C.C. Elgot and G. Mezei. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9:47--65, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algebras automatically have closure under projection and product, in addition to the Boolean operations. In the case of the model S len above, this algebra is not new: in fact, the definable n ary relations are exactly the ones recognizable under a natural notion of automaton running over n tuples [19, 29]. We will refer to these automata definable relations as the regular relations: the formal definition is given in subsection 3.1.1. We show here that by taking restrictions of the model S len , one gets new algebras of regular relations which behave better, in many ways, than the full algebra of ....

....are a subclass of recursive structures [43] and were introduced as a generalization of automatic groups [30] In an automatic structure M = h Sigma i, every predicate in Omega is definable by a finite automaton. More precisely, an n ary predicate P is given by a letter to letter n automaton [29, 34]. These structures were also studied in [45] in connection with decidability questions for first order theories. It is known [19, 14] that a structure is automatic iff it can be interpreted in the structure S len ; hence S len is in some sense the universal automatic structure. The first part of ....

C. Elgot and J. Mezei. On relations defined by generalized finite automata. IBM J. Res. Develop. 9 (1965), 47--68.

....independence relations) which contain just the pairs of commuting letters. The most important class of trace monoids are direct products of free monoids; already in the beginnings of computability theory, certain subsets of these monoids (such as rational and enumerable relations) were studied [34, 20]. General partial commutations were probably first employed in 1969 by Cartier and Foata [11] as a tool for the study of Mobius functions. In the following decades trace monoids appeared in connection with different research fields and a self contained theory of traces has also gradually ....

C.C. Elgot, J.E. Mezei, On relations defined by generalized finite automata, IBM J. Res. Dev. 9(1) (1965) 47--68.

....that (explicitly or implicitly) searches this graph in order to find all the n tuplcs in R with some interesting property, say those corresponding 1 The present set can be increased ahnost exponentially by adding new derivatlonal affixes. aThe name and the basic idea of these automata come from [5]. For simplicity of exposition we gloss over various authors attempts to distinguish variously among machines, aatomata and transducerz, as well as the profusion of precursors and de endants in ( 15] 16] 2] etc. Our notation is eclectic. i 277 to forms whose surface spelling is the ....

Elgot, C.C. and J.E. Mezei 1965. On Relations Defined by Generalized Finite Automata, IBM Journal Res. 9, pp. 47-68.

....criteria, namely that automatic structures are countable and that their first order theory is decidable, not much is known. The only non trivial criterion that is available at present use growth rates for the length of the encodings of elements of definable sets. Proposition 6. 1 (Elgot and Mezei [17]) Let A be an automatic structure with injective presentation (#, d) and let f : A A be a function of A. Then there is a constant m such that #(f(a) #(a) m for all a . The same is true if we replace f by a relation R where for all a there are only finitely many values b such ....

C. C. Elgot and J. E. Mezei, On relations defined by generalized finite automata, IBM J. Res. Develop., 9 (1965), pp. 47--68.

....family of rational languages turns out to be one of the most important classes within the Chomsky hierarchy. Finite automata that are the main object for studying rational languages are now used in most domains of computer science. Rational transductions introduced by C. C. Elgot and J. E. Mezei [6] are a natural extension of rational languages and were very useful to represent several kinds of computations. For instance, the addition of two integers in any base can be realized by a rational transduction and the study of these tools has led to efficient parallel algorithms (see [1, 19] for ....

Elgot, C. C., and Mezei, J. E. On relations defined by generalized finite automata. IBM Journal of Research and Development 9 (1965), 47--68.

....of rational languages turns out to be one of the most important classes within the Chomsky hierarchy. Finite automata that are the main object for studying rational languages are now used in the most domains of computer science. Rational transductions introduced by C. C. Elgot and J. E. Mezei [5] are a natural extension of rational languages and were very useful to represent several kinds of computations. For instance, the addition of two integers in any base can be realized by a rational transduction and the study of these tools has led to efficient parallel algorithms (see [2, 15] for ....

..... By the same way, right sequential transducers are defined. The input word is then read from right to left. For more details, see [3, 4] for example. A sequential transducers are kind of deterministic transducers without accepting states (all of them are accepting) The decomposition theorem [1, 4, 5, 17], characterizes rational functions by using sequential functions: a rational function including ( is the composition of a right sequential transduction followed by a left sequential transduction. We use this theorem in order to deal with deterministic functions and then we can prove the ....

Elgot, C. C., and Mezei, J. E. On relations defined by generalized finite automata. IBM Journal of Research and Development 9 (1965), 47--68.

