this study was inspired by a reading of [17, Chapter III] where Salomaa gives a text book presentation of results ....

A. Salomaa, Two complete axiom systems for the algebra of regular events, J. Assoc. Comput. Mach., 13 (1966), pp. 158--169.

No context found.

Arto Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, January 1966.

No context found.

Arto Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, January 1966.

No context found.

Arto Salomaa. Two complete axiom systems for the algebra of regular events. Journal of the ACM, 13(1):158--169, January 1966.

No context found.

A. Salomaa. Two complete axiom systems for the algebra of regular events. Journal of ACM, 13:158--169, 1966.

No context found.

A. Salomaa. Two complete axiom systems for the algebra of regular events. J. ACM, 13(1):158--169, Jan. 1966.

No context found.

A. Salomaa, Two complete axiom systems for the algebra of regular events, J. Assoc. Comput. Mach., 13 (1966), pp. 158--169.

No context found.

A. Salomaa. Two complete axiom systems for the algebra of regular events. J. Assoc. Comput. Mach., 13:158-169, 1966.

No context found.

Salomaa, A. (1966), Two complete axiom systems for the algebra of regular events, Journal of the ACM, 13(1):158--169.

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