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Makanin's algorithm is not primitive recursive

1998
*
Theoretical Computer Science
*

We prove that, contrary to the common belief, the

doi:10.1016/s0304-3975(96)00321-0
fatcat:4e6ddxuxrvhntnx43z4zeskc4u
*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*. ...##
###
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,

doi:10.1016/s0304-3975(97)80149-1
fatcat:rlax7guf4va2te3x4nevocrkmm
*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. ...##
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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]

2015
*
arXiv
*
pre-print

So, the title of the present article does

arXiv:1507.03215v1
fatcat:v434pjrdsrg2nmntkoqbwwuuda
*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] . ...##
###
Equations in Free Semigroups with Anti-involution and Their Relation to Equations in Free Groups
[chapter]

2000
*
Lecture Notes in Computer Science
*

A free SGA

doi:10.1007/10719839_38
fatcat:c5htkei2qbgstobto46qjapt6y
*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*. ...##
###
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

1989
*
Journal of symbolic computation
*

Pure associative unification

doi:10.1016/s0747-7171(89)80056-2
fatcat:lqsxwqa3ijhr5h3eaioyh3l4me
*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. ...##
###
Equations in free semigroups with involution and their relation to equations in free groups

2003
*
Theoretical Computer Science
*

A free semigroup with involution (FSI)

doi:10.1016/s0304-3975(02)00642-4
fatcat:u5eovx2tu5ba5cef5cl2tfigqm
*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, MatemÃ aticas Aplicadas. ...##
###
A NOTE ON THE EXISTENTIAL THEORY OF EQUATIONS IN PLAIN GROUPS

2002
*
International journal of algebra and computation
*

Based on a PSPACE-completeness result for free monoids with involution [4] it

doi:10.1142/s0218196702000870
fatcat:7hkbao6ievaobcmyuedcexj7ii
*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. ...##
###
Solving Word Equations (And Other Unification Problems) by Recompression (Invited Talk)

2020
*
Annual Conference for Computer Science Logic
*

The first

doi:10.4230/lipics.csl.2020.3
dblp:conf/csl/Jez20
fatcat:fkksdckdnjhs5ekjqqzpizxnvy
*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. ...##
###
Complexity of unification in free groups and free semi-groups

*
Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
*

Moreover we give some evidence that, contrary to the common belief, the

doi:10.1109/fscs.1990.89605
dblp:conf/focs/KoscielskiP90
fatcat:jncrappatvbtphyzraqdma4r24
*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*...##
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Z3-str: a z3-based string solver for web application analysis

2013
*
Proceedings of the 2013 9th Joint Meeting on Foundations of Software Engineering - ESEC/FSE 2013
*

Z3-str treats strings as a

doi:10.1145/2491411.2491456
dblp:conf/sigsoft/ZhengZG13
fatcat:jsr7kptgw5erxbtw77e5krqhyu
*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. ...##
###
Codes and equations on trees

2001
*
Theoretical Computer Science
*

Moreover, we propose a new

doi:10.1016/s0304-3975(99)00317-5
fatcat:7og4auczyvbmjlhu35aimkkgbu
*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. ...
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