....for the next input symbol, and always terminates in a final state. Because T 1 is functional, it can be factorized into a left sequential 1 and a right sequential FST, T 11 and T 12 , that jointly constitute a bimachine (Schutzenberger, 1961) using an existing factorization algorithm (Elgot and Mezei, 1965; Berstel, 1979; Reutenauer and Schutzenberger, 1991; Roche and Schabes, 1997) 2 T 11 , T 12 , and T 2 together represent a trimachine. An input string is processed by this trimachine, first deterministically from left to right, then deterministically from right to left, and finally ambiguously ....

Elgot, C. C. and J. E. Mezei. 1965. On relations defined by generalized finite automata. IBM Journal of Research and Development, pages 47--68, January.

....0 ; ff 0 ) be two generating systems for M and fi a description of (X; ff) By [Sak87, Proposition 2.1] there exist homomorphisms fl : X X 0 and ffi : X 0 X such that fi 0 : fl ffi fi ffi ffi is a description of (X 0 ; ff 0 ) Now assume fi to be rational. Then, by [EM65] (cf. Sak87, Proposition A.16] fi 0 is rational. Thus, the existence of a rational description does not depend on the actual generating system. Nevertheless, it might be easier to describe such a description in an alternative generating system as we will do later. The key property of ....

.... Delta Delta a n of the monoid M . For a computation with first state p 0 , last state p n and label a, we will usually write p 0 a p n without mentioning the intermediate states. The behavior of A is the subset jAj of M consisting of labels of successful computations in A. Theorem 3. 5 ([EM65]) Let M be a monoid. A set L M is rational iff it is the behavior of a finite automaton over M . If the monoid M is a direct product M 1 Theta M 2 of two monoids, it is convenient to think of M 1 as the input and of M 2 as the output of the automaton. Then the automaton computes from an input ....

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C.C. Elgot and G. Mezei. On relations defined by generalized finite automata. IBM J. Res. Develop., 9:47--65, 1965.

....from words have the property that the difference of height of any couple of trees (the input tree being a word) is bounded. We establish the fact that the tree transformations induced by such transducers have some good closure properties. 1 Introduction We extend here a result of Elgot and Mezei [4] about rational relations with the property that the difference of length of two words in relation is bounded. These relations can be seen as the sets obtained by means of computations of 2 tape automata with bounded delay which are also equivalent with letter toletter 2 automata with terminal ....

C.C. Elgot and J.E. Mezei. On relations defined by generalized finite automata. In IBM J. Res. Develop.. Nber 9, pp 47--68, 1965.

....criteria, namely that automatic structures are countable and that their first order theory is decidable, not much is known. The only non trivial criterion that is available at present use growth rates for the length of the encodings of elements of definable sets. Proposition 4. 1 (Elgot and Mezei [8]) Let A be an automatic structure with injective presentation (n;d) and let f : A n A be a function of A. Then there is a constant m such that l d ( f ( a) l d ( a) m for all a 2 A n . The same is true if we replace f by a relation R where for all a there are only finitely ....

C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM J. Res. Develop., 9:47--68, 1965.

....family of subsets of M containing the finite subsets and closed under union, product and the star operation is denoted by RatM ; its elements are the rational sets of M . The following generalization of Kleene s theorem is due to Elgot and Mezei (cf. 16] for more details) Theorem 1. 1 [5] A subset of M is rational if and only if it is the behaviour of a finite automaton over M , the labels of the edges of the automaton being taken in any set of generators of M . 6 Representation of deterministic 2 tape automata The set E of labelled edges of an automaton A = Q;M;E; I ; T is ....

C.C. Elgot and J.E. Mezei, On relations defined by generalized finite automata. IBM Journal Res. and Dev. 9, 1965, 47--68.

.... three subsets Q 0 , QA , and QB , E 0 , EA , and EB , respectively, such that: E 0 Q 0 Theta A Theta B Theta Q EA QA Theta A Theta ffflg Theta QA EB QB Theta ffflg Theta B Theta QB (4) The class Syn(A Theta B ) of synchronized rational relations over A Theta B [3] (see also [4] is defined as the class of relations accepted by synchronized rational transducers. The following proper inclusions, where A and B are assumed non empty, are well known (where Rat(A ) Theta Rat(B ) stands for the family of subsets of the form X Theta Y with X 2 Rat(A ) ....

