Degrees of Streams. Jörg Endrullis Dimitri Hendriks Jan Willem Klop. Streams Seminar Nijmegen, 20th April Vrije Universiteit Amsterdam

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1 Degrees of Streams Jörg Endrullis Dimitri Hendriks Jan Willem Klop Vrije Universiteit Amsterdam Streams Seminar Nijmegen, 20th April 2010

2 Complexity of streams Complexity measures for infinite streams: Subword complexity Kolmogorov complexity Comparing streams by transforming them into each other: Recursion theoretic degrees of unsolvability (transformation via Turing machines)

3 Complexity of streams Complexity measures for infinite streams: Subword complexity Kolmogorov complexity Comparing streams by transforming them into each other: Recursion theoretic degrees of unsolvability (transformation via Turing machines) We envisage: infinitary view on information content complexity invariant under exchange of finitely many elements capture the intrinsic, invariant infinite pattern of streams We propose a comparison via finite state transducers (FSTs).

4 Subword complexity Definition Subword complexity is a measure on streams σ, that records as a function of n, how many of the finite words of length n occur in σ. Examples: Sturmian words: n + 1 morphic words: linear automatic sequences: quadratic

5 Subword complexity Definition Subword complexity is a measure on streams σ, that records as a function of n, how many of the finite words of length n occur in σ. Examples: Sturmian words: n + 1 morphic words: linear automatic sequences: quadratic Interesting, but... Even non-computable streams can have linear subword complexity.

6 Kolmogorov complexity Definition The Kolmogorov complexity K (w) of a word w is the length of the shortest program computing w. (in a fixed universal programming system, e.g., Turing machines) Examples: Thue Morse: K (M) 6 (Turing machine needs 6 states)

7 Kolmogorov complexity Definition The Kolmogorov complexity K (w) of a word w is the length of the shortest program computing w. (in a fixed universal programming system, e.g., Turing machines) Examples: Thue Morse: K (M) 6 (Turing machine needs 6 states) Interesting, but... The Kolmogorov complexity can be increased arbitrarily by: prefixing a finite word, changing the encoding (0 I am a zero!, 1 Here is a one!)

8 Finite state transducers Definition A finite state transducer (FST) is a deterministic finite automaton with: output words w Σ along the edges, a transition function δ : Q Σ Q, an output function λ : Q Σ Γ. The following automaton computes the diff of a stream: 0 ε q q ε q Thus it reduces Thue Morse to Toeplitz:

9 Partial order of stream degrees Definition (Equivalence of streams) We write M N if there exists an FST that transforms M into N. := We use σ := {τ σ τ} to denote the equivalence class of σ. Note that: is reflexive and transitive ( ) implies a partial order on the equivalence classes w.r.t.. We are interested in the hierarchy of streams created by.

10 Hierarchy of streams sup? descending sequence of degrees M = T =? S ascending sequence of degrees? prime degree 0 eventually periodic streams (wuuu...) A stream M is prime if there is no N strictly in-between M and 0.

11 Hierarchy of streams: degrees are countable We can enumerate all FSTs (and hence all reducts of a stream). Hence: Theorem Every degree is countable. Theorem Every degree has only a countable number of degrees below it.

12 Hierarchy of streams: upper bounds upper bound Lemma zip n,m (σ,τ) σ zip n,m (σ,τ) τ

13 Hierarchy of streams: upper bounds upper bound Lemma zip n,m (σ,τ) σ zip n,m (σ,τ) τ Theorem A set A of streams has an upper bound A is countable. Proof. Let A = {σ 1,σ 2,...} be a set of streams. We define: τ n = zip(σ n,τ n+1 ) Then τ 1 is an upper bound, that is, τ 1 σ n for all n.

14 Hierarchy of streams: least upper bounds sup? Theorem Not every pair of streams has a supremum. Proof. For suitable σ and τ: no common reduct of zip 1,1 (σ,τ) and zip 1,2 (σ,τ) is an upper bound for σ and τ.

15 Hierarchy of streams: infinite ascending sequences ascending sequence of degrees Theorem There exist infinite ascending sequences. Proof. Take any stream σ. The degree σ is countable. There exist uncountably many streams. Hence there exists τ such that σ τ. Then zip(σ,τ) σ but not σ zip(σ,τ).

16 Hierarchy of streams: primes prime degree 0 eventually periodic streams (wuuu...) Definition A stream M is prime if there exists no N such that: M N 0 N N M, that is N is strictly in-between M and 0. Theorem The following stream is prime: P = =

17 A prime stream: P = Heuristic evidence 1: q 0 q This FST deletes every second 1, that is, it reduces P to: P 1 = =

18 A prime stream: P = Heuristic evidence 1: q 0 q This FST deletes every second 1, that is, it reduces P to: P 1 = = We can transform P 1 back to P by:

19 A prime stream: P = Heuristic evidence 2: q 0 q This FST deletes every second 1, that is, it reduces P to: P 2 = =

20 A prime stream: P = Heuristic evidence 2: q 0 q This FST deletes every second 1, that is, it reduces P to: P 2 = = We can transform P 2 back to P by compressing blocks of zeros: 0 n 0 n+1 6

21 A prime stream: P = Heuristic evidence 2: q 0 q This FST deletes every second 1, that is, it reduces P to: P 2 = = We can transform P 2 back to P by compressing blocks of zeros: 0 n 0 n+1 6 FSTs can perform arbitrary linear compressions.

22 A prime stream: P = w 1 w 2 w 2 Lemma Let Z be the least common multiple of all 0-loops in the FST. For all n > Q and states s there exist w 1,w 2 Γ s.t. for all i N: δ(s,10 n+i Z ) = δ(s,10 n ) λ(s,10 n+i Z ) = w 1 w2 i Proof. Analogous to the pumping lemma for regular languages.

23 A prime stream: P = Lemma For every FST A there exist n N, w,w j,1,w j,2 Γ such that: A(P) = w i=0 n w j,1 wj,2 i j=0 Proof. By the pigeonhole principle we find blocks 10 m 1 and 10 m 2 in P s.t.: Q < m 1 < m 2, m 1 m 2 mod Z, the FST A enters 10 m 1 and 10 m 2 with the same state q. Define n = m 2 m 1. Then we have: A also leaves 10 m 1 and 10 m 2 with the same state q, and m m mod Z,... The w j,1,w j,2 are derived from the previous lemma.

24 Questions and open problems Is Sierpinsky interreducible with Morse? Is Morse prime? How many primes are out there? Are there interesting invariants for FST-transductions?