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Pattern Languages Seminar Algorithmic Learning Theory, SS 2015 Michael Krause RWTH Aachen 07.05.2015 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 1 / 44 Basic ideas 1 Finding Patterns Common to a Set of Strings 2 Other

  1. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit Repetition: Gold’s model Objects: formal languages Presentation: sequence of strings from a language, where each string appears at least once (a text) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 11 / 44

  2. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit Repetition: Gold’s model Objects: formal languages Presentation: sequence of strings from a language, where each string appears at least once (a text) The learner outputs hypotheses after receiving a string Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 11 / 44

  3. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit Repetition: Gold’s model Objects: formal languages Presentation: sequence of strings from a language, where each string appears at least once (a text) The learner outputs hypotheses after receiving a string The learner learns the language, if, after some finite amount of time, the hypotheses are correct and remain the same Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 11 / 44

  4. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit In our case Assuming a learner is presented with a text s 1 , s 2 , s 3 , . . . of some pattern language Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 12 / 44

  5. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit In our case Assuming a learner is presented with a text s 1 , s 2 , s 3 , . . . of some pattern language The hypothesis space is the set of all patterns Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 12 / 44

  6. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit In our case Assuming a learner is presented with a text s 1 , s 2 , s 3 , . . . of some pattern language The hypothesis space is the set of all patterns The hypotheses are patterns descriptive of the strings seen so far Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 12 / 44

  7. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit In our case Assuming a learner is presented with a text s 1 , s 2 , s 3 , . . . of some pattern language The hypothesis space is the set of all patterns The hypotheses are patterns descriptive of the strings seen so far Assuming there exists an algorithm to find descriptive patterns Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 12 / 44

  8. Finding Patterns Common to a Set of Strings Learning pattern languages in the limit In our case Assuming a learner is presented with a text s 1 , s 2 , s 3 , . . . of some pattern language The hypothesis space is the set of all patterns The hypotheses are patterns descriptive of the strings seen so far Assuming there exists an algorithm to find descriptive patterns Then paper by Angluin shows: Pattern languages can be learned in the limit from positive data Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 12 / 44

  9. Finding Patterns Common to a Set of Strings Finding descriptive patterns Basic ideas 1 Finding Patterns Common to a Set of Strings 2 Learning pattern languages in the limit Finding descriptive patterns Properties of pattern languages Finding descriptive one-variable patterns Other results 3 Conclusion 4 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 13 / 44

  10. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  11. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  12. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S Test for each pattern if its language contains S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  13. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S Test for each pattern if its language contains S From all patterns that pass the test: Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  14. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S From all patterns that pass the test: Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  15. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Theorem (3.6, Angluin) The membership problem for pattern languages is NP-complete Test for each pattern if its language contains S From all patterns that pass the test: Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  16. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Theorem (3.6, Angluin) The membership problem for pattern languages is NP-complete Test for each pattern if its language contains S → NP-complete From all patterns that pass the test: Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  17. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest → exponential growth string in S Theorem (3.6, Angluin) The membership problem for pattern languages is NP-complete Test for each pattern if its language contains S → NP-complete Theorem (5.1, Jiang et al.) The inclusion problem for arbitrary pattern languages is undecidable From all patterns that pass the test: Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  18. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our first attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest → exponential growth string in S Theorem (3.6, Angluin) The membership problem for pattern languages is NP-complete Test for each pattern if its language contains S → NP-complete Theorem (5.1, Jiang et al.) The inclusion problem for arbitrary pattern languages is undecidable From all patterns that pass the test: → � ! Select one which is minimal with regards to inclusion Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 14 / 44

  19. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our second attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S → NP-complete Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 15 / 44

  20. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our second attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S → NP-complete Corollary (3.4, Angluin) Let p , q be patterns with the same length. Then the language of q includes the language of p iff there is a substitution from q to p Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 15 / 44

  21. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our second attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S → NP-complete Corollary (3.4, Angluin) Let p , q be patterns with the same length. Then the language of q includes the language of p iff there is a substitution from q to p From all patterns that pass the test select the longest Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 15 / 44

  22. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our second attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S → NP-complete Corollary (3.4, Angluin) Let p , q be patterns with the same length. Then the language of q includes the language of p iff there is a substitution from q to p From all patterns that pass the test select the longest From the resulting set of patterns, output any which cannot be gained by substituting from another Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 15 / 44