C. Elgot and J. Mezei. On Relations Defined by Generalized Finite Automata. IBM J. Res. Develop., 9:47--68, 1965.

....a remark on finite transducers 1 . He first defined a transducer to be the composition of what we call now a left sequential function by a right sequential functions. And he proved that such mappings from a free monoid into another one are closed under composition. Few years later, in a paper ([4]) that received less attention that it deserved 2 Elgot and Mezei proved that rational relations are closed under composition and, moreover, that the transducer defined by Schutzenberger is indeed the model of computation for the rational functions i.e. Theorem 1 [4] Decomposition Theorem. ....

.... years later, in a paper ( 4] that received less attention that it deserved 2 Elgot and Mezei proved that rational relations are closed under composition and, moreover, that the transducer defined by Schutzenberger is indeed the model of computation for the rational functions i.e. Theorem 1 [4] Decomposition Theorem. Any rational function is the product of a left sequential function by a right sequential function. 3 To tell the truth, the original proof of Decomposition Theorem in [4] is rather hard to follow. It has thus been completely reworked by Eilenberg and Schutzenberger who ....

[Article contains additional citation context not shown here]

C.C. Elgot and J.E. Mezei, On relations defined by generalized finite automata. IBM Journal Res. and Dev. 9, 1965, 47--68.

....criteria, namely that automatic structures are countable and that their first order theory is decidable, not much is known. The only non trivial criterion that is available at present use growth rates for the length of the encodings of elements of definable sets. Proposition 4. 1 (Elgot and Mezei [8]) Let A be an automatic structure with injective presentation (#, d) and let f : A n # A be a function of A. Then there is a constant m such that # d (f(a) # # d (a) m for all a # A n . The same is true if we replace f by a relation R where for all a there are only finitely ....

C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM J. Res. Develop., 9:47--68, 1965.

.... as we have already noted, the lengths of f and of g are approximately equal the difference of these lengths is indeed bounded by 1 and this property implies that S is a rational set in f0; 1g Theta f0; 1g if, and only if, it is a rational set in (f0; 1g Theta f0; 1g) cf. [7, 6, 10]) Such a statement will be made more intelligible by means of the following convention. Every element of J = f0; 1g Theta f0; 1g will be written as a vertical double digit : J = f 0 0 ; 0 1 ; 1 0 ; 1 1 g . Any element of J can be read as the superposition of two words of equal length, ....

....issue dedicated to Marcel Paul Schutzenberger Thus, in both cases, N mff k 1 = Gamma 1) It follows that N k p holds, with p = blog (mff = Gamma 1) c 2. And then, recalling that k uN N , it holds : jf j Gamma 1 j ;D (f)j jf j p . It is then a known result (cf. [7], 10, Cor. 2.5] that a relation with bounded length difference that is realized by a finite two tape automaton is realized by a letter to letter finite two tape automaton. And the proof of Theorem 2 assuming Theorem 3 is thus complete. The results established in this section call for ....

C.C. Elgot and J.E. Mezei, On relations defined by generalized finite automata. IBM Journal Res. and Dev. 9, 1965, 47--68.

....of rational languages turns out to be one of the most important classes within the Chomsky hierarchy. Finite automata that are the main object for studying rational languages are now used in the most domains of computer science. Rational transductions introduced by C. C. Elgot and J. E. Mezei [5] are a natural extension of rational languages and were very useful to represent several kinds of computations. For instance, the addition of two integers in any base can be realized by a rational transduction and the study of these tools has led to efficient parallel algorithms (see [2, 15] for ....

..... By the same way, right sequential transducers are defined. The input word is then read from right to left. For more details, see [3, 4] for example. A sequential transducers are kind of deterministic transducers without accepting states (all of them are accepting) The decomposition theorem [1, 4, 5, 17], characterizes rational functions by using sequential functions: a rational function including ( is the composition of a right sequential transduction followed by a left sequential transduction. We use this theorem in order to deal with deterministic functions and then we can prove the ....

Elgot, C. C., and Mezei, J. E. On relations defined by generalized finite automata. IBM Journal of Research and Development 9 (1965), 47--68.

....and H is a finite subset of fQ Theta ( Sigma ) n Theta Qg, where ( Sigma ) n is the set of n tuples of (possibly empty) words over Sigma. Thus A defines a labeled directed graph, whose nodes are elements of Q, and whose edges are elements 1 See Rabin Scott [28] and Elgot Mezei [8]. of H. Each edge is labeled with an n tuple of (possibly empty) words. The component wise concatenation of labels along every path that begins in q 1 and ends in an element of F defines a set of n tuples, R ( Sigma ) n , which is the relation transduced by A. The information content of ....

