J. A. Brzozowski. Derivatives of regular expressions. J. ACM, 11(4):481--494, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can impose a mixed automaton structure on the set of all mixed languages with type A #. We take as states mixed languages of type #. A state is accepting if the empty string # is in the language. It remains to define the transitions between states; we adapt the idea of Brzozowski derivatives [1]. Our definition of derivative depends on whether we are taking the derivative with respect to a program element or a literal. If the mixed language L has type # B and p is a primitive program, define D p (L) p L . If the mixed language L has type A B (for A #) and l ....

Brzozowski, J. A., Derivatives of regular expressions, Journal of the ACM 11 (1964), pp. 481--494.

.... out a better set of primitives and made much better sense of instantaneous control propagation and communication, which is the key to get Esterel right[11, 24] The author rediscovered the beautiful derivative algorithm of Brzozowski that translates any kind of regular expression into automata [20] and that can be extended to any finite state language described by SOS rules. Using this algorithm, Laurent Cosserat wrote the first Esterel v1 prototype compiler to automata. That compiler was entirely rewritten by Philippe Couronn e and the author in 1985 86, and the new Esterel v2 system was ....

J. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4), 1964.

....is familiar both from a regular expression context and from CSP. We will use the CSP notation for it. 2.1.3 De nition: If q E is a set of event sequences, and 2 E ; q= f j 2 qg: In CSP, q= is called q after . For regular expressions, this is the Brzozowski derivative [Brz64]. It strips o all elements of q that had as a pre x, leaving the tails. The derivative operation has some obvious elementary properties which we will use without proof, such as (q= q= 2.1.4 De nition: event system acceptor M is an acceptor for an event system S = E; I ; O; T ) ....

J. A. Brzozowski, \Derivatives of Regular Expressions," J. ACM, Vol. 11, No. 4 (Oct. 1964), pp. 481-494

.... out a better set of primitives and made much better sense of instantaneous control propagation and communication, which is the key to get Esterel right[11, 24] The author rediscovered the beautiful derivative algorithm of Brzozowski that translates any kind of regular expression into automata [20] and that can be extended to any nite state language described by SOS rules. Using this algorithm, Laurent Cosserat wrote the rst Esterel v1 prototype compiler to automata. That compiler was entirely rewritten by Philippe Couronn# and the author in 1985 86, and the new Esterel v2 system was ....

J. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4), 1964.

....it is denoted by a regular expression. The Kleene s Theorem states that the class of regular languages Reg( is identical to Rec( 2. 2 Residual languages and RFSAs For any language L and any word u over , the residual language of L associated with u (also called Brzozowski derivative [Brz64] or left quotient) is de ned by u L = fv 2 j uv 2 Lg and we say that u is a characterizing word for u L. The number of residual languages of a language L is nite if and only if L is regular (Myhill Nerode theorem) A residual language is composite if it is equal to the union of the ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11:481-494, 1964.

....functionallogic programming languages [7] in the implementation of the type checker. Using this approach, we proceed as follows. Instead of precompiling the regular expressions to nite automata and hardcoding them in the type structure, we take up the idea of derivatives of regular expressions [2] to avoid the explicit construction of a nite automaton. Brzozowski [2] shows that starting from a regular expression r and an input symbol a it is possible to compute another regular expression d(r; a) the a derivative of r) such that L(d(r; a) a n L(r) fw j aw 2 L(r)g. Iterating this ....

....type checker. Using this approach, we proceed as follows. Instead of precompiling the regular expressions to nite automata and hardcoding them in the type structure, we take up the idea of derivatives of regular expressions [2] to avoid the explicit construction of a nite automaton. Brzozowski [2] shows that starting from a regular expression r and an input symbol a it is possible to compute another regular expression d(r; a) the a derivative of r) such that L(d(r; a) a n L(r) fw j aw 2 L(r)g. Iterating this construction yields a nite set of regular expressions that is closed under ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-494, 1964.

