The Final Deterministic Automaton on Streams

Transcription

1 The Final Deterministic Automaton on Streams Helle Hvid Hansen Clemens Kupke Jan Rutten Joost Winter Radboud Universiteit Nijmegen & CWI Amsterdam Brouwer seminar, 29 April 2014

2 Overview 1. Automata, streams and language. 2. Final k-automaton of streams. 3. Linear automata and streams. 4. Characterisation of k-regular (and k-automatic) sequences

3 Automata, Streams and Languages (S is a set) deterministic 1-automata X S X (streams) deterministic k-automata X S X k (languages)

4 Automata, Streams and Languages (S is a set) (S is a semiring) deterministic 1-automata X S X (streams) linear 1-automata X S S(X ) (streams) deterministic k-automata X S X k (languages)

5 Automata, Streams and Languages (S is a set) (S is a semiring) deterministic 1-automata X S X (streams) linear 1-automata X S S(X ) (streams) deterministic k-automata X S X k (languages) linear k-automata X S S(X ) k (power series)

6 1-Automata and Streams Let B be a set. A (deterministic) 1-automaton (with output in B) and state space X is a map X B X Streams over B: B ω = {σ : N B}. We write: σ = (σ(0), σ(1), σ(2),...) head or initial value : hd(σ) = σ(0), tail or derivative : tl(σ) = σ = (σ(1), σ(2),...),

7 The 1-Automaton of Streams Streams are a final 1-automaton: o,d X B X [[ ]] id B [[ ]] where behaviour map is B ω hd,tl B B ω x X : [[x]](0) = o(x) [[x]] = [[d(x)]] x X : [[x]](n) = o(d n (x)) n N. Simple stream differential equation (SDE) format (B = {0, 1}), example: x(0) = 1, x = y y(0) = 0, y = x z(0) = 1, z = y Fact: A stream σ is a solution to a finite system of simple SDEs iff σ is eventually periodic.

8 k-automata and Languages A (deterministic) k-automaton (with output in B) with state space X is a map: X B X A where the alphabet A is any set of cardinality k. B-valued languages over alphabet A: L := B A = {L: A B} The (left-)derivative of language L wrt a A is L a L where L a (w) = L(aw) for all w A

9 The Final k-automaton of Languages The k-automaton of languages... is final: λ : L B L A L L(ε), a L a X o,d B X A [[ ]] L λ id B [[ ]] A B L A x X, a A : [[x]](ε) = o(x) [[x]] a = [[d(x)(a)]] where behaviour map is: x X : [[x]](w) = o(d (x)(w)) w A.

10 The Final k-automaton of Languages The k-automaton of languages... is final: λ : L B L A L L(ε), a L a X o,d B X A [[ ]] L λ id B [[ ]] A B L A x X, a A : [[x]](ε) = o(x) [[x]] a = [[d(x)(a)]] where behaviour map is: x X : [[x]](w) = o(d (x)(w)) w A.

11 Zip Define k-automaton structure on streams. Zipping two streams together: zip(σ, τ) = (σ(0), τ(0), σ(1), τ(1),...). Zipping k streams together: zip k (σ 0,..., σ k 1 )(i + nk) = σ i (n) n, i N, 0 i < k or equivalently (by the stream differential equation): zip k (σ 0,..., σ k 1 )(0) = σ 0 (0) zip k (σ 0,..., σ k 1 ) = zip k (σ 1,..., σ k 1, σ 0 ).

12 Zip Define k-automaton structure on streams. Zipping two streams together: zip(σ, τ) = (σ(0), τ(0), σ(1), τ(1),...). Zipping k streams together: zip k (σ 0,..., σ k 1 )(i + nk) = σ i (n) n, i N, 0 i < k or equivalently (by the stream differential equation): zip k (σ 0,..., σ k 1 )(0) = σ 0 (0) zip k (σ 0,..., σ k 1 ) = zip k (σ 1,..., σ k 1, σ 0 ).

13 Zip Define k-automaton structure on streams. Zipping two streams together: zip(σ, τ) = (σ(0), τ(0), σ(1), τ(1),...). Zipping k streams together: zip k (σ 0,..., σ k 1 )(i + nk) = σ i (n) n, i N, 0 i < k or equivalently (by the stream differential equation): zip k (σ 0,..., σ k 1 )(0) = σ 0 (0) zip k (σ 0,..., σ k 1 ) = zip k (σ 1,..., σ k 1, σ 0 ).

14 Unzip Unzipping a stream into k streams: For k, j N with j < k: unzip j,k (σ)(n) = σ(j + nk) n N For k = 3, unzip 0,3 (σ) = (σ(0), σ(3), σ(6),...) unzip 1,3 (σ) = (σ(1), σ(4), σ(7),...) unzip 2,3 (σ) = (σ(2), σ(5), σ(8),...) We have: zip k (unzip 0,k (σ),..., unzip k 1,k (σ)) = σ unzip i,k (zip k (σ 0,..., σ k 1 )) = σ i i < k In other words, zip k : (S N ) k S N is a bijection with inverse unzip k = unzip 0,k,..., unzip k 1,k : S N (S N ) k.