C. C. Elgot and J.E. Mezei. On Relations Defined by Generalized Finite Automata, IBM Journal Res. 9:47-68, 1965.

....was introduced by Roche (1993) MERL TR 94 07. Version 3.0 March 9 transducer. This corresponds to the formal operation of composition defined on transducers. The formalization of this notion and an algorithm for computing the composed transducer are well known and are described originally by Elgot and Mezei (1965). Returning to our running example of Section 2, the transducer obtained by composing the local extension of T 2 (right in Figure 5) with the local extension of T 1 (right in Figure 4) is shown in Figure 6. np np vbd vbn vbd vbd 0 vbn vbn vbn vbd vbd vbn vbd vbd np np 1 by by 2 ....

Elgot, C. C. and J. E. Mezei. 1965. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9:47--65, January.

....several finite state transducers. In particular, all the lattices in the above ASR cascade can be considered to be finite state transducers, and each process in the cascade performs a generalized composition operation between its inputs. Berstel et al. 3, 4] Eilenberg [15] and Elgot and Mezei [16] give extensive treatments of the theory of rational transductions and languages and its correspondence to automata. For now, it should be clear, again, that smaller lattices will lead to faster compositions and thus faster ASR. Analogues to classical automata determinization and minimization have ....

C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9:47--68, 1965.

....and the relation it defines. The family of rational transductions is closed by ffl inversion (simply reverse the elements of each digram) ffl concatenation (to build f:g, connect the terminal states of f to the initial states of g by edges bearing the null digram hffl; ffli. ffl composition [EM65] Many subfamilies of fsa and gsa have been defined in the literature: deterministic fsa, length preserving gsa, rational functions, etc. We will have no use for such special cases in this work. 1.4 The Control Domain of Recursive Programs Let us consider the following contrived example (the ....

....that x and y are iterations of S k and S l which are generated by one and the same call to foo, 3) and (4) expressing the fact that both x and y access location w. The first step is to eliminate w, giving hx; yi 2 k = g Gamma1 ffi f . k is a rational transduction by Elgot and Mezei s theorem [EM65] We thus see that the pair hx; yi belongs to the intersection of the two transductions h and k. Deciding whether the intersection of two transductions is empty is a well known undecidable problem [Ber79] Post correspondence problem can be reduced to it. Nevertheless, it is possible to define a ....

C.C. Elgot and J.E. Mezei. On relations defined by generalized finite automata. IBM J. of Research and Development, pages 47--68, 1965.

....sliding block map which defines the same map between orbits of bi infinite sequences. The goal of the paper is to synchronize transducers while keeping the local property of its input automaton. The question of the synchronization of transducers goes back to the paper of Elgot and Mezei [10] about rational relations realized finite automata, and the result of Eilenberg and Schutzenberger [9] which states that a length preserving rational relation of A Theta B is a rational subset of (A Theta B) or, equivalently, is realized by a synchronous automaton (labeled in A Theta ....

Elgot, C. C., and Mezei, J. E. On relations defined by generalized finite automata. IBM Journal Res. and Dev. 9 (1965), 47--68.

....the system. Section 6 illustrates how the implementationcan be used in compiling rewrite rules into automata. Finally, section 7 gives concluding remarks. 2 Multi tape Automata and Regular Relations 2. 1 Definition Multi tape finite state machines were first described by (Rabin and Scott, 1959) and (Elgot and Mezei, 1965). Here, we adopt a definition of multi tape machines based on traditional transducers. An n tape finite state automaton is a 5tuple M = Q; Sigma ; ffi; q 0 ; F ) where Q is a finite set of states, Sigma is a finite input alphabet, ffi is a transition function mapping Q Theta Sigma ffl ....

....regular n relation if and only if it is accepted by an ffl free FST. Further, they demonstrate that regular n relations are closed under composition, whilst the subclass of samelength regular n relations is closed under intersection and complementation. 4 Also called transductions by (Elgot and Mezei, 1965; Nivat, 1968) and rational relations by (Eilenberg, 1974) and other earlier writers. The computational linguistics community uses the term regular relations (Kaplan and Kay, 1994) In what follows, we base our discussion on (Kaplan and Kay, 1994) 3 A Finite State Calculus For practical ....

Elgot, C. and J. Mezei. 1965. On relations defined by generalized finite automata.

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C.C. Elgot and G. Mezei. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9:47--65, 1965.

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C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM J. Research and Development, 9:47 -- 68, 1965.

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C. C. Elgot and J. E. Mezei, On relations defined by generalized finite automata, IBM Journal of Research Development, 9 (1965), 47--68.

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