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Makanin's algorithm is not primitive recursive

Antoni Kościelski, Leszek Pacholski
1998 Theoretical Computer Science  
We prove that, contrary to the common belief, the algorithm deciding sat&ability of equations in free groups, given by Makanin, is not primitive recursive.  ...  Moreover, the fact that the bounds are not primitive recursive does not imply that the algorithm itself is not primitive recursive.  ...  f, whose complexity is not primitive recursive.  ... 
doi:10.1016/s0304-3975(96)00321-0 fatcat:4e6ddxuxrvhntnx43z4zeskc4u

Page 1484 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
Herrmann (D-HUMB-PM,; Berlin) 99c:03065 03D40 03D15 03D80 20F10 68Q40 Koscielski, Antoni (PL-WROC-C; Wroclaw) ; Pacholski, Leszek (PL-WROC-C; Wroclaw) Makanin’s algorithm is not primitive recursive.  ...  Mat. 46 (1982), no. 6, 1199-1273, 1344; MR 84m:20040] indicated that the function which gives an upper bound on the number of iterations of elementary steps in Makanin’s algorithm is primitive recursive  ... 

Author index volume 191 (1998)

1998 Theoretical Computer Science  
Pacholski, Makanin's algorithm is not primitive recursive Latteux, M., see V. Bruyere Li, M., see R. Beige1 Meyer, B., see 0.  ...  Meyer, Computational indistinguishability: algorithms vs. circuits (Note) Kautz, S.M., An improved zeroone law for algorithmically random sequences (Note) Koicielski, A. and L.  ... 
doi:10.1016/s0304-3975(97)80149-1 fatcat:rlax7guf4va2te3x4nevocrkmm

Page 6174 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
We also provide an analysis of Makanin’s proof and point out why estimates of the complexity of Makanin’s algorithm and of the length of a minimal solution of an equation in a free group are not primitive  ...  recursive.”  ... 

Page 3972 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
Makanin’s algorithm decides whether a word equation has a solution. The aim of this paper is to give an introduction to the algorithm and a general description of our work.  ...  Since first-order logic is unde- cidable, most effort is expended on improving the efficiency of the successful test of a theorem, not on widening the decidability of the prover.  ... 

More Than 1700 Years of Word Equations [article]

Volker Diekert
2015 arXiv   pre-print
So, the title of the present article does not seem to be justified at all.  ...  Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations.  ...  Kościelski and Pacholski published a result that the scheme of Makanin's algorithm for free groups is not primitive recursive [14] .  ... 
arXiv:1507.03215v1 fatcat:v434pjrdsrg2nmntkoqbwwuuda

Equations in Free Semigroups with Anti-involution and Their Relation to Equations in Free Groups [chapter]

Claudio Gutiérrez
2000 Lecture Notes in Computer Science  
A free SGA is essentially the set of words over a given alphabet plus an operator which reverses words.  ...  The main result of the paper is the reduction of the problem of satisfiability of equations in free groups to the satisfiability of equations in free semigroups with anti-involution (SGA), by a non-deterministic  ...  On the other hand, the analysis of the complexity of Makanin's algorithm for groups was done by Koscielski and Pacholski [8] , who showed that it is not primitive recursive.  ... 
doi:10.1007/10719839_38 fatcat:c5htkei2qbgstobto46qjapt6y

Page 121 of Mathematical Reviews Vol. , Issue 95a [page]

1995 Mathematical Reviews  
The algorithms described are primitive recur-  ...  As ‘a consequence, it is shown that the isomorphism problem for finitely generated virtually free groups is primitive recursive, thus improving the decidability result of S. A. Krstié [J.  ... 

