A. Ko'scielski and L. Pacholski. Complexity of Makanin's algorithm. Journal of the Association for Computing Machinery, 43(4):670--684, 1996. Preliminary version in Proc. of the 31st IEEE FOCS, Los Alamitos, 1990, 824--829.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be an equation with constraints. The exponent of periodicity of E is also denoted by exp(E) It is defined by exp(E) infff exp(oe(L) j oe is a solution of E g [ f1g g: By definitions we have exp(E) 1 if and only if E is solvable. Here we show that the well known result from word equations [13] transfers to the situation here. The exponent of periodicity of a solvable equation can be bounded by a singly exponential function. Thus, in the following sections we shall assume that if E 0 is solvable, then exp(E 0 ) 2 2 . This is the content of the next proposition. Proposition 7 Let E = ....

....fl k by some k 1, fi 1 1, and fl k 1 such that still fl k = ff k fi 1 c, then we obtain new words u , v , and w with the same images under h in M and still the identity u v = w . What follows then is completely analogous to what has been done in detail in [13, 10, 11, 3]. Using the p stable normal form we can associate with an equation L = R of denotational length d together with its solution oe some linear Diophantine system of d equations in at most 3d variables. The variables range over natural numbers since zeros are substituted. In fact the number of ....

[Article contains additional citation context not shown here]

Antoni Ko'scielski and Leszek Pacholski. Complexity of Makanin's algorithm. Journal of the Association for Computing Machinery, 43(4):670-- 684, 1996.

....an n fold copy. This again simplifies the description of the algorithm sacrificing efficiency. SCU makes use of a lemma on the exponent of periodicity of a minimal solution of context unification problems, proved in [SSS98] which is a generalization of a similar result for string unification [Mak77,KP96]. An experimental implementation of stratified context unification (with exponents) was done in [Hoh97] The following result is proved in this paper: Theorem: Stratified context unification is decidable. A corollary following from [NTT00] is: Theorem: Satisfiability of one step rewrite ....

Antoni Ko'scielski and Leszek Pacholski. Complexity of Makanin's algorithms. Journal of the Association for Computing Machinery, 43:670--684, 1996.

....called Markov s problem by mathematicians in eastern countries. It is called Lob s problem by mathematicians in western countries, for example by A. Lentin and M.P. Schutzenberger [11] A solution to the string unification problem was found by Makanin [13] in 1977. Subsequent papers on this topic [18, 9, 23, 10] were concerned with finding a better description of Makanin s algorithm, closing small gaps in the proof of correctness, and studying its complexity. Context Unification. Context unification is a subproblem of linear second order unification (see below) and a generalization of string ....

A. Ko'scielski and L. Pacholski. Complexity of makanin's algorithm. Journal of the ACM, 43(4), July 1996.

....is devoted to prove some pumping like theorems of expressible languages. These are achieved by using tools of the previous sections, namely the method of filling positions of variables and synchronizing F factorizations. The latter involves methods to generalize techniques from [4] cf. also [13], which were used to prove an upper bound for the index of the periodicity of a minimal solution of a word equation. Now we are ready to state our first tool to show the nonexpressibility. Theorem 16 Let L be expressible by an equation e and let F be a property defining a synchronizing ....

....exists a pattern p(X 1 ; X s ) with s = nF (w) Gamma k variables such that it synchronizes with the F factorization of w and satisfies p( Sigma ) L. Now, we move to a proof of our second main theorem. It is based on Theorem 17 and additional considerations, similar to those in [4, 13], dealing with a particular F factorization. Let Q be a primitive word. Then it is well known that each word w 2 Sigma can be uniquely written in the form w = w 0 Q x1 w 1 : Q w k ; 3) where, for all i, ffl w i does not contain Q as a subword, ffl Q is a proper prefix of w i ....

[Article contains additional citation context not shown here]

Koscielski, A., and Pacholski, L., Complexity of Makanin's algorithm, J. ACM 43(4), 670-684, 1996.

....and E contains x Delta x = x, we call Delta an ACI symbol (I stands for idempotence) Also, Delta is called an AC1 symbol (or an ACI1 symbol) if Delta is an AC symbol (an ACI symbol respectively) and E contains the equation x Delta 1 = x where 1 is a constant belonging to L. Theorem 8. 1 ([96, 11, 17, 86]) Let E be an equational theory defining a function symbol Delta in L as an associative symbol (E contains all logical consequences of x Delta (y Delta z) x Delta y) Delta z and no other equations) The following upper and lower bounds on the complexity of the E unification problem hold: ....

....and lower bounds on the complexity of the E unification problem hold: i) this problem is in 3 NEXPTIME, ii) this problem is NP hard. Basically, all algorithms for unification under associativity are based on Makanin s algorithm for word equations [96] The 3 NEXPTIME upper bound is obtained in [86]. The following theorem characterizes other popular kinds of equational theories. Theorem 8.2 ( 82, 83] Let E be an equational theory defining some symbols as AC symbols or ACI symbols or AC1 symbol or ACI1 symbols (there can be one or more of these kinds of symbols) The theory E is assumed to ....