15 Unzip Unzipping a stream into k streams: For k, j N with j < k: unzip j,k (σ)(n) = σ(j + nk) n N For k = 3, unzip 0,3 (σ) = (σ(0), σ(3), σ(6),...) unzip 1,3 (σ) = (σ(1), σ(4), σ(7),...) unzip 2,3 (σ) = (σ(2), σ(5), σ(8),...) We have: zip k (unzip 0,k (σ),..., unzip k 1,k (σ)) = σ unzip i,k (zip k (σ 0,..., σ k 1 )) = σ i i < k In other words, zip k : (S N ) k S N is a bijection with inverse unzip k = unzip 0,k,..., unzip k 1,k : S N (S N ) k.

16 The k-automaton of Streams (Case k = 3) Define k-automaton structure on B ω, β = B ω hd,tl = B B ω id B unzip 3 B (B ω ) {1,2,3} = 1 σ σ(0) 2 unzip 0,3 (σ ) unzip 2,3 (σ ) unzip 3,3 (σ ) 3... By finality of (L, λ), there is unique h : (B ω, β) (L, λ)

17 The k-automaton of Streams (Case k = 3) Define k-automaton structure on B ω, β = B ω hd,tl = B B ω id B unzip 3 B (B ω ) {1,2,3} = 1 σ σ(0) 2 unzip 0,3 (σ ) unzip 2,3 (σ ) unzip 3,3 (σ ) 3... By finality of (L, λ), there is unique h : (B ω, β) (L, λ)

18 The k-automaton of Streams (Case k = 3) Define k-automaton structure on B ω, β = B ω hd,tl = B B ω id B unzip 3 B (B ω ) {1,2,3} = 1 σ σ(0) 2 unzip 0,3 (σ ) unzip 2,3 (σ ) unzip 3,3 (σ ) 3... By finality of (L, λ), there is unique h : (B ω, β) (L, λ)

19 Bijective k-adic Numeration Concretely, h(σ)(w) = σ(ν(w)) w {1, 2, 3} where ν : {1, 2, 3} = N is bijective 3-adic numeration (least significant digit first, reverse). defined by: ν(ε) = 0, ν(aw) = a + 3 ν(w)

20 Bijective k-adic Numeration Concretely, h(σ)(w) = σ(ν(w)) w {1, 2, 3} where ν : {1, 2, 3} = N is bijective 3-adic numeration (least significant digit first, reverse). defined by: ν(ε) = 0, ν(aw) = a + 3 ν(w) w ν(w) ε ν(ε) = ν(ε) = ν(ε) = ν(1) = ν(1) = ν(1) = ν(2) = 7..

21 The Final k-automaton of Streams ν bijective h : B ω L bijective h is isomorphism of k-automata (B ω, β) is also final. X [[ ]] B ω o,d hd,unzip 3 tl B X {1,2,3} id B [[ ]] {1,2,3} B (B ω ) {1,2,3} x X : [[x]](0) = o(x) unzip 0,3 ([[x]] ) = [[d(x)(1)]] unzip 1,3 ([[x]] ) = [[d(x)(2)]] unzip 2,3 ([[x]] ) = [[d(x)(3)]] or equivalently (using zip-unzip isomorphism): [[x]](0) = o(x), [[x]] = zip 3 ([[d(x)(1)]], [[d(x)(2)]], [[d(x)(3)]])

22 The Final k-automaton of Streams ν bijective h : B ω L bijective h is isomorphism of k-automata (B ω, β) is also final. X [[ ]] B ω o,d hd,unzip 3 tl B X {1,2,3} id B [[ ]] {1,2,3} B (B ω ) {1,2,3} x X : [[x]](0) = o(x) unzip 0,3 ([[x]] ) = [[d(x)(1)]] unzip 1,3 ([[x]] ) = [[d(x)(2)]] unzip 2,3 ([[x]] ) = [[d(x)(3)]] or equivalently (using zip-unzip isomorphism): [[x]](0) = o(x), [[x]] = zip 3 ([[d(x)(1)]], [[d(x)(2)]], [[d(x)(3)]])

23 The Final k-automaton of Streams ν bijective h : B ω L bijective h is isomorphism of k-automata (B ω, β) is also final. X [[ ]] B ω o,d hd,unzip 3 tl B X {1,2,3} id B [[ ]] {1,2,3} B (B ω ) {1,2,3} x X : [[x]](0) = o(x) unzip 0,3 ([[x]] ) = [[d(x)(1)]] unzip 1,3 ([[x]] ) = [[d(x)(2)]] unzip 2,3 ([[x]] ) = [[d(x)(3)]] or equivalently (using zip-unzip isomorphism): [[x]](0) = o(x), [[x]] = zip 3 ([[d(x)(1)]], [[d(x)(2)]], [[d(x)(3)]])

