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Axel Thues 1914 paper: An early completion algorithm: Axel Thues 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,
Axel Thue’s 1914 paper:
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
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
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
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
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”
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
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
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
A Thue system: A finite sequence of corresponding pairs of strings over some fixed alphabet: A1 ↔ B1 A2 ↔ B2 · · · An ↔ Bn 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 Ai for Bi (or vice versa).
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
A Thue system: A finite sequence of corresponding pairs of strings over some fixed alphabet: A1 ↔ B1 A2 ↔ B2 · · · An ↔ Bn 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 Ai for Bi (or vice versa). Nowadays: derivations using an unrestricted grammar
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
Hilbert Godel Church Kleene Post T uring 1950s 1920s 1930s 1940s
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
Hilbert Godel Church Kleene Post T uring 1950s Moore Mealy Rabin & Scott Chomsky 1920s 1930s 1940s Post
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
Hilbert Godel Church Kleene Post T uring 1914 1950s Moore Mealy Rabin & Scott Chomsky Thue 1920s 1930s 1940s Post
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
Hilbert Godel Church Kleene Post T uring 1914 1950s Moore Mealy Rabin & Scott Chomsky Thue 1920s 1930s 1940s Post
Q1: Why was Thue studying these systems in 1914?
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Introduction
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 (1863-1922) Biography: 1863: Born Tönsberg, Norway 1889: Degree at Univ. of Oslo 1891-2: visited Leipzig and Berlin 1894: teacher of mechanics, Trondheim technical college 1903 Prof. of Applied Mechanics,
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 (1863-1922) Published papers on: Diophantine approximations geometry and mechanics combinatorics (patterns in infinite strings)
From “Axel Thue” by Viggo Brun, Selected Mathematical Papers, 1977
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
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
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
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
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
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
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
Q1: Why was Thue studying these systems in 1914? Q2: What is the rest of Thue’s paper about?
1
Decision Problems
2
Overlapping strings
3
Thue’s algorithms
4
Meta-properties
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
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 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
Example from group theory: Presentation of a group: generators: a, b relation: aba−1b = 1
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example from group theory: Presentation of a group: generators: a, b relation: aba−1b = 1 Can derive equations from the identity element: aba−1b = 1 ba−1b = a−1 a−1b = b−1a−1 b = ab−1a−1 1 = b−1ab−1a−1
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
The identity problem: Suppose some element of a group is given in terms of the composition of its generators, we want to give a method to decide in a finite number of steps, whether it is the identity element or not. The transformation problem: Given any two elements of a group S and T, we want a method to answer the question whether S and T can be transformed into one another.
Max Dehn, On infinite discontinuous groups Mathematische Annalen 71(3), 1911.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
But Thue’s paper is not explicitly algebraic From §1: In this paper I will deal with a problem concerning the transformation of symbol sequences using rules. This problem, [...] is a special case of one of the most fundamental problems that can be posed Since this task seems to be extensive and of the utmost difficulty, I must be satisfied with only treating the question in a piecewise and fragmentary manner.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Q1: Why was Thue studying these systems in 1914?
Geometry Syllogisms Abstract Algebra Symbolic Logic Word problem Decision problem Language theory
19th century: abstraction Early 20th century: meta-level
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Q1: Why was Thue studying these systems in 1914?
Geometry Syllogisms Abstract Algebra Symbolic Logic Word problem Decision problem Language theory
19th century: abstraction Early 20th century: meta-level
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Thue’s 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.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Thue’s 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. Example: Let R ≡ abc, and decide if cabcbaabccb =R abccbabcacbabc
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Thue’s 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. Example: Let R ≡ abc, and decide if cabcbaabccb =R abccbabcacbabc
cabcbaabccb → cbaabccb → cbacb
abccbabcacbabc → cbabcacbabc → cbacbabc → cbacb
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Problem: What do we do if there are overlaps? Example: let R ≡ abcab.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Problem: What do we do if there are overlaps? Example: let R ≡ abcab. Reduce the string: ...abcabcab... Two choices: ...abcabcab... → ...cab... ...abcabcab... → ...abc...
c.f. critical pairs
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Problem: What do we do if there are overlaps? Example: let R ≡ abcab. Reduce the string: ...abcabcab... Two choices: ...abcabcab... → ...cab... ...abcabcab... → ...abc...