A. Koscielski and L. Pacholski. Complexity of Makanin's algorithm. JACM, 43(4):670--684, 1996.

....of the most complicated algorithms existing in the literature. There were several attempts to simplify it [2, 12] The algorithm has been implemented [1] During last 20 years its complexity has been improved several times: 4 NEXPTIME (composition of four exponential functions) 7, 17] 3NEXPTIME [9], 2 EXPSPACE [4] EXPSPACE [8] The exact complexity of the algorithm is still not known. Current version of the algorithm, the full version of which can be found in [5] is still very complicated. Recently, another algorithm has been proposed in [15] It works nondeterministically in time ....

.... can be concluded from the last version of the algorithm is triple exponential (Corrollary 1 in [8] Very recently a double exponential estimation for N has been proved [14] The proof does not use Makanin s approach but uses the optimal bound for the index of periodicity of a minimal solution [9]. With this estimation the algorithm in [15] places the problem in NEXPTIME. We propose the third algorithm. The full version of the algorithm requires only a proof of the upper bound for index of periodicity of a minimal solution [9] Our algorithm is the first one which is proved to work in ....

[Article contains additional citation context not shown here]

Koscielski A., Pacholski L., Complexity of Makanin's Algorithm, Journal of the ACM 43(4), 670-684.

....time for constant size alphabet. Its parallel version works in O(log 2 n) time and uses n processors. The algorithm can be extended to report all primitive squares and maximal repetitions. Our notion of order of repetition of a word allows fractional repetitions contrary to earlier definitions [8, 16]. An order of repetition of a word w is a maximal number ae such that there is a factor of w of the form u i u 0 with u 0 a prefix of u and such that ae = ju i u 0 j juj . In other words the order of repetition of w is a maximum number in form jvj period(v) where v is a factor of w ....

Koscielski A., and Pacholski L., Complexity of Makanin's algorithm, J. ACM 43(4), 670-684, 1996.

.... of the Diophantine equation in unknowns n and m (c 1 n c 2 )m c 3 n c 4 = k: A function f is D 1 type if there is a divisor type function g such that f = Theta(g) where g(n) max 1in g(i) Before stating the announced result, we need the notion of a P factorization (as defined in [KoPa] and [KMP] Consider a primitive word P 2 Sigma . It is well known that any word w 2 Sigma can be uniquely written in the form w = w 1 P k 1 w 2 P k 2 Delta Delta Delta w n P kn w n 1 (7) where n 0; k i 0, for any 1 i n and w i does not contain P 2 as a factor, for any ....

Koscielski, A. and Pacholski, L., Complexity of Makanin's algorithm, Journal of the ACM, 43(4), 670-684, 1996.

.... with w (and write u w) i there is a Thus pattern P occurs at position i in text T i P T [i: i m 1] 2 String matching with variables Matching strings with variables corresponds to word equations, whose solution is generally very hardly computable, it seems to be harder than NP hard, see [11, 10]. However word equations resulting in our situation are much simpler since values of variables are single symbols or strings whose lengths are given in unary. The equality of two strings with variables whose lengths are given in unary can be done easily in linear time, see e.g. 10] however ....

.... hard, see [11, 10] However word equations resulting in our situation are much simpler since values of variables are single symbols or strings whose lengths are given in unary. The equality of two strings with variables whose lengths are given in unary can be done easily in linear time, see e.g. [10], however pattern matching in such setting is more dicult and was not considered before. Assume initially that the values of variables are single letters, we drop this assumption later. Example. Assume P and T are as shown in the Figure 2, where x; y; z are variables and a; b; c; d; e are ....

A. Koscielski, and L. Pacholski, Complexity of Makanin's algorithm, J. ACM 43(4), 670-684, 1996.

....but several authors have contributed to lower the complexity of the original algorithm which was an exponential function of height 5. Actually, this complexity depends on the complexity of computing the minimal solutions of diophantine equations. We refer the interested reader to [Ab1] Do] and [KoPa] for the latest results on this topic. Several sofware packages have been produced which work relatively well up to length, see e.g. Ab2] 5.2 The rank of an equation One of the most direct consequences of Makanin s result is the fact that the rank of an equation may be effectively computed, ....

A. Koscielski and L. Pacholski, Complexity of Makanin's algorithm, JACM 43, 1996.

.... Makanin s algorithm for solving word equations, see [8] The time complexity of the algorithm is too high, its most efficient version works in 2 2 p(n) nondeterministic time where p(n) is the maximal index of periodicity of word equations of length n (p(n) is a singly exponential function) see [6]. The descriptional complexity is also too high. As a side effect of our results we present a much simpler algorithm. It is known that the solvability problem for word equations is NP hard, even if we consider (short) solutions with the length bounded by a linear function and the right side of ....

Koscielski, A., and Pacholski, L., Complexity of Makanin's algorithm, J. ACM 43(4), 670-684, 1996.

No context found.

A. Ko'scielski and L. Pacholski. Complexity of Makanin's algorithm. Journal of the Association for Computing Machinery, 43(4):670--684, 1996. Preliminary version in Proc. of the 31st IEEE FOCS, Los Alamitos, 1990, 824--829.

No context found.

A. Koscielski, L. Pacholski, Complexity of Makanin's Algorithm, Journal of the ACM, Vol. 43, No. 4, July 1996, pp. 670-684.

No context found.

A. Ko'scielski, L. Pacholski, Complexity of Makanin's algorithm, J. Assoc. Comput. Mach. 43 (1996) 670-684.

No context found.

Koscielski, A., and Pacholski, L., Complexity of Makanin's algorithm, J. ACM 43(4), 670--684, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC