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Axel Thue’s 1914 paper: An early completion algorithm: Axel Thue’s 1914 paper on the transformation of symbol sequences James Power Department of Computer Science, National University of Ireland Maynooth CiE, June 24, 2014 James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Axel Thue’s 1914 Paper Problems on the transformation of sequences of symbols according to given rules, Axel Thue, Christiana Videnskabs-Selskabs Skrifter , No 10, 1914. James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Axel Thue’s 1914 Paper Problems on the transformation of sequences of symbols according to given rules, Axel Thue, Christiana Videnskabs-Selskabs Skrifter , No 10, 1914. James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Reference by Post (1947) James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Reference by Post (1947) First example of an undecidable problem outside the GoF from 1936/7 “a particular problem whose credentials as being of genuine and independent mathematical interest were quite unimpeachable” - Martin Davis (1993) James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Reference by Post (1947) Terminology: canonical systems of this “Thue type” a system of “semi-Thue type” James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Reference by Post (1947) Terminology: canonical systems of this “Thue type” a system of “semi-Thue type” Also: work on “Turing machines” James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Canonical systems of this “Thue type” A Thue system: A finite sequence of corresponding pairs of strings over some fixed alphabet: A 1 ↔ B 1 A 2 ↔ B 2 · · · A n ↔ B n Equivalence: For any strings P and Q , we write P → Q when ∗ ← P can be transformed into Q by a series of operations, each involving a substitution of some A i for B i (or vice versa). James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Canonical systems of this “Thue type” A Thue system: A finite sequence of corresponding pairs of strings over some fixed alphabet: A 1 ↔ B 1 A 2 ↔ B 2 · · · A n ↔ B n Equivalence: For any strings P and Q , we write P → Q when ∗ ← P can be transformed into Q by a series of operations, each involving a substitution of some A i for B i (or vice versa). Nowadays: derivations using an unrestricted grammar James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Models of Computation: 1936 - the annus mirabilis Church Hilbert T uring Godel Kleene Post 1920s 1930s 1940s 1950s James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Models of Computation: automata & languages Church Hilbert T uring Mealy Moore Godel Kleene Rabin & Scott Post Chomsky Post 1920s 1930s 1940s 1950s James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Models of Computation: 1914 - Thue Church Hilbert T uring Mealy Moore Godel Kleene Rabin & Scott Thue Post Chomsky Post 1914 1920s 1930s 1940s 1950s James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Models of Computation: Church Hilbert T uring Mealy Moore Godel Kleene Rabin & Scott Thue Post Chomsky Post 1914 1920s 1930s 1940s 1950s Q1: Why was Thue studying these systems in 1914? James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Post uses only 1-2 pages of Thue’s paper James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Introduction Post uses only 1-2 pages of Thue’s paper Q2: What is the rest of Thue’s paper about? James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Background Biography: 1863: Born Tönsberg, Norway 1889: Degree at Univ. of Oslo 1891-2: visited Leipzig and Berlin 1894: teacher of mechanics, Trondheim technical college Axel Thue 1903 Prof. of Applied Mechanics, (1863-1922) Univ. of Oslo From “Axel Thue” by Viggo Brun, Selected Mathematical Papers , 1977 James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Background Published papers on: Diophantine approximations geometry and mechanics combinatorics (patterns in infinite strings) Axel Thue (1863-1922) From “Axel Thue” by Viggo Brun, Selected Mathematical Papers , 1977 James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Works on symbol-sequences Thue published four papers on symbol-sequences: 1906 Über unendliche Zeichenreihen. 1910 Die Lösung eines Spezialfalles eines generellen logischen Problems . 1912 Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. 1914 Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. All published in Christiana Videnskabs-Selskabs Skrifter James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Works on symbol-sequences Thue published four papers on symbol-sequences: 1906 About infinite sequences of symbols. 1910 Die Lösung eines Spezialfalles eines generellen logischen Problems . 1912 On the relative position of equal parts in certain sequences of symbols. 1914 Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. All published in Christiana Videnskabs-Selskabs Skrifter Axel Thue’s papers on repetitions in words: a translation Jean Berstel, Publications du LaCIM, 1995. James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Works on symbol-sequences Thue published four papers on symbol-sequences: 1906 About infinite sequences of symbols. 1910 The solution of a special case of a general logical problem. 1912 On the relative position of equal parts in certain sequences of symbols. 1914 Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln. All published in Christiana Videnskabs-Selskabs Skrifter Trees and term rewriting in 1910: On a paper by Axel Thue M. Steinby and W. Thomas, EATCS Bull., 2000. James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Axel Thue: Works on symbol-sequences Thue published four papers on symbol-sequences: 1906 About infinite sequences of symbols. 1910 The solution of a special case of a general logical problem. 1912 On the relative position of equal parts in certain sequences of symbols. 1914 Problems on the transformation of sequences of symbols according to given rules. All published in Christiana Videnskabs-Selskabs Skrifter Thue’s 1914 paper: a translation arXiv:1308.5858 James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Themes in Thue’s paper Q1: Why was Thue studying these systems in 1914? Q2: What is the rest of Thue’s paper about? James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Themes in Thue’s paper Q1: Why was Thue studying these systems in 1914? Q2: What is the rest of Thue’s paper about? Decision Problems 1 Overlapping strings 2 Thue’s algorithms 3 Meta-properties 4 James Power, NUI Maynooth CiE 2014

1. Decision problems

Axel Thue’s 1914 paper: Some background Thue and decision problems Thue poses two decision problems : Problem I: For any arbitrary given sequences A and B , to find a method, where one can always decide in a predictable number of operations, whether or not two arbitrary given symbol sequences are equivalent in respect of sequences A and B . Problem II: Given an arbitrary sequence R , to find a method where one can always decide in a finite number of investigations whether or not two arbitrary given sequences are equivalent with respect to R . (Here R is the null sequence) James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Thue and decision problems Thue poses two decision problems : Problem I: For any arbitrary given sequences A and B , to find a method, where one can always decide in a predictable number of operations, whether or not two arbitrary given symbol sequences are equivalent in respect of sequences A and B . = conjugacy problem for semi-groups Problem II: Given an arbitrary sequence R , to find a method where one can always decide in a finite number of investigations whether or not two arbitrary given sequences are equivalent with respect to R . (Here R is the null sequence) = word problem for monoids James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Context: algebra Example from group theory: Presentation of a group: � generators: a , b aba − 1 b = 1 relation: James Power, NUI Maynooth CiE 2014

Axel Thue’s 1914 paper: Some background Context: algebra Example from group theory: Presentation of a group: � generators: a , b aba − 1 b = 1 relation: Can derive equations from the identity element: aba − 1 b 1 = ba − 1 b a − 1 = a − 1 b b − 1 a − 1 = ab − 1 a − 1 b = b − 1 ab − 1 a − 1 1 = James Power, NUI Maynooth CiE 2014