..... The Kleene theorem proves that the class of regular languages Reg( is identical to Rec( 2. 2 Residual languages and RFSA For any language L and any word u over , we note u L = fv 2 j uv 2 Lg the residual language of L associated with u (also called Brzozowski derivative [Brz64]) The set of distinct residual languages of any regular language is nite (Myhill Nerode theorem) A residual language is composed if it is equal to the union of the residual languages it strictly contains i.e. u L is composed if and only if u L = fv L j v L ( u Lg . A residual ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11:481-494, 1964.

....a canonical construction for the deterministic automaton corresponding to a regular language. Its states are languages of the form wnR, and the transition function is (R; a) 7 anR. Cognoscenti will recognise this as the construction of a deterministic automaton via derivatives, due to Brzozowski [4]. Given a pattern S , name this nite automaton S . Given another language R, mark all states in S that are reachable from an initial state via a path that is in R. By the above construction, each state is in fact a language. Take the intersection of all such languages X , where X is a ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11:481-494, 1964.

....expressions (REGs) over M . We denote by [ the language associated to a regular expression , and we write ; for the regular expression denoting an empty set of words. To compactly represent the result of receiving a message from the front of a channel we will use the notion of derivatives [8]. Formally, given a message m 2 M , we define the derivative operator m : REG(M) REG(M) as follows: if = m ; if = or if = m 2 M with m 6= m ( 2 ) if = 1 2 ( 1 ) 1 if = Proposition 3.1. ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....to its small size. However, we need a final confirmation step to find the actual matching strings. It will be interesting to compare the performance of these two approaches. Finite automata There is a large body of literature studying how to match a regex to a string (see the textbook [17] and [22, 9, 25, 6] for instance) The approach is to first convert a regex into an equivalent deterministic finite automaton (DFA) and then use the DFA to match the regex. To expedite the matching, most systems allow the user to save the constructed DFA, so that the user can reuse it when she wants to match the ....

.... = is a zero length string [2] While (expand is not empty) 3] k grams : all k grams in database whose (k 1) prefix expand [4] expand : 5] For each gram x in k grams [6] If sel(x) c Then check selectivity [7] insert(x, index) the gram is useful [8] Else [9] expand : expand # x [10] k : k 1 Figure 4. Construction of a multigram index Theorem 3.9 Let X be the set of grams indexed by algorithm 3.1. 1. If x X , then x is useful. 2. Conversely, if x is useful, then either x X or there is a unique prefix x # of x such that x # X . 3. X ....

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Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....therefore serve as basis for an ESTEREL interpreter. However, this interpreter would not be fast enough for actual real time applications. Our next step is to compile ESTEREL programs into sequential automata. This is the purpose of section 7. We use a variant of Brzozowski s derivative algorithm [17, 10], which was originally designed to transform regular expressions into automata. The idea is to formally iterate the computational semantic calculations, building a graph whose nodes are ESTEREL terms and whose arcs bear the action sequences. Starting from a node bearing the initial program, we ....

....program stops; its body becomes the statement halt that accepts input but never produces output. The derivative technique transforms a temporal problem into two instantaneous ones: find the instan taneous reaction on an input and find the derivative. The technique was introduced by Brzozowski [17] to compute the automaton recognizing the language generated by a regular expression. 6.3. Inductive rules The relation between programs is deduced from a similar relation between statements, which is defined by deduction rules that determine the transition of any ESTEREL construct from the ....

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J. A. BRZOZOWSKI, Derivatives of Regular Expressions, JACM, vol. 11, no. 4 (964).

....at runtime, and hence is not a serious issue in practice. We are currently working on techniques that can avoid unbounded growth by restricting the class of patterns permitted in ASL. The starting points for our algorithm for generating EFSA from ASL patterns are the seminal papers by Brzozowski [Brzozowski64] and Berry and Sethi [Berry86] However, these papers address regular expressions and classical FSA, whereas we must address conditions on event arguments and state variables that can be complex data structures. Our earlier work on first order term matching [Sekar95] provides the starting point ....

J.A. Brzozowski, Derivatives of Regular Expressions, Journal of ACM Vol. 11, No.4, pp. 481-494, 1964.

