Enumeration and Decidable Properties of Automatic Sequences

A pure morphic sequence is a right-infinite, symbolic sequence obtained by iterating a letter-to-word substitution. For instance, the Fibonacci sequence and the Thue–Morse sequence, which play an important role in theoretical computer science, are pure morphic. Define a coding as a letter-to-letter substitution. The image of a pure morphic sequence under a coding is called a morphic sequence.A sequence x is called uniformly recurrent if for each finite subword u of x there exists an integer l such that u occurs in every l-length subword of x.The paper mainly focuses on the problem of deciding whether a given morphic sequence is uniformly recurrent. Although the status of the problem remains open, we show some evidence for its decidability: in particular, we prove that it can be solved in polynomial time on pure morphic sequences and on automatic sequences.In addition, we prove that the complexity of every uniformly recurrent, morphic sequence has at most linear growth: here, complexity is understood as the function that maps each positive integer n to the number of distinct n-length subwords occurring in the sequence.

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ON THE NUMBER OF ABELIAN BORDERED WORDS (WITH AN EXAMPLE OF AUTOMATIC THEOREM-PROVING)

International Journal of Foundations of Computer Science ◽

10.1142/s0129054114400267 ◽

2014 ◽

Vol 25 (08) ◽

pp. 1097-1110 ◽

Cited By ~ 3

Author(s):

DANIEL GOČ ◽

NARAD RAMPERSAD ◽

MICHEL RIGO ◽

PAVEL SALIMOV

Keyword(s):

Theorem Proving ◽

Triangular Lattice ◽

Automatic Theorem Proving ◽

Combinatorial Objects ◽

Letter Alphabet ◽

Automatic Sequences ◽

Motzkin Paths ◽

Automatic Theorem

In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in ℤ not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue–Morse word using some kind of automatic theorem-proving provided by a logical characterization of the k-automatic sequences.

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