V.N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Zh., 16:120--126, 1964. (in Russian).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Partial decidability results for certain kinds of equality problems over the tropical and equatorial semirings are studied in [23] Another classic question for the language of regular expressions, with or without multiplicities, is the study of complete axiom systems for them (see, e.g. [9, 21, 32]) Along this line of research, Bonnier Rigny and Krob have o#ered a complete system of identities for one letter regular expressions with multiplicities in the tropical semiring [6] However, to the best of our knowledge, there has not been a systematic investigation of the equational theory of ....

V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

....question going back to the original paper of [Kleene, 1956] A completeness theorem for relational algebras was given in an extended language by [Ng, 1984; Ng and Tarski, 1977] Axiomatization is a central focus of the monograph of [Conway, 1971] but 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 ....

V. N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Z., 16:120--126, 1964. In Russian.

....fragment of XPath are examined in [26] for tree patterns, 1, 20] present algorithms for achieving minimization, which can be viewed as a certain normal form for tree patterns. The issue of axiomatizing expression equivalence has been investigated for a number of formalisms related to XPath: [21] shows that there can be no finite axiom system for regular expressions, while [11] gives ax iom systems for propositional dynamic logic with converse. Elimination of inverse roles (upward modality) has been studied for description logics [4] The results of propositional dynamic logic and ....

.... Yes Yes Yes Yes Yes No Yes complementation under = No No No No No No No No under =r No No No No No No No No Figure 4: The closure properties of the fragments axiom system when the inference rules are restricted to the ones given in Section 6, even when the alphabet consists of a single symbol [21]. Although r is a large class of regular expressions, we speculate that E0 is finitely axiomatizable for r. We are currently investigating this issue for r terms, which may contain free variables. Another topic for future work is to study the expressiveness and closure properties of other ....

V. Redko. On defining relations for the algebra of regu- lar events. Ukrainskii Matematicheskii Zhurnal, 16:120-126, 1964. (Russian). 12

....Lang . The same result holds if one considers iterated shuffle defined by: x : 1 x x x Omega x Omega x : But there is nothing surprising here, Theorem 7. 2 follows from the fact that the regular sets (over the one letter alphabet) do not have a finite axiomatization, cf. [19], 9] The proof of Theorem 7.2 using this fact is standard. 8 OPEN PROBLEMS 11 There are other operations of interest. In contrast with the star operation, the equational laws of reversal(mirror image) can be captured by the involution axioms relative to any variety considered above: x y) ....

V. N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Z., 16:120-- 126, 1964. (in Russian)

....of a model in which some of the inequivalences I:n, and some of the equivalences E:n, fail. The model we use for this purpose is based on an adaptation of a beautiful construction due to Conway (cf. 20, Thm. 2, page 105] who used it to obtain a new proof of a theorem, originally due to Redko [58] (see also [62, Chapter 3 x6] and the references therein) to the effect that equality of regular expressions cannot be axiomatized using a finite number of equations. 3 The construction of our model relies heavily on the use of prime numbers, as do related arguments presented in, e.g. 20, 26, ....

....to a presentation of a proof of that result. 17 4.1 A proof of Thm. 4.5 The proof of Thm. 4.5 we now proceed to present is based on an adaptation of a beautiful argument due to Conway (cf. 20, Thm. 2, page 105] In op. cit. Conway offers two proofs of a theorem, originally due to Redko [58], to the effect that equality of regular expressions cannot be axiomatized using a finite number of equations. The argument we present below is inspired by the second of those proofs (cf. 20, Pages 105 107] and is model theoretic in nature. In order to show Thm. 4.5, for every finite set of ....

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V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

.... of Redko s, whose proof was simplified and corrected by Pilling [6, Chapter 11] gives an infinite, complete system of identities for commutative regular expressions [15] An infinite equational axiomatization of the theory of regular expressions over a singleton alphabet was given by Redko in [14] (cf. also [6, Chapter 4] Variations on the aforementioned results of Redko s that apply to regular expressions over a singleton alphabet with multiplicities over the tropical semiring may be found in [5] The construction of a complete equational axiomatization for regular expressions over an ....

....Conway [6] in his proof of a result to the effect that infinitely many equations are needed to axiomatize equality of regular expressions over a countably infinite alphabet. The nonexistence of a finite equational axiomatization for the algebra of regular expressions was originally shown by Redko [14]. Redko s proof theoretic argument shows that that the equational theory of regular expressions over an alphabet containing at least two letters is not finitely based, cf. Thm. 6.2 in [17] Our analysis of the model, however, needs to be more refined than the one provided by Conway ibidem (cf. ....