....regular expressions should not take long. A lazy TDFA generating algorithm might also be acceptable, but would be much more complex and use a lot more memory, so I decided to go ahead with a TNFA implementation. There are numerous methods for converting regular expressions to nite automata [8, 9, 10, 46, 36], making an NFA matcher run faster [2, 41] reducing the space requirements for the transition tables [4, 5, 12, 17, 52] and other useful methods and tricks [18, 42, 53] Most of these are probably applicable to TNFAs and TDFAs perhaps with slight modi cations. 4.1 Sacri cing Complexity Any NFA ....

J. A. Brzozowski. Derivatives of regular expressions. J. ACM, 11(4):481494, Oct. 1964.

....until all states, when rewritten with every possible set of input signals, take transitions to other established states. When this process is completed, the state labels can be discarded. At each step, in 8 effect, the derivative of the state machine with respect to some input symbols is taken[7]. To keep the number of states within reason, all data dependent actions are treated separately. When a transition (the execution of a program in an instant) affects memory, perhaps by evaluating an expression, the transition is labeled with that expression. At run time, the expression is ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the Association for Computing Machinery, 11(4):481--494, October 1964.

....a regular expression that speci es in which sequence elements can legally be added. Initially, this expression is the content speci cation from the DTD. Whenever a new element is added to the contents, the add function computes the next state by taking the derivative of the regular expression [4] (this approach uses regular expressions as the set of states of a nite automaton) In addition, the code checks that the new content element is complete by demanding that its state is a nal state. The following code fragment illustrates the idea: 20 add (ELT state e ) ELT state e ) ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-494, 1964.

....for acceptance testing and DFA construction from regular expressions are implemented in the UNIX operating system[26] Throughout this thesis our model of computation is a uniform cost sequential RAM [1] We report the following six results. 1. Berry and Sethi[5] use results of Brzozowski[8] to formally derive and improve McNaughton and Yamada s algorithm[18] for turning regular expressions into NFA s. NFA s produced by this algorithm have fewer states than NFA s produced by Thompson s algorithm[30] and are believed to outperform Thompson s NFA s for acceptance testing. Berry and ....

Brzozowski, J., "Derivatives of Regular Expressions", JACM, Vol. 11, No. 4., Oct. 1964, pp. 481-494.

....schemas into first normal form causes an additional blow up of jN jjGj. We use the derivative dE dX of a regular expression E by a symbol X to give a new expression F such that L(F ) fy : Xy 2 L(E)g and fXgL(F ) L(E) The derivative of a regular expression was introduced by Brzozowski [3] who defined it inductively. Now, given a schema EA , we obtain its derivatives for each symbol X 2 N [ Sigma. When we catenate X with its derivative we obtain one of the terms in the first normal form. Since G is null free and unit free, the only derivative that can cause exponential blow up is ....

J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11:481--494, 1964.

....authors seem to provide their own variant of the construction. Of these, Berry and Sethi [BS86] have shown that the construction of an ffl free NFA due to to Glushkov [Glu61] is a natural representation of the regular expression, because it can be described in terms of the Brzozowski derivatives [Brz64] of the expression. Moreover, the Glushkov construction also plays a significant role in the document processing area: The SGML standard [ISO86] now widely adopted by publishing houses and government agencies for the syntactic specification of textual markup systems, uses deterministic regular ....

.... whose states correspond to the occurrences of symbols in E and whose transitions connect positions that can be consecutive on a path through E [BEGO71, ASU86] Recently, Berry and Sethi have shown that the Glushkov con2 struction ME is related in a natural way to the Brzozowski derivatives of E [Brz64, BS86]. None of the cited papers considers, however, the time complexity of constructing ME . A straightforward implementation takes time cubic in the size of the expression, as opposed to the quadratic time of the standard construction. In this paper we provide a quadratic time algorithm (Theorem 3.6) ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

.... resulting NFA into a deterministic finite automaton (DFA) HU79] The intermediate step can be avoided by directly constructing an ffl free automaton [BEGO71, ASU86] It has been claimed [BS86] that this NFA is the canonical representation because it has a natural connection with the derivatives [Brz64] of the original expression. Since it takes exponential time in the worst case to convert an NFA into a DFA, it is natural to ask for which regular expressions E the canonical NFA ME is already deterministic. Such expressions are exactly what we have called deterministic above. It can be tested in ....

