Springer Nature version time Lempel-Ziv compression complexity properties 2022-08-04T17:21 binaries approach polynomial time problem solvability length https://scigraph.springernature.com/explorer/license/ length of variables function first main result 1998 relation https://doi.org/10.1007/bfb0055097 LZ intricate algorithms encoding main results variables new approach words second main result Application of Lempel-Ziv encodings to the solution of word equations compression first solution exponential function false terms 1998-01-01 applications deterministic time 731-742 One of the most intricate algorithms related to words is Makanin's algorithm solving word equations. The algorithm is very complicated and the complexity of the problem of solving word equations is not well understood. Word equations can be used to define various properties of strings, e.g. general versions of pattern-matching with variables. This paper is devoted to introduce a new approach and to study relations between Lempel-Ziv compressions and word equations. Instead of dealing with very long solutions we propose to deal with their Lempel-Ziv encodings. As our first main result we prove that each minimal solution of a word equation is highly compressible (exponentially compressible for long solutions) in terms of Lempel-Ziv encoding. A simple algorithm for solving word equations is derived. If the length of minimal solution is bounded by a singly exponential function (which is believed to be always true) then LZ encoding of each minimal solution is of a polynomial size (though the solution can be exponentially long) and solvability can be checked in nondeterministic polynomial time. As our second main result we prove that the solvability can be tested in polynomial deterministic time if the lengths of all variables are given in binary. We show also that lexicographically first solution for given lengths of variables is highly compressible in terms of Lempel-Ziv encodings. size word equations results polynomial size general version solution paper simple algorithm minimal solutions string nondeterministic polynomial time long solutions equations chapters chapter Makanin's algorithm properties of strings algorithm Mathematical Sciences Sven Skyum Winskel Glynn Springer Nature - SN SciGraph project pub.1026190890 dimensions_id Wojciech Plandowski 978-3-540-68681-1 978-3-540-64781-2 Automata, Languages and Programming Department of Computer Science, University of Liverpool, UK Department of Computer Science, University of Liverpool, UK Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02-097, Warszawa, Poland Turku Centre for Computer Science and Department of Mathematics, Turku University, 20 014, Turku, Finland Turku Centre for Computer Science and Department of Mathematics, Turku University, 20 014, Turku, Finland Larsen Kim G. Pure Mathematics Rytter Wojciech doi 10.1007/bfb0055097