c.f. critical pairs
But: this is only possible because R itself has an overlap: R ≡ abcab abcab ≡ R
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
In his 1906 and 1912 papers Thue was interested in producing
In this paper we present some investigations of the theory of sequences of symbols, a theory that has some connections with number theory. We expect that the results of such investigations have applications to usual mathematical problems. As an example, the existence of nonperiodic decimal developments proves that irrational numbers exist Example: the Thue-Morse sequence 01101001 10010110 10010110 01101001 . . . is overlap-free
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
(Back in the 1914 paper) Thue asks: what kind of strings overlap themselves? R ≡ abcab abcab ≡ R
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
(Back in the 1914 paper) Thue asks: what kind of strings overlap themselves? R ≡ abcab abcab ≡ R They must have the general form R ≡ CU UD ≡ R
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
(Back in the 1914 paper) Thue asks: what kind of strings overlap themselves? R ≡ abcab abcab ≡ R They must have the general form R ≡ CU UD ≡ R But then C R ≡ C(UD) ≡ (CU)D ≡ R D
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
(Back in the 1914 paper) Thue asks: what kind of strings overlap themselves? R ≡ abcab abcab ≡ R They must have the general form R ≡ CU UD ≡ R But then C R ≡ C(UD) ≡ (CU)D ≡ R D That is, C =R D
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
(Back in the 1914 paper) Thue asks: what kind of strings overlap themselves? R ≡ abcab abcab ≡ R They must have the general form R ≡ CU UD ≡ R But then C R ≡ C(UD) ≡ (CU)D ≡ R D That is, C =R D In the above example abc =R cab
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
In §VIII Thue describes an algorithm based on creating alternating sets of null sequences Si and equations Ei
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
In §VIII Thue describes an algorithm based on creating alternating sets of null sequences Si and equations Ei Given a null sequence R: Start with the set S0 = {R}. Form a set Ei from Si by adding equations of the form C ↔ D where CU ≡ UD ≡ r for some r ∈ Si. Form a set Si+1 from Ei by applying the equations in Ei as substitutions in S0, . . . Si.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
In §VIII Thue describes an algorithm based on creating alternating sets of null sequences Si and equations Ei Given a null sequence R: Start with the set S0 = {R}. Form a set Ei from Si by adding equations of the form C ↔ D where CU ≡ UD ≡ r for some r ∈ Si. Form a set Si+1 from Ei by applying the equations in Ei as substitutions in S0, . . . Si. Termination: Note that in a rule P ↔ Q, the strings P and Q have the same length. Thus all elements of Si have the same length as the
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab}
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc}
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc} E1: bcabab bcabab
CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc} E1: bcabab bcabab
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc} E1: bcabab bcabab
abbcab bcabab
CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc} E1: bcabab bcabab
abbcab bcabab
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Example: suppose the given null string is R ≡ abbcab. S0: {abbcab} E0: abbcab abbcab
S1: {bcabab, ababbc} E1: bcabab bcabab
abbcab bcabab
S2: {cababb, bbcaba}
NB: last two equations are not independent
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
Q: Given a set of equations derived from some null sequence R, can I use the equations systematically to show P =R Q for any strings P and Q? A: Yes, if you can show the set of equations are: Complete: (vollständiges) For any symbol z we can prove zR
∗
← → Rz Perfect: (vollkommenes) For any strings A and B, whenever we can prove RA
∗
← → RB then we can prove A
∗
← → B.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
§VI: Theorem Suppose we have a system of equations of the form: x1P1 ↔ y1Q1 x2P2 ↔ y2Q2 . . . . . . . xkPk ↔ ykQk, where each yi is different from each other yj and xj. Then for any symbol z and strings A, B, if zA
∗
← → zB then it is also the case that A
∗
← → B.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
§VI: Theorem Suppose we have a system of equations of the form: x1P1 ↔ y1Q1 x2P2 ↔ y2Q2 . . . . . . . xkPk ↔ ykQk, where each yi is different from each other yj and xj. Then for any symbol z and strings A, B, if zA
∗
← → zB then it is also the case that A
∗
← → B. Corollary: This is a perfect system of equations i.e. if RA
∗
← → RB then A
∗
← → B.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
If we start with R ≡ abbcab and derive the equations abbc ↔ bcab abbca ↔ cabab
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Some background
If we start with R ≡ abbcab and derive the equations abbc ↔ bcab abbca ↔ cabab These equations are: Perfect as they fit the format for the §VI theorem. Complete because: aR ≡ aabbcab ↔ abcabab ↔ ababbca ↔ abbcaba ≡ Ra bR ≡ babbcab ↔ bcababb ↔ abbcabb ≡ Rb cR ≡ cabbcab ↔ cababbc ↔ abbcabc ≡ Rc.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Summary
Themes in Thue’s 1914 paper:
1
Decision Problems
2
Overlapping strings
3
Thue’s algorithms
4
Meta-properties
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Summary
Contributions of Thue’s 1914 paper:
1
Clearly specifies an interesting decision problem
2
Made the leap in abstraction from algebra to language theory.
3
Provides an early example of a “completion algorithm”.
4
Studies topics that were later finessed through grammars and parsing.
James Power, NUI Maynooth CiE 2014
Axel Thue’s 1914 paper: Summary
Contributions of Thue’s 1914 paper:
1
Clearly specifies an interesting decision problem
2
Made the leap in abstraction from algebra to language theory.
3
Provides an early example of a “completion algorithm”.
4
Studies topics that were later finessed through grammars and parsing.
James Power, NUI Maynooth CiE 2014