....the language of F , and let v be a word that leads from the initial state to q in M F . Since M F is closed within ME , the orbit language of q in ME is also the orbit language of q in M F , which in turn is vnL : fw 2 Sigma j vw 2 Lg: The language vnL is known as the derivative of L by v [Brz64]. Thus, a non trivial orbit language of ME is a derivative of a language denoted by a maximal starred subexpression of E. The proof of the next proposition is in the full paper. Proposition 3.7 The derivative of a deterministic regular language is also deterministic. As a corollary, we have: ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....wants to compute a function f(E 1 ; E 2 ; E n ) which approximates E. As far as the authors know, there are two methods for computing such a function f which approximates E from below. The first one of Conway [Con71] is based on the derivatives of regular expressions introduced by Brzozowski [Brzo64], which provide the ground for the development of an algebraic theory of factorization in the regular algebra [BL80] which in turn gives the tools for computing the approximating function. The second method by Calvanese et al. [CGLV99] is automata based. Both methods are equivalent in the sense ....

J. A. Brzozowski. Derivatives of Regular Expressions. JACM 11(4) 1964, pp. 481-494 13

....the sets of streams, languages, and formal power series each carry a final automaton structure, plays a central role. In all cases, the transitions of the final automaton are determined by the notion of input derivative (and initial value) This notion, which already occurs in work of Brzozowski [Brz64] and Conway [Con71] can be understood, in a very precise sense, as an abstract generalisation of the analytical notion of function derivative. Finality gives rise to both a coinduction definition and a coinduction proof principle, formulated in terms of derivatives. It is exactly this use of ....

....power series, which was a generalisation of [Rut98a] where automata and languages were treated coinductively. General references on the coalgebraic approach are [Rut96] and [JR97] Our notion of input derivative for power series generalises Brzozowski s original definition for regular expressions [Brz64, Con71]. In addition to our own earlier work, the presentation of the calculus for streams has been influenced by [PE98] which gives a coinductive treatment of analytic functions in terms of their Taylor series, and by [McI99] which treats power series (in one variable) as lazy lists in the programming ....

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J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, 1964.

....expression R accepts the empty word [ next(R; S; R1) holds iff S is a symbol in fa; bg and for every word W in fa; bg , W is accepted by the regular expression R iff there exists a word Z accepted by the regular expression R1 such that W = S; Z. Thus, R1 is the derivative of R w.r.t. S [Brzozowski 64] R1 is computed as the following three steps specify. i) First the predicate transf (R; TR) transforms the regular expression R into an equivalent regular expression TR in the set TReg defined as follows: TReg : void j void j a j b j (a; Reg) j (b; Reg) j (TReg TReg) where void is the ....

Brzozowski, J. A.: Derivatives of Regular Expressions. JACM 11 (4) (1964) 481--494

....automaton (NFA) GE that recognizes the language of E. They show that E is unambiguous if and only if GE is unambiguous. Berry and Sethi [BS86] show that this NFA is the canonical representation of the corresponding regular expression, because it has a natural connection with the derivatives [Brz64] of the regular expression. Regular expressions are built with the usual operators , Delta, and . SGML, however, deals with model groups that may also contain the operators , and . E denotes L(E ffl) F G denotes L(FG GF ) and E denotes L(EE ) Whereas the transformations ....

....multiplicity 1, denotes only words of multiplicity 1; that is, if and only if its language is unambiguous in the sense of Eilenberg. One Unambiguous Regular Languages 7 3 Derivatives of 1 unambiguous languages We now prove that the family of 1 unambiguous languages is closed under derivatives [Brz64]. This result is essential for characterizing the 1 unambiguous languages. The proof makes use of a linear time algorithm to convert regular expressions into star normal form [BK92a, BK93c] We use the same technique, in Section 4, to obtain a Kleene theorem for 1 unambiguous languages. ....

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J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....definitions and proofs by coinduction, which is the coalgebraic counterpart of induction. Coinductive definitions of operators on formal power series take the shape of what we have called behavioural di#erential equations , since they are formulated in terms of (a generalization of) Brzozowski s [Brz64] notion of input derivative: the input derivative # a of a series # can intuitively be understood as the specification of the behaviour of # after the input a has been accepted. For instance, the following behavioural di#erential equation defines the input derivative (# # #) a of the so called ....

....which deals with languages and regular expressions. Our way of solving di#erential equations is essentially based on the processes as terms methodology, used in [RT94] The notion of input derivative of formal power series, generalizes Brzozowski s original definition for regular expressions [Brz64, Con71]. Its relation with function derivatives f # of functions f on IR will be explained by invoking an example from [PE98] where a coinductive treatment of analytic functions in terms of their Taylor expansions is given. Our present theory generalizes the settings of [Rut98] k = IB and A is ....

J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, 1964.

....all authors seem to provide their own variant of the construction. Of these, Berry and Sethi [BS86] have shown that the construction of an ffl free NFA due to Glushkov [Glu61] is a natural representation of the regular expression, because it can be described in terms of the Brzozowski derivatives [Brz64] of the expression. Moreover, the Glushkov construction also plays a significant role in the document processing area: The SGML standard [ISO86] now widely adopted by publishing houses and government agencies for the syntactic specification of textual markup systems, uses deterministic regular ....

.... E, whose states correspond to the occurrences of symbols in E and whose transitions connect positions that can be consecutive on a path through E [BEGO71, ASU86] Recently, Berry and Sethi have shown that the Glushkov construction ME is related in a natural way to the Brzozowski derivatives of E [Brz64, BS86]. None of the cited papers considers, however, the time complexity of constructing ME . A straightforward implementation takes time cubic in the size of the expression, as opposed to the quadratic time of the standard construction. In this paper we provide a quadratic time algorithm that is ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....role in the characterization of the automaton L of partial languages . These partial languages are actually pairs of languages (subsets of words) and represent what in control theory are called the marked behaviour and the closed behaviour of automata. Using the notion of input derivative [Brz64, Con71], the set L can be supplied with an automaton structure, which is final among all automata. The finality of L gives rise to both definitions and proofs by coinduction. Since this will be our main instrument for tackling the problems of controllability, relatively much time is spent on the ....

....in coalgebra. Sections 5 through 7 repeat and extend parts of [Rut98] There the use of a derivatives in coinductive definition and proof principles for (ordinary) languages L # A # was introduced, again following the general coalgebraic approach but also inspired by Brzozowski s paper [Brz64], where a derivatives seem to appear first, and Conway s book [Con71] Although [Rut98] briefly mentions partial automata, the present characterization of a final partial automaton in terms of partial languages in Section 5 is new, as is the coinductive definition of the supervised product. Our ....

J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-- 494, 1964.

....that lead from a regular expression to its Glushkov automaton. The first route is to construct, for each regular expression, an NFA for it inductively [Mir66, Lei81] Champarnaud [Cha92] proves that the NFA thus obtained is isomorphic to the Glushkov automaton. The second route is via derivatives [Brz64]. Berry and Sethi [BS86] prove that this method also leads to the Glushkov automaton. Finally, there is Thompson s well known construction [Tho68, ASU86] of an NFA with ffl transitions from a regular expression; there are other similar methods [HU79, AO83, Woo87, SS88] that also result in NFAs ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....and all authors seem to provide their own variant of the construction. Of these, Berry and Sethi [4] have shown that the construction of an ffl free NFA due to Glushkov [10] is a natural representation of the regular expression, because it can be described in terms of the Brzozowski derivatives [7] of the expression. Moreover, the Glushkov construction also plays a significant role in the document processing area: The SGML standard [13] now widely adopted by publishing houses and government agencies for the syntactic specification of textual markup systems, uses deterministic regular ....

