G. S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Proceedings of Word Equations and Related Topics (IWWERT '90), volume 572 of LNCS, pages 1-11, Berlin;Heidelberg;New York, 1992. Springer.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92, PR98]. It has been shown, however, that any non xed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non xed size equations on bit vectors built up from concatenation, ....

G. S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Proceedings of Word Equations and Related Topics (IWWERT '90), volume 572 of LNCS, pages 1-11, Berlin;Heidelberg;New York, 1992. Springer.

....of this specific problem. For a more elaborate overview confer to [GHR93] unification theory, chapter 5. The positive result that L (t 1 =t2 ) is decidable, derives from the decidability result for arbitrary single equations in a free monoid G.S. Makanin presented in 1977. For a description see [Mak92] More interesting in our context is the result of Jaffar [Jaf90] where an algorithm for construction of all models is given. The actual complexity class of L (t 1 =t2 ) is unknown, but known to be NP hard [Benanav et al. 1985] Independently, this fact was shown in 1996 by the author, confer ....

G. S. Makanin. Investigations on equations in a free group. In Schulz [Sch92b], pages 1--11.

....(unification under associativity) also called word equations was initiated by Markov at the end of the 1950s in connection with the then still unsolved tenth Hilbert problem. The decidability of the word equations has been proved only in 1977 by Makanin [27] The definition may be found e.g. in [28]. Theorem 4.2 The word equation problem is polynomial time reducible to monadic simultaneous rigid E unification. 7 Proof. The complete proof may be found in our technical report [7] Here we only show how to represent the concatenation of words. Suppose that x i , x j and x k are different ....

G. Makanin. Investigations on equations in a free group. In K. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science, pages 1--12, T ubingen, Germany, Oct. 1990.

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92,PR98]. It has been shown, however, that any non fixed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non fixed size equations on bit vectors built up from concatenation, ....

G. S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Proceedings of Word Equations and Related Topics (IWWERT '90), volume 572 of LNCS, pages 1--11, Berlin;Heidelberg;New York, 1992. Springer.

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92,PR98]. It has been shown, however, that any non xed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non xed size equations on bit vectors built up from concatenation, ....

G. S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Proceedings of Word Equations and Related Topics (IWWERT '90), volume 572 of LNCS, pages 1-11, Berlin;Heidelberg;New York, 1992. Springer.

.... under associativity) was initiated by Markov at the end of the 1950s in connection with the then still unsolved Hilbert a tenth problem [43] The problem happened to be very hard and its decidability has been proven only in 1977 by Makanin [38] The precise definition may be found e.g. in [39, 40], we shall give here an informal definition of the word equation problem. 13 Section 2. Solutions Given an equality V W of words V; W in the alphabet ff 1 ; f n g [ fx 1 ; xm g, is there a substitution oe = fU 1 =x 1 ; Um =xm g of words for variables such that U 1 ; ....

G.S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science, pages 1--12, Tubingen, Germany, October 1990.

.... under associativity) was initiated by Markov at the end of the 1950s in connection with the then still unsolved Hilbert a tenth problem [43] The problem happened to be very hard and its decidability has been proven only in 1977 by Makanin [38] The precise definition may be found e.g. in [39, 40], we shall give here an informal definition of the word equation problem. Given an equality V W of words V; W in the alphabet ff 1 ; f n g[fx 1 ; xm g, is there a substitution oe = fU 1 =x 1 ; Um=xm g of words for variables such that U 1 ; Um are words in the ....

G.S. Makanin. Investigations on equations in a free group. In K.U. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science, pages 1--12, Tubingen, Germany, October 1990.

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