  23. Finding Patterns Common to a Set of Strings Finding descriptive patterns Our second attempt Let S be a sample Enumerate all patterns of shorter or equal length of the shortest string in S → exponential growth Test for each pattern if its language contains S → NP-complete Corollary (3.4, Angluin) Let p , q be patterns with the same length. Then the language of q includes the language of p iff there is a substitution from q to p From all patterns that pass the test select the longest From the resulting set of patterns, output any which cannot be gained by substituting from another → NP-complete Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 15 / 44

  24. Finding Patterns Common to a Set of Strings Finding descriptive patterns Results so far Let S be a sample Theorem (4.2) If P � = NP then there is no polynomial-time algorithm to find a pattern of maximum possible length descriptive of S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 16 / 44

  25. Finding Patterns Common to a Set of Strings Finding descriptive patterns Results so far Let S be a sample Theorem (4.2) If P � = NP then there is no polynomial-time algorithm to find a pattern of maximum possible length descriptive of S We may still solve this efficiently in special cases! Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 16 / 44

  26. Finding Patterns Common to a Set of Strings Properties of pattern languages Basic ideas 1 Finding Patterns Common to a Set of Strings 2 Learning pattern languages in the limit Finding descriptive patterns Properties of pattern languages Finding descriptive one-variable patterns Other results 3 Conclusion 4 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 17 / 44

  27. Finding Patterns Common to a Set of Strings Properties of pattern languages Comparison to other language types The pattern language L ( xx ) is not context-free Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 18 / 44

  28. Finding Patterns Common to a Set of Strings Properties of pattern languages Comparison to other language types The pattern language L ( xx ) is not context-free The regular language L (0 | 1) = { 0 , 1 } is not a pattern language Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 18 / 44

  29. Finding Patterns Common to a Set of Strings Properties of pattern languages Comparison to other language types The pattern language L ( xx ) is not context-free The regular language L (0 | 1) = { 0 , 1 } is not a pattern language Theorem (3.4, Jiang) Every pattern language is context-sensitive Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 18 / 44

  30. Finding Patterns Common to a Set of Strings Properties of pattern languages Comparison to other language types The pattern language L ( xx ) is not context-free The regular language L (0 | 1) = { 0 , 1 } is not a pattern language Theorem (3.4, Jiang) Every pattern language is context-sensitive Language Membership Emptiness Equivalence Inclusion Context-sens. D U U U Context-free D D U U Regular D D D D Pattern lang. D D D U Table: D=decidable, U=undecidable Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 18 / 44

  31. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Basic ideas 1 Finding Patterns Common to a Set of Strings 2 Learning pattern languages in the limit Finding descriptive patterns Properties of pattern languages Finding descriptive one-variable patterns Other results 3 Conclusion 4 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 19 / 44

  32. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Overview Introduce necessary conditions for one-variable patterns that 1 could generate a string Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 20 / 44

  33. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Overview Introduce necessary conditions for one-variable patterns that 1 could generate a string Bound the number of one-variable patterns that could generate 2 every string in a sample Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 20 / 44

  34. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Overview Introduce necessary conditions for one-variable patterns that 1 could generate a string Bound the number of one-variable patterns that could generate 2 every string in a sample Construct automata that recognize exactly these patterns 3 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 20 / 44

  35. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Overview Introduce necessary conditions for one-variable patterns that 1 could generate a string Bound the number of one-variable patterns that could generate 2 every string in a sample Construct automata that recognize exactly these patterns 3 Finally, select a specific automaton that recognizes descriptive 4 one-variable patterns Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 20 / 44

  36. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  37. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  38. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  39. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  40. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p k is the position of the first occurence of x in p Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  41. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p k is the position of the first occurence of x in p A pattern p can only generate s , if τ ( p ) is feasible for s Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  42. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p k is the position of the first occurence of x in p A pattern p can only generate s , if τ ( p ) is feasible for s Let S = { s 1 , . . . , s m } a sample Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  43. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p k is the position of the first occurence of x in p A pattern p can only generate s , if τ ( p ) is feasible for s Let S = { s 1 , . . . , s m } a sample Let F be the set of all triples feasible for every string in S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  44. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Feasible triples Let p be a one-variable pattern and s a string of constants We define a mapping τ ( p ) = ( i , j , k ) where i is the number of constants in p j is the number of occurences of x in p k is the position of the first occurence of x in p A pattern p can only generate s , if τ ( p ) is feasible for s Let S = { s 1 , . . . , s m } a sample Let F be the set of all triples feasible for every string in S We can bound | F | = O ( l 2 log l ) where l is the length of the shortest string in S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 21 / 44

  45. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  46. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  47. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  48. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F For each string s ∈ S Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  49. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F For each string s ∈ S Construct automaton which recognizes patterns p that - fulfill τ ( p ) = f - generate s Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  50. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F For each string s ∈ S Construct automaton which recognizes patterns p that - fulfill τ ( p ) = f - generate s Intersect these automata Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  51. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F For each string s ∈ S Construct automaton which recognizes patterns p that - fulfill τ ( p ) = f - generate s Intersect these automata From the resulting set of automata: discard those whose language is empty Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  52. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Anguin’s algorithm for finding descriptive one-variable patterns Let S be a sample Construct F by enumerating all feasible triples For each triple f ∈ F For each string s ∈ S Construct automaton which recognizes patterns p that - fulfill τ ( p ) = f - generate s Intersect these automata From the resulting set of automata: discard those whose language is empty Lemma (6.3) Any pattern accepted by an automaton built from a triple that maximizes i + j is descriptive of S among one variable patterns Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 22 / 44

  53. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Example Let S = { s 1 , s 2 , s 3 } a sample with s 1 = 1101011 , s 2 = 10011 , s 3 = 11111 We construct F through enumeration We get: F = { (1 , 1 , k ) , (1 , 2 , k ) , (2 , 1 , k ) , (3 , 1 , k ) , (3 , 2 , k ) , (4 , 1 , k ) } 1 ≤ k ≤ i + 1 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 23 / 44

  54. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Example Let S = { s 1 , s 2 , s 3 } a sample with s 1 = 1101011 , s 2 = 10011 , s 3 = 11111 We construct F through enumeration We get: F = { (1 , 1 , k ) , (1 , 2 , k ) , (2 , 1 , k ) , (3 , 1 , k ) , (3 , 2 , k ) , (4 , 1 , k ) } 1 ≤ k ≤ i + 1 We construct three automata per triple in F Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 23 / 44

  55. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Example Let S = { s 1 , s 2 , s 3 } a sample with s 1 = 1101011 , s 2 = 10011 , s 3 = 11111 We construct F through enumeration We get: F = { (1 , 1 , k ) , (1 , 2 , k ) , (2 , 1 , k ) , (3 , 1 , k ) , (3 , 2 , k ) , (4 , 1 , k ) } 1 ≤ k ≤ i + 1 We construct three automata per triple in F In this example we do this for: (3 , 2 , 2) ∈ F Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 23 / 44

  56. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  57. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  58. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 (0 , 0) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  59. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  60. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x (1 , 1) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  61. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x (1 , 1) x (1 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  62. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x (1 , 1) x 1 (1 , 2) (2 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  63. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x (1 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  64. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x 1 (1 , 1) (2 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  65. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x 1 0 (1 , 1) (2 , 1) (3 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  66. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x 1 0 1 (1 , 1) (2 , 1) (3 , 1) (4 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  67. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 1 = 1101011 Substring starts at position 2, length: ( | s 1 | − 3) / 2 = 2 Substring: x = 10 1 (0 , 0) (1 , 0) x 1 0 1 1 (1 , 1) (2 , 1) (3 , 1) (4 , 1) (5 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 24 / 44

  68. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 2 = 10011, Substring length: ( | s 2 | − 3) / 2 = 1 Substring: x = 0 1 (0 , 0) (1 , 0) x 0 1 1 (1 , 1) (2 , 1) (3 , 1) (4 , 1) x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 25 / 44

  69. Finding Patterns Common to a Set of Strings Finding descriptive one-variable patterns Triple: (3 , 2 , 2), String: s 3 = 11111, Substring length: ( | s 3 | − 3) / 2 = 1 Substring: x = 1 1 (0 , 0) (1 , 0) x 1 1 1 (1 , 1) (2 , 1) (3 , 1) (4 , 1) x x x 1 1 (1 , 2) (2 , 2) (3 , 2) Michael Krause (RWTH Aachen) Pattern Languages 07.05.2015 26 / 44

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