.... of E, whose states correspond to the occurrences of symbols in E and whose transitions connect positions that can be consecutive on a path through E [5, 2] Recently, Berry and Sethi have shown that the Glushkov construction ME is related in a natural way to the Brzozowski derivatives of E [7, 4]. None of the cited papers considers, however, the time complexity of constructing ME . A straightforward implementation takes time cubic in the size of the expression, as opposed to the quadratic time of the standard construction. In this paper we provide a quadratic time algorithm (Theorem 3.9) ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

.... resulting NFA into a deterministic finite automaton (DFA) HU79] The intermediate step can be avoided by directly constructing an ffl free automaton [BEGO71, ASU86] It has been claimed [BS86] that this NFA is the canonical representation because it has a natural connection with the derivatives [Brz64] of the original expression. Since it takes exponential time in the worst case to convert an NFA into a DFA, it is natural to ask for which regular expressions E the canonical NFA ME is already deterministic. Such expressions are exactly what we have called deterministic above. It can be tested in ....

....language of F , and let v be a word that leads from the initial state to q in M F . Since M F is closed within ME , the orbit language of q in ME is also the orbit language of q in M F , which in turn is vnL : fw 2 Sigma j vw 2 Lg: The language vnL is known as the derivative of L by v [Brz64]. Thus, a non trivial orbit language of ME is a derivative of a language denoted by a maximal starred subexpression of E. The proof of the next proposition is in the full paper. Proposition 3.7 The derivative of a deterministic regular language is also deterministic. As a corollary, we have: ....

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....Its application to languages and regular expressions, in Sections 6 and 10, is to the best of our knowledge new. The calculation rules for a derivatives (Section 5) of regular combinations of languages are well known, have been reinvented several times, and are originally due to Brzozowski [Brz64] (see also [Con71] and [BS86] Both Brzozowski s paper [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 ....

....Sections 6 and 10, is to the best of our knowledge new. The calculation rules for a derivatives (Section 5) of regular combinations of languages are well known, have been reinvented several times, and are originally due to Brzozowski [Brz64] see also [Con71] and [BS86] Both Brzozowski s paper [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] ....

J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-- 494, 1964.

....expressions (REGs) over M . We denote by [ the language associated to a regular expression , and we write ; for the regular expression denoting an empty set of words. To compactly represent the result of receiving a message from the front of a channel we will use the notion of derivatives [7]. Formally, given a message m 2 M , we define the derivative operator m 1 : REG(M) REG(M) as follows: m 1 ( 8 : if = m ; if = or if = m 0 with m 0 6= m m 1 ( 1 ) 2 if = 1 2 and 62 [ m 1 ( 2 ) m 1 ( 1 ) 2 if = 1 ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

....wants to compute a function f(E1 ; E2 ; En) which approximates E. As far as the authors know, there are two methods for computing such a function f which best approximates E from below. The first one of Conway [12] is based on the derivatives of regular expressions introduced by Brzozowski [7], which provide the ground for the development of an algebraic theory of factorization in the regular algebra [8] which in turn gives the tools for computing the approximating function. The second method by Calvanese et al. [9] is automata based. Both methods are equivalent in the sense that they ....

J. A. Brzozowski. Derivatives of Regular Expressions. J. ACM 11(4) 1964, pp. 481--494

....acceptance testing and DFA construction from regular expressions are implemented in the UNIX operating system[17] Throughout this paper our model of computation is a uniform cost sequential RAM [1] We report the following four results. 1. Recently Berry and Sethi[5] used results of Brzozowski[6] to formally derive and improve McNaughton and Yamada s algorithm[13] for turning regular expressions into NFA s. NFA s produced by this algorithm have fewer states than NFA s produced by Thompson s algorithm[18] and in practice they are known to outperform Thompson s NFA s for acceptance ....

Brzozowski, J., "Derivatives of Regular Expressions", JACM, Vol. 11, No. 4., Oct. 1964, pp. 481-494.

....pieces of length one from the extended regular expression. Let Delta be an alphabet. We define, for each a 2 Delta, a function D a : E Delta 2 Delta by D a (e) fx 2 Delta j ax 2 L(e)g: Here, D a (e) is a restricted version of Brzozowski s derivatives of extended regular expressions [1]. We also denote, for e 2 E Delta , e) if 2 L(e) if 62 L(e) Let e 1 ; e 2 2 E Delta and a 2 Delta. The following equalities are clear: i) D a ( ii) D a ( iii) D a (a Delta e 1 ) L(e 1 ) iv) D a (b Delta e 1 ) for b 2 Delta and b 6= a; v) D a ....