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V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

....EF with the conventions of sum and product respectively in algebra. Copi, Elgot Wright [CEW58] proposed a simplification of Kleene s setting, 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 ....

V.N. Redko. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16:120--126, 1964. In Russian.

.... BCCS for most of the other behavioural equivalences in the linear time branching time spectrum that are reviewed by van Glabbeek in [8] Only in the case of additional, more complex operators, such as iteration, are these equivalences known to lack a finite equational axiomatization; see, e.g. [3, 6, 7, 20, 21]. Of special relevance for concurrency theory are Moller s results to the e#ect that the process algebras ACP and CCS (without the auxiliary left merge operator from [5] do not have a finite equational axiomatization modulo bisimulation equivalence [17, 18] Aceto, Esik and Ingolfsdottir [2] ....

V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

....should be noted that the singulary and binary star operations are interdefinable. 34, page 195] This contradicts Kleene s original argument in [53, page 50] that the length of an event is at least one, and that 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 ....

....and Ing olfsd ottir construct a model A p for these equations in which equation E:p fails, for some prime number p. The model that is used for this purpose is based on an adaptation of a construction due to Conway [35] who used it to obtain a new proof of a theorem, originally due to Redko [69], saying that BPA (A) is not finitely based modulo language equivalence. Let a be an action. For p a prime number, the carrier A p of the algebra A p consists of non empty formal sums of a 0 ; a 1 ; a p Gamma1 , together with the formal symbol a , that is, f X i2I a i j ; ....

V.N. Redko. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16:120--126, 1964. In Russian.

....relation X 2 and the algebra Eqv E of subsets of an arbitrary equivalence relation E does not come up here since we no longer have either intersection or a top element. Any representable relation algebra with star becomes a representable Kleenean algebra when complement is dropped. Redko [Red64] has shown that the equational theory of the representable Kleenean algebras without converse is not finitely axiomatizable. Moreover Conway [Con71] has enumerated several finite models of this theory which do not satisfy axiom (S3) given in the section on star for Boolean monoids) expressing ....

....have is a Boolean algebra and a monoid. Too close and they interfere destructively. Brink [Bri81] argues that Boolean modules are relatively well behaved compared to relation algebras. I make a similar point in the context of regular algebras versus dynamic algebras [Pra79a, Pra79b, Pra80a] Redko [Red64] has shown that the equational theory of regular algebras has no finite basis. Conway [Con71] has observed that this theory has a three element model in which x 0 x 1 . x n is constant with increasing n 0 yet x is not that constant, a discontinuity we refer to as Conway s Leap. ....

V.N. Redko. On defining relations for the algebra of regular events (Russian). Ukrain. Mat. Z., 16:120--126, 1964.

....for Computing Machinery, Inc. ACM) To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and or a fee. c fl 1999 ACM 0164 0925 99 0100 0111 00.75 2 Delta Dexter Kozen 1994; Krob 1991; Kuich and Salomaa 1986; Pratt 1990; Redko 1964; Sakarovitch 1987; Salomaa 1966] In semantics and logics of programs, Kleene algebra forms an essential component of Propositional Dynamic Logic (PDL) Fischer and Ladner 1979] in which it is mixed with Boolean algebra and modal logic to give a theoretically appealing and practical system for ....

Redko, V. N. 1964. On defining relations for the algebra of regular events. Ukrainian Mathematical Journal 16, 120--126. In Russian.

....in which some of the inequivalences I:n, and a fortiori some of the equivalences E:n, fail. The model we use for this purpose is based on an adaptation of a beautiful construction due to Conway (cf. 18, Thm. 2, page 105] who used it to obtain a new proof of a theorem, originally due to Redko [50] (see also [52, Chapter 3 x6] and the references therein) to the effect that equality of regular expressions cannot be axiomatized using a finite number of equations. The construction of our model relies heavily on the use of prime numbers, as do related arguments presented in, e.g. 18, 23, ....

