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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
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 ...
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
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.” ...
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. ...
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  . ...arXiv:1507.03215v1 fatcat:v434pjrdsrg2nmntkoqbwwuuda
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  , who showed that it is not primitive recursive. ...doi:10.1007/10719839_38 fatcat:c5htkei2qbgstobto46qjapt6y
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. ...
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
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, MatemÃ aticas Aplicadas. ...doi:10.1016/s0304-3975(02)00642-4 fatcat:u5eovx2tu5ba5cef5cl2tfigqm
Based on a PSPACE-completeness result for free monoids with involution  it is shown that the existential theory of equations with rational constraints in plain groups is PSPACE-complete. ... In this case Kościelski and Pacholski  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
The first algorithm for the group case was given by Makanin [37, 38], his algorithm was not primitively-recursive . ... 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
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 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
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  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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