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Solving word equations St ep an Holub Department of Algebra MFF UK, Prague Prague Gathering of Logicians, February 13, 2016 1/23 St ep an Holub Solving word equations Outline Algorithms Compactness Independent systems

  1. Solving word equations ˇ Stˇ ep´ an Holub Department of Algebra MFF UK, Prague Prague Gathering of Logicians, February 13, 2016 1/23 ˇ Stˇ ep´ an Holub Solving word equations

  2. Outline Algorithms Compactness Independent systems and their size 2/23 ˇ Stˇ ep´ an Holub Solving word equations

  3. Outline Algorithms Compactness Independent systems and their size (running background question: combinatorics or algebra?) 2/23 ˇ Stˇ ep´ an Holub Solving word equations

  4. Decidable? 3/23 ˇ Stˇ ep´ an Holub Solving word equations

  5. Decidable? 4/23 ˇ Stˇ ep´ an Holub Solving word equations

  6. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  7. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) Joxan Jaffar, Minimal and complete word unification. J. ACM , 1990 (4-NEXPTIME, all solutions) 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  8. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) Joxan Jaffar, Minimal and complete word unification. J. ACM , 1990 (4-NEXPTIME, all solutions) Klaus U. Schulz, Makanin’s Algorithm for Word Equations - Two Improvements and a Generalization. IWWERT 1990 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  9. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) Joxan Jaffar, Minimal and complete word unification. J. ACM , 1990 (4-NEXPTIME, all solutions) Klaus U. Schulz, Makanin’s Algorithm for Word Equations - Two Improvements and a Generalization. IWWERT 1990 Antoni Ko´ scielski and Leszek Pacholski, Complexity of Makanins algorithm. J. ACM , 1996 (3-NEXPTIME) 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  10. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) Joxan Jaffar, Minimal and complete word unification. J. ACM , 1990 (4-NEXPTIME, all solutions) Klaus U. Schulz, Makanin’s Algorithm for Word Equations - Two Improvements and a Generalization. IWWERT 1990 Antoni Ko´ scielski and Leszek Pacholski, Complexity of Makanins algorithm. J. ACM , 1996 (3-NEXPTIME) Claudio Guti´ errez, Satisfiability of word equations with constants is in exponential space. FOCS , 1998 (EXPSPACE) 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  11. Makanin’s algorithm G. S. Makanin, The problem of solvability of equations in a free semigroup, Mat. Sb. , 1977 (6-NEXPTIME ?) Joxan Jaffar, Minimal and complete word unification. J. ACM , 1990 (4-NEXPTIME, all solutions) Klaus U. Schulz, Makanin’s Algorithm for Word Equations - Two Improvements and a Generalization. IWWERT 1990 Antoni Ko´ scielski and Leszek Pacholski, Complexity of Makanins algorithm. J. ACM , 1996 (3-NEXPTIME) Claudio Guti´ errez, Satisfiability of word equations with constants is in exponential space. FOCS , 1998 (EXPSPACE) Volker Diekert, Makanin’s algorithm. In Algebraic Combinatorics on Words , 2002 (rational constraints) 5/23 ˇ Stˇ ep´ an Holub Solving word equations

  12. Some ideas : Length type xay = zbzb 6/23 ˇ Stˇ ep´ an Holub Solving word equations

  13. Some ideas : Length type xay = zbzb ( | x | , | y | , | z | ) = (1 , 4 , 2) 6/23 ˇ Stˇ ep´ an Holub Solving word equations

  14. Some ideas : Length type xay = zbzb ( | x | , | y | , | z | ) = (1 , 4 , 2) x 1 y 1 y 2 y 3 y 4 a z 1 z 2 z 1 z 2 b b 6/23 ˇ Stˇ ep´ an Holub Solving word equations

  15. Some ideas : Length type xay = zbzb ( | x | , | y | , | z | ) = (1 , 4 , 2) x 1 y 1 y 2 y 3 y 4 a z 1 z 2 z 1 z 2 b b 6/23 ˇ Stˇ ep´ an Holub Solving word equations

  16. Some ideas : Length type xay = zbzb ( | x | , | y | , | z | ) = (1 , 4 , 2) x 1 y 1 y 2 y 3 y 4 a z 1 z 2 z 1 z 2 b b x �→ a y �→ baab z �→ aa 6/23 ˇ Stˇ ep´ an Holub Solving word equations

  17. Some ideas : Elementary transformations xay = zbzb 7/23 ˇ Stˇ ep´ an Holub Solving word equations

  18. Some ideas : Elementary transformations xay = zbzb | x | < | z | 7/23 ˇ Stˇ ep´ an Holub Solving word equations

  19. Some ideas : Elementary transformations xay = zbzb | x | < | z | z �→ xz 7/23 ˇ Stˇ ep´ an Holub Solving word equations

  20. Some ideas : Elementary transformations xay = zbzb | x | < | z | z �→ xz xay = xzbxzb 7/23 ˇ Stˇ ep´ an Holub Solving word equations

  21. Some ideas : Elementary transformations xay = zbzb | x | < | z | z �→ xz xay = xzbxzb x 1 y 1 y 2 y 3 y 4 y 1 y 2 y 3 y 4 a a �→ z 1 z 2 z 1 z 2 z 1 x 1 z 1 b b b b 7/23 ˇ Stˇ ep´ an Holub Solving word equations

  22. Bound on the exponent of periodicity φ ( u ) = φ ( v ) = pw e s 8/23 ˇ Stˇ ep´ an Holub Solving word equations

  23. Bound on the exponent of periodicity φ ( u ) = φ ( v ) = pw e s Makanin: double exponential 8/23 ˇ Stˇ ep´ an Holub Solving word equations

  24. Bound on the exponent of periodicity φ ( u ) = φ ( v ) = pw e s Makanin: double exponential scielski and Pacholski: O (2 1 . 07 d ) Ko´ 8/23 ˇ Stˇ ep´ an Holub Solving word equations

  25. Different concept of transformations: Compression Wojciech Plandowski, Wojciech Rytter, Application of Lempel-Ziv encodings to the solution of word equations. ICALP 1998 9/23 ˇ Stˇ ep´ an Holub Solving word equations

  26. Different concept of transformations: Compression Wojciech Plandowski, Wojciech Rytter, Application of Lempel-Ziv encodings to the solution of word equations. ICALP 1998 Wojciech Plandowski, Satisfiability of word equations with constants is in NEXPTIME. STOC 1999. 9/23 ˇ Stˇ ep´ an Holub Solving word equations

  27. Different concept of transformations: Compression Wojciech Plandowski, Wojciech Rytter, Application of Lempel-Ziv encodings to the solution of word equations. ICALP 1998 Wojciech Plandowski, Satisfiability of word equations with constants is in NEXPTIME. STOC 1999. Wojciech Plandowski, Satisfiability of word equations with constants is in PSPACE. J. ACM 2004. 9/23 ˇ Stˇ ep´ an Holub Solving word equations

  28. Different concept of transformations: Compression Wojciech Plandowski, Wojciech Rytter, Application of Lempel-Ziv encodings to the solution of word equations. ICALP 1998 Wojciech Plandowski, Satisfiability of word equations with constants is in NEXPTIME. STOC 1999. Wojciech Plandowski, Satisfiability of word equations with constants is in PSPACE. J. ACM 2004. Wojciech Plandowski, An efficient algorithm for solving word equations. STOC , 2006. (Graph representing all solutions) 9/23 ˇ Stˇ ep´ an Holub Solving word equations

  29. Lempel - Ziv compression aacaacabcabaaac (0,0,a) (1,1,c) (3,4,b) (3,3,a) (12,3,$) 10/23 ˇ Stˇ ep´ an Holub Solving word equations

  30. Lempel - Ziv compression aacaacabcabaaac (0,0,a) (1,1,c) (3,4,b) (3,3,a) (12,3,$) xay = zbzb 10/23 ˇ Stˇ ep´ an Holub Solving word equations

  31. Artur Je˙ z: Recompression Approximation of Grammar-Based Compression via Recompression. CPM 2013 Artur Je˙ z, Recompression: a simple and powerful technique for word equations. STACS 2013. Recompression: Word Equations and Beyond. Developments in Language Theory 2013 The Complexity of Compressed Membership Problems for Finite Automata. Theory Comput. Syst. 2014 Approximation of grammar-based compression via recompression. Theor. Comput. Sci. 2015 Faster Fully Compressed Pattern Matching by Recompression. ACM Transactions on Algorithms 2015 One-Variable Word Equations in Linear Time. Algorithmica 2016 11/23 ˇ Stˇ ep´ an Holub Solving word equations

  32. Artur Je˙ z: Recompression Guess letters at the beginning and the end of variables Compress chosen pairs of letters 12/23 ˇ Stˇ ep´ an Holub Solving word equations

  33. Artur Je˙ z: Recompression Guess letters at the beginning and the end of variables Compress chosen pairs of letters xay = zbzb 12/23 ˇ Stˇ ep´ an Holub Solving word equations

  34. Artur Je˙ z: Recompression Guess letters at the beginning and the end of variables Compress chosen pairs of letters xay = zbzb bxbaayb = azbbazb 12/23 ˇ Stˇ ep´ an Holub Solving word equations

  35. Artur Je˙ z: Recompression Guess letters at the beginning and the end of variables Compress chosen pairs of letters xay = zbzb bxbaayb = azbbazb bxbaayb = azbbazb 12/23 ˇ Stˇ ep´ an Holub Solving word equations

  36. Artur Je˙ z: Recompression Guess letters at the beginning and the end of variables Compress chosen pairs of letters xay = zbzb bxbaayb = azbbazb bxbaayb = azbbazb bxcayb = azbczb 12/23 ˇ Stˇ ep´ an Holub Solving word equations

  37. General strategy of all algorithms Transformation rules (non-deterministic) boundary equations (Makanin) exponential expressions (Plandowski) ordinary equations (Je˙ z) Terminating condition based on bounds on length of the minimal solution size of the transformed equation 13/23 ˇ Stˇ ep´ an Holub Solving word equations

  38. Current knowledge exponent of periodicity: O (2 cn ) (tight) 14/23 ˇ Stˇ ep´ an Holub Solving word equations

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