....be devoted to a presentation of a proof of that result. 4.1 A proof of Thm. 4.5 The proof of Thm. 4.5 we now proceed to present is based on an adaptation of a beautiful argument due to Conway (cf. 18, Thm. 2, page 105] In op. cit. Conway offers two proofs of a theorem, originally due to Redko [50], to the effect that equality of regular expressions cannot be axiomatized using a finite number of equations. The argument we present below is inspired by the second of those proofs (cf. 18, Pages 105 107] and is model theoretic in nature. In order to show Thm. 4.5, for every finite set of ....

V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

....appeared as [16] Kleene [13] who posed axiomatization as an open problem. Salomaa [28] gave two complete axiomatizations of the algebra of regular events in 1966, but these axiomatizations depended on rules of inference that are not sound in general under nonstandard interpretations. Redko [25] proved in 1964 that no finite set of equational axioms could characterize the algebra of regular events. The algebra of regular events and its axiomatization is the subject of the extensive monograph of Conway [8] The bulk of Conway s treatment is infinitary. In 1981, we gave a complete ....

V. N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Z., 16:120--126, 1964. In Russian.

....# b # Y = # # X # b # Y = ab) # H 3 # (S 3 ) 1 (SPL) 2 5. false Example 6. The following family of cyclic identities C k : a # = a k ) # (# a a 2 . a k 1 ) 39) for all k 0, forms a set of equations in Reg which is not derivable from any finite et of equational axioms [23, 6]. Consider the inference of C 3 produced by TR: i S i H i (R i ) 1. a # = a 3 ) # (# a a 2 ) true (SPL) 2. a # = a(a(a 3 ) # (# a a 2 ) #) # S 1 (SPL) 3. a # = a(a 3 ) # (# a a 2 ) # H 2 # S 2 (SPL) 4. a # = a 3 ) # (# a a 2 H 3 # S 3 (IND) 5. true ....

....related work. At the end, we shall consider possible improvements and extensions of our approach. 5. 1 Related Work There has been much research on the axiomatization of Reg[A] whose (ground) equational theory is not finitely based for alphabets with more than one letter, as proved by Redko [23] and Conway [6] cf. also Salomaa [25] Infinite equational axiomatizations were first provided by Conway [6] and shown to be complete by Krob [17] To obtain a finite axiomatization, several approaches have been explored: Using special (non logical) inference rules: Salomaa [24] gave two ....

V. N. Redko, On defining relations for the algebra of regular events, Ukrainian Mat. Z. 16 (1964) 120-126.

....variant of Propositional Dynamic Logic (PDL, 21, 7] The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ( 15, 26, 17, 4] and algebraic translations of the Segerberg ( 27] axioms for PDL. Kleene algebras are not finitely axiomatizable ([25, 6]) so our result does not give us a finite axiomatization of test algebra: in fact, no finite equational axiomatization exists. We also present a single sorted version of test algebra, using the notion of dynamic negation ( 9, 2, 11] to which the previous results carry over. 1 Introduction ....

....of PDL that has test free programs only. This implies that the valid program equations of dynamic algebra coincide with those of Kleene algebra. The valid proposition equations are given by algebraic translations of the Segerberg axioms for PDL ( 27] Kleene algebra is not finitely axiomatizable ([25, 6]) hence dynamic algebra is not either. A true algebraic counterpart to PDL is test algebra ( 24, 18, 29] These are simply dynamic algebras that are equipped with the mode of turning a proposition into its test. Now the valid program equations will strictly include those of Kleene algebra, as ....

[Article contains additional citation context not shown here]

V.N. Redko. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16:120--126, 1964. In Russian.

....the Completeness of the Equations for the Kleene Star in Bisimulation Wan Fokkink Utrecht University, Department of Philosophy Heidelberglaan 8, 3584 CS Utrecht, The Netherlands fokkink phil.ruu.nl Abstract. A classical result from Redko [20] says that there does not exist a complete finite equational axiomatization for the Kleene star modulo trace equivalence. Fokkink and Zantema [13] showed, by means of a term rewriting analysis, that there does exist a complete finite equational axiomatization for the Kleene star up to strong ....

....it can express recursion, while on the other hand it can be captured in equational laws. Hence, one does not need meta principles such as the Recursive Specification Principle from Bergstra and Klop [8] Kleene formulated several equations for his operator, notably x y = x(x y) y. Redko [20] (see also Conway [10] proved that there does not exist a complete finite equational axiomatization for the Kleene star in language theory. We observe that Redko s proof can be transposed to the binary Kleene star in Basic Process Algebra, denoted by BPA , modulo trace equivalence. This ....