....to produce the equational representation Eqr(e) of an AFA equivalent to a given arbitrary extended regular expression e 2 E Delta (containing also operators. Naturally, in the general case, the corresponding AFA will not be loop free. The construction relies on the below result from [1], which is a variant of the well known Myhill Nerode Theorem [14] For e 2 E Delta , we define the set of derivatives of e as Der(e) fD a 1 ( Delta Delta Delta (D a k (e) Delta Delta Delta) j k 0; a 1 ; a k 2 Deltag: Theorem 4.2 ( 1] Theorem 4.3) For each e 2 E Delta , ....

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J.A. Brzozowski, "Derivatives of Regular Expressions", Journal of the ACM 11:4 (1964) 481494.

....of traversal speci cation. While this is more general than their earlier work, the relation to our speci cations is not clear. 2 The algebraic approach is based on a notion of derivatives which is closely related to quotients of formal languages [9] and to derivatives of regular expressions [6, 7, 4]. However, traversal speci cations di er from standard regular expressions, so our derivatives are novel to this work. 1.2 Contribution of this work The algebraic foundations of adaptive programming are based on the algebraic properties of traversal speci cations. Exploiting the algebraic laws, ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-494, 1964.

....spread over the code implementing the traversal. They employ a di erent notion of traversal speci cation than in their earlier work. The algebraic approach is based on a notion of derivatives which is closely related to quotients of formal languages [9] and to derivatives of regular expressions [6, 7, 4]. The formal underpinnings of this approach are elaborated in a companion paper [23] Evaluation algorithms for attribute grammars [13, 14] also rely on traversals of tree structures. These traversals are usually speci ed indirectly as dependences 2 between attribute values. So the paths are ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481-494, 1964.

....) and computing a sequence b 0 ; b 1 ; b n ; of Booleans, such that b n is true if and only if the word s 0 s 1 : s n belongs to L 2 . This paper addresses the problem of building a grep machine for languages described by regular expressions. This problem is rather classical [4, 11, 10, 3, 1, 2]. We propose a solution which, to our knowledge, is new: Informally, it consists of building, from a regular expression E, a circuit (or Boolean data flow network) exploring all the branches of a non deterministic automaton recognizing L(E) The relations between regular languages and ....

.... Delta Y 1 : an Delta Yn [ all the symbols a i are different, the automaton is deterministic, and each Y i is said to be the a i derivative of X . Such a deterministic system can be built by computing derivatives of regular expressions. Brzozowski has formally defined this construction [4]. The size of the resulting deterministic automaton (and thus the cost of the construction) is, in the worst case, exponential with respect to the size of the regular expression (number of operators in the regular expression) Non deterministic automaton (NFA) If in each equation X = a 1 Delta ....

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

No context found.

J. A. Brzozowski. Derivatives of regular expressions. J. ACM, 11(4):481--494, 1964.

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Janusz A. Brzozowski. Derivatives of regular expressions. JACM, 11(4):481--494, 1964.

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J. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4), 1964.

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J.A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, 1964.

No context found.

J.A. Brzozowski. Derivatives of Regular Expressions. Journal of the ACM, 11(4):481--494, October 1964.

No context found.

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, October 1964.

No context found.

J. A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, 1964.

No context found.

J. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4), 1964.

No context found.

Janusz A. Brzozowski. Derivatives of regular expressions. Journal of the ACM, 11(4):481--494, 1964.

No context found.

J.A. Brzozowski. Derivatives of regular expressions. Journal of the Association for Computing Machinery, 11:481--494, 1964.

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J. A. Brzozowski, `Derivatives of regular expressions', J. ACM, 11, (4), 481--494 (1964).

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J. A. Brzozowski. "Derivatives of Regular Expressions," Journal of the ACM, Vol. 11, No. 4, pages 481-494. October 1964.

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J.A. Brzozowski. Derivatives of regular expressions. Journal of the Association for Computing Machinery, 11:481--494, 1964.

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