Solving word equations

Habib Abdulrab, Jean-Pierre Pécuchet
1989 Journal of symbolic computation  
Pure associative unification is equivalel~t to the resolution of word equations. We give a short survey of known results and algorithms in this field and describe the central algorithm of Makanin.  ...  An algebra equipped with a single associative law is a semigroup. It is a m o n o i d when it has a unit. The free monoid generated by the set A (also called a l p h a b e t ) is denoted by A*.  ...  This set is presented as a language accepted by a deterministic automaton (not necessarily finite). The main results of the proof of Makanin's algorithm are set out in this paper.  ... 
doi:10.1016/s0747-7171(89)80056-2 fatcat:lqsxwqa3ijhr5h3eaioyh3l4me

Equations in free semigroups with involution and their relation to equations in free groups

Claudio Gutiérrez
2003 Theoretical Computer Science  
A free semigroup with involution (FSI) is essentially the set of words over a given alphabet plus an operator which reverses words.  ...  The main result of the paper is the reduction of the problem of satisÿability of equations in free groups to the satisÿability of equations in FSI by a non-deterministic polynomial time transformation.  ...  Acknowledgements Volker Diekert is thanked for useful comments. The author was supported by FON-DAP, Matemà aticas Aplicadas.  ... 
doi:10.1016/s0304-3975(02)00642-4 fatcat:u5eovx2tu5ba5cef5cl2tfigqm


2002 International journal of algebra and computation  
Based on a PSPACE-completeness result for free monoids with involution [4] it is shown that the existential theory of equations with rational constraints in plain groups is PSPACE-complete.  ...  In this case Kościelski and Pacholski [11] were able to prove that the scheme of Makanin is not primitive recursive.  ...  Due to the technical difficulty of Makanin's proof it is not easy to give precise bounds on the complexity of the algorithm developed by Makanin.  ... 
doi:10.1142/s0218196702000870 fatcat:7hkbao6ievaobcmyuedcexj7ii

Solving Word Equations (And Other Unification Problems) by Recompression (Invited Talk)

Artur Je?, Michael Wagner
2020 Annual Conference for Computer Science Logic  
The first algorithm for the group case was given by Makanin [37, 38], his algorithm was not primitively-recursive [28].  ...  Since Makanin's original solution much effort was put into extending his algorithm to other structures.  ...  Makanin's algorithm, so that it workd in EXPSPACE.  ... 
doi:10.4230/lipics.csl.2020.3 dblp:conf/csl/Jez20 fatcat:fkksdckdnjhs5ekjqqzpizxnvy

Complexity of unification in free groups and free semi-groups

A. Koscielski, L. Pacholski
Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science  
Moreover we give some evidence that, contrary to the common belief, the algorithm deciding satisfiability of equations in free groups, given by Makanin in not primitive recursive.  ...  The exponent of periodicity is an important factor in estimates of complexity of word-unification algorithms.  ...  Since, as we believe, the notion of an abstract Makanin's algorithm describes the properties of the Makanin's algorithm, this gives some evidence that the algorithm given in [MA21 is not primitive recursive  ... 
doi:10.1109/fscs.1990.89605 dblp:conf/focs/KoscielskiP90 fatcat:jncrappatvbtphyzraqdma4r24

Z3-str: a z3-based string solver for web application analysis

Yunhui Zheng, Xiangyu Zhang, Vijay Ganesh
2013 Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering - ESEC/FSE 2013  
Z3-str treats strings as a primitive type, thus avoiding the inherent limitations observed in many existing solvers that encode strings in terms of other primitives.  ...  The application of Z3-str in remote code execution detection shows that its support of a wide spectrum of string operations is key to reducing false positives.  ...  A key choice in the design is to support strings as a primitive type so that we do not need to convert them to other representations, such as bit-vectors.  ... 
doi:10.1145/2491411.2491456 dblp:conf/sigsoft/ZhengZG13 fatcat:jsr7kptgw5erxbtw77e5krqhyu

Codes and equations on trees

Sabrina Mantaci, Antonio Restivo
2001 Theoretical Computer Science  
Moreover, we propose a new algorithm to test whether a ÿnite set of trees is a tree code. The running time of the algorithm is polynomial in the size of the input.  ...  The main problem emerging in this approach is to decide the satisÿability of tree equations: it is a special case of second-order uniÿcation, and it remains still open.  ...  Acknowledgements We wish to thank the authors of [14] for pointing out that our original deÿnition of stability was not correct and providing the correct one.  ... 
doi:10.1016/s0304-3975(99)00317-5 fatcat:7og4auczyvbmjlhu35aimkkgbu
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