V.N. Redko. On defining relations for the algebra of regular events. Ukrainskii Matematicheskii Zhurnal, 16:120--126, 1964. In Russian.

....b Y = X b Y = ab) H 3 (S 3 ) 1 (SPL) 2 5: false Example 6. The following family of cyclic identities C k : a = a k ) a a 2 : a k Gamma1 ) 38) for all k 0, forms a set of equations in Reg which is not derivable from any finite set of equational axioms [24, 7]. Consider the inference of C 3 produced by TR: i S i H i (R i ) 1: a = a 3 ) a a 2 ) true (SPL) 2: a = a(a(a 3 ) a a 2 ) S 1 (SPL) 3: a = a(a 3 ) a a 2 ) H 2 S 2 (SPL) 4: a = a 3 ) a a 2 ) H 3 S 3 (IND) 5: true ....

....related work. At the end, we shall consider possible improvements and extensions of our approach. 5. 1 Related Work There has been much research on the axiomatization of Reg[A] whose (ground) equational theory is not finitely based for alphabets with more than one letter, as 14 proved by Redko [24] and Conway [7] cf. also Salomaa [26] Infinite equational axiomatizations were first provided by Conway [7] and shown to be complete by Krob [18] To obtain a finite axiomatization, several approaches have been explored: ffl Using special (non logical) inference rules: Salomaa [25] gave two ....

V. N. Redko, On defining relations for the algebra of regular events, Ukrainian Mat. Z. 16 (1964) 120--126.

....from which each of these applications can be derived as a special case. Besides equations, the axiomatization of Kleene algebras contains the two equational implications ax x ) a x x (1) xa x ) xa x : 2) It is known that no finite equational axiomatization exists over this signature [23] (although well behaved infinite equational axiomatizations have been given [13, 2] Pratt [22] argues that this is due to an inherent nonmonotonicity associated with the operator. This nonmonitonicity is handled in Kleene algebras with the equational implications (1) and (2) In light of the ....

....infinite equational axiomatizations have been given [13, 2] Pratt [22] argues that this is due to an inherent nonmonotonicity associated with the operator. This nonmonitonicity is handled in Kleene algebras with the equational implications (1) and (2) In light of the negative result of [23], it is quite surprising that the essential properties of should be captured purely equationally. Pratt [22] shows that this is possible over an expanded signature. He augments the regular operators with two residuation operators and , which give a kind of weak left and right inverse to the ....

V. N. Redko, "On defining relations for the algebra of regular events," Ukrain. Mat. Z. 16 (1964), 120--126 (in Russian).

....rules such as the approximation induction principle of ACP or the rule arising from domain theoretic models [Hen88] Care must be taken when comparing the results as there are differing definitions, for example of the star expressions. Looking first at the equational face, it was shown by Redko [Red64, completed by Pilling] and Conway [Con71] that no finite system exists for language equality of star expressions. The latter and the present work can both be seen as stemming from the impossibility of equationally expressing arbitrary internal unfoldings of iteration recursion. It would be nice ....

.... [ Esi93] using the rule fix u : P: tuu = fix u : P: t 0 uu fix u : P: tuu = fix v : P: t(fix u : P: t 0 uv)v p [Mil84] impure) p [ Esi93] implicational p [Sal66] impure) p [AG87] p [Bof90; Kro91] p [Koz91] Theta[Sew] p [FZ93] Theta[Sew] bisimulation equational Theta[Red64] Theta[Con71] p [Yan] for events 1) language star expressions expressions derived from work on iteration theories. Finally we should mention that various illuminating infinite but simple equational systems exist. Discussion of these may be found for example for language equivalence of star ....

V.N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Zh., 16:120--126, 1964. (in Russian).

No context found.

V.N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Zh., 16:120--126, 1964. (in Russian).

No context found.

V. N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Zh., 16:120--126, 1964. (in Russian).

No context found.

V. N. Redko. On defining relations for the algebra of regular events. Ukrain. Mat. Zh., 16:120--126, 1964. (in Russian).

No context found.

V. REDKO, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

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V. N. Redko. On defining relations for the algebra of regular events (Russian). Ukrain. Mat. Z., 16:120--126, 1964.

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V. Redko, On defining relations for the algebra of regular events, Ukrainskii Matematicheskii Zhurnal, 16 (1964), pp. 120--126. In Russian.

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