String Topology and Infinite Strings

1
String Topology and Infinite Strings
By Sean Falconer
Supervisor Oleg Golubitsky
2
Outline

• Introduction
• Definitions and Lemmas
• Infinite Insertion String
• Canonical Representative Path
• Countability of Insertion Strings
• Conclusions
• Bibliography

3
Introduction

• Evolving Transformation System Model
• Conventional Strings
• Insertion Strings

4
What is bad about the conventional string?

• abbababbbaabbaaa

• Does not say anything about its construction

• Can create string in many different ways

• Not a good representation, has no historical
  structure

5
Idea of an Insertion String

• Ø, a, ab, aba, abba, abbaa, abbaba, abbabab,

• String is constructed through a series of
  insertions

• Has a clearly defined formative history

• Is a better representation of the final object

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Ancestors and Descendants

• Definition If a string s is a subsequence of a
  string q, we will call s an ancestor of q and q a
  descendant of s.

• Example s aabbaab is an ancestor of
• q aababaab.

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Closed Sets

• Definition A set of strings X is closed, if for
  each string s ? X, all descendants of s also
  belong to X.

• Example Consider a string s abba. A closed
  set X, containing s, must contain all strings
  that have abba in them, not necessarily
  continuously.

• Definition The closure of a set X, is the
  intersection of all closed sets that contain X.

• Lemma 1 The only finite closed set is the empty
  set .

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Open Sets

• Definition A set of strings Y is open, if its
  complement, S \ Y is closed.

• Lemma 2 If a string s belongs to an open set Y,
  then all ancestors of s must belong to Y.

• Example Consider a set Y Ø , a, aa, aaa.
  Then, Y is open, since for every string s ? Y,
  contains all its ancestors in Y.

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Insertion Paths

• Definition A path is any sequence of strings, in
  which each next string is a result of an
  insertion of a letter in the previous string.

• Examples
• P1 Ø, a, ab, aba, abab, ababa, ,
• P2 Ø, a, ab, aab, aabb, aaabb, ,
• P3 Ø, a, aa, aaa, aaaa, aaaaa, ,

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Path Equivalence

• Definition Two paths, P and Q, will be called
  equivalent, if they end in the same open sets.

• Lemma 3 Two finite paths are equivalent if and
  only if their final strings coincide.

• Definition If every string in a path P has an
  ancestor in the path Q, then P is an ancestor of
  Q and Q is a descendant of P.

• Lemma 4 Two paths, P and Q, are equivalent if
  and only if P is an ancestor and descendant of Q.

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Path Equivalence Example

• Finite Path Example
• P1 Ø, a, ab, aab, aaba, aabaa.
• P2 Ø, a, aa, aaa, aaaa, aabaa.

• It is clear that paths P1 and P2 are equivalent
  since both the terminal strings of P1 and P2 are
  the same, thus they construct the same string
  aabaa.

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Infinite Insertion Strings

• Definition A string is an equivalence class of
  paths, denoted p.

• This definition is possible because of the path
  equivalence definitions previously introduced.

• Example
• s Ø, a, ab, aab, aaab, aaabb, aaabbb,
• q Ø, a, aa, aab, aabb, aaabb, aaabab,

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Canonical Representative Path

• Lemma 5 Every string s satisfies
• (ab) ? s
• (ab) not ? s
• where (ab) a, ab, aba, abab,

• If (ab) ? s, then the canonical path p of s is
  equivalent to (ab).

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Canonical Representative Path

• If (ab) not ? s, then for all p ? s every string
  uj ? p has the form
• uj (bjk1)(ajk2)(bjk3)(ajk4)(bjk(2n1))(ajk(2n2
  ))
• where n is the largest n such that (ab)n is an
  ancestor of uj.

• Example
• s abbab, then n 2, and any given
• uj (bjk1)(ajk2)(bjk3)(ajk4)(bjk5)(ajk6)

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Canonical Path Procedure

• Let Ki maxj kij, if maximum is infinite then Ki
  8

• Let li 0, i 1, 2, , 2n2 r1 0 j 0.
• while there exists li lt Ki
• for i 1 to 2n2
• if li lt Ki then
• li li 1, j j1,
• rj (bl1)(al2)(bl3)(bl(2n1))(al(2n2))
• end
• end
• end

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Example

• Consider string s abbbaba, then n 2, so
  (ab)2 is an ancestor of s.

• Therefore, every uj (bjk1)(ajk2)(bjk3)(ajk4)(bjk
  5)(ajk6)

• Also, for this string K1 0, K2 1, K3 3, K4
  1, K5 8, K6 1.

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Example Continued

• Originally, r1 b0a0b0a0b0a0 Ø

• Since, there exists li lt Ki, Loop

i 1, since l1 is not less than K1, then skip bl1
i 2, since l2 lt K2, then al2 a1, and r2
b0a1b0a0b0a0 a
i 3, since l3 lt K3, then bl3 b1, and r3
b0a1b1a0b0a0 ab
i 4, since l4 lt K4, then al4 a1, and r4
b0a1b1a1b0a0 aba
i 5, since l5 lt K5, then bl5 b1, and r5
b0a1b1a1b1a0 abab
i 6, since l6 lt K6, then al6 a1, and r6
b0a1b1a1b1a1 ababa
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Example Continued

• Loop is done, but there still exists li lt Ki, so
  continue to loop

i 1, i 2, r8 b0a1b2a1b2a1 abbabba

• After 2 iterations, our canonical path so far is
  p 0, a, ab, aba, abab, ababa, abbaba, abbabba

• After 3 iterations p Ø, a, ab, aba, abab,
  ababa, abbaba, abbabba, abbbabba, abbbabbba

• After m iterations p Ø, a, ab, aba, abab,
  ababa, abbaba, abbabba, abbbabba, abbbabbba, ,
  abbbabma

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Countability of Insertion Strings

• This procedure is parameterized by a finite
  vector of integers, K1, K2, K3, , K2n2, whose
  components are positive integers or 8.

• Therefore, the set of possible path constructed
  by the procedure is countable. Hence, insertion
  strings is countable.

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Conclusion

• The topology of infinite insertion strings is
  completely different than that of real numbers or
  binary sequences.

• A canonical representative path exists for all
  strings.

• An infinite insertion string provides a better
  representation of a string than the conventional
  definition.

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Bibliography

• T. Becker, V. Weispfenning, Grobner Basis,
  Springer-Verlag New York Inc., New York, New
  York, USA, 1993
• R. V. Book, F. Otto, String-Rewriting Systems,
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  York, USA, 1993
• M. Burgin, Super-recursive algorithms as a tool
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  Diego, California, USA, 1999
• F. Capra, The Web Of Life, Anchor Books, New
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• L. Goldfarb, O. Golubitsky, What Is a Structural
  Measurement Process, Technical Report TR01-147,
  Faculty of Computer Science, U.N.B., October 2001
• L. Goldfarb, O. Golubitsky, D. Korkin, What is
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  TR00-137, Faculty of Computer Science, U.N.B.,
  November 2001
• O. Golubitsky, On the generating process and the
  class typicality measure, Technical Report
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• K. Kuratowski, Topology, Academic Press Inc., New
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• K. H. Rosen, Discrete Mathematics and Its
  Applications, McGraw Hill, New York, New York,
  USA, 1999