24 The Final k-automaton of Streams ν bijective h : B ω L bijective h is isomorphism of k-automata (B ω, β) is also final. X [[ ]] B ω o,d hd,unzip 3 tl B X {1,2,3} id B [[ ]] {1,2,3} B (B ω ) {1,2,3} x X : [[x]](0) = o(x) unzip 0,3 ([[x]] ) = [[d(x)(1)]] unzip 1,3 ([[x]] ) = [[d(x)(2)]] unzip 2,3 ([[x]] ) = [[d(x)(3)]] or equivalently (using zip-unzip isomorphism): [[x]](0) = o(x), [[x]] = zip 3 ([[d(x)(1)]], [[d(x)(2)]], [[d(x)(3)]])

25 Zip-Stream Differential Equations Example of a (finite) system of zip 3 -SDEs over X = {x, y, z} with B = {a, b}: x(0) = a, x = zip 3 (y, x, x), y(0) = b, y = zip 3 (x, y, x), z(0) = b, z = zip 3 (y, z, y), Lemma Every system of zip k -SDEs has a unique stream solution.

26 Zip-Stream Differential Equations Example of a (finite) system of zip 3 -SDEs over X = {x, y, z} with B = {a, b}: x(0) = a, x = zip 3 (y, x, x), y(0) = b, y = zip 3 (x, y, x), z(0) = b, z = zip 3 (y, z, y), Lemma Every system of zip k -SDEs has a unique stream solution.

27 From Deterministic to Linear Automata (X is a set) (S is an semiring) deterministic 1-automata X S X (streams) (eventually periodic) deterministic k-automata X S X k (languages/streams) (rational languages/?) linear 1-automata X S S(X ) (streams) (rational streams) linear k-automata X S S(X ) k (power series/streams) (rational/?)

28 Linear k-automata Let S = (S, +,, 0, 1) be a semiring. An S-linear k-automaton with state space X is a map: where X S S(X ) k S(X ) = {φ: X S supp(φ) finite} = {s 1 x s n x n s 1,..., s n S, x 1,..., x n X } BDE format, example (S = N, k = {a, b}): x(0) = 1, x a = 2x + y, x b = 3x y(0) = 0, y a = x + 3y, y b = y

29 Linear k-automata Let S = (S, +,, 0, 1) be a semiring. An S-linear k-automaton with state space X is a map: where X S S(X ) k S(X ) = {φ: X S supp(φ) finite} = {s 1 x s n x n s 1,..., s n S, x 1,..., x n X } BDE format, example (S = N, k = {a, b}): x(0) = 1, x a = 2x + y, x b = 3x y(0) = 0, y a = x + 3y, y b = y

30 Language Semantics of Linear k-automata S-weighted languages are formal power series (in k noncommuting variables with coefficients in S) S k = {ρ: k S} Language semantics via determinisation : X η S(X ) S k o,d S S(X ) k ô, ˆd λ S S k k Note: ô, ˆd : S(X ) S S(X ) k is deterministic k-automaton in category of S-semimodules. Note: S k is S-semimodule with pointwise operations.

31 Stream Semantics of Linear k-automata Stream semantics via determinisation and k-adic numeration: X η S(X ) [[ ]] S S ω o,d S S(X ) k ô, ˆd S (S ω ) k hd,unzip k tl [[ ]] S is the unique S-linear map such that η(x) S (0) = o(x) η(x) S = zip k ( δ(x)(1) S,..., δ(x)(k) S ).

32 Stream Specification Streams specified in linear BDE format: x(0) = 1, x 1 = 2x + y, x 2 = 3x y(0) = 0, y 1 = x + 3y, y 2 = y equivalent with linear even-odd specification: x(0) = 1, even(x ) = 2x + y, odd(x ) = 3x y(0) = 0, even(y ) = x + 3y, odd(y ) = y equivalent with linear zip-stream differential equations: x(0) = 1, x = zip 2 (2x + y, 3x) y(0) = 0, y = zip 2 (x + 3y, y) Lemma A system of linear zip-sdes has a unique solution.

33 Stream Specification Streams specified in linear BDE format: x(0) = 1, x 1 = 2x + y, x 2 = 3x y(0) = 0, y 1 = x + 3y, y 2 = y equivalent with linear even-odd specification: x(0) = 1, even(x ) = 2x + y, odd(x ) = 3x y(0) = 0, even(y ) = x + 3y, odd(y ) = y equivalent with linear zip-stream differential equations: x(0) = 1, x = zip 2 (2x + y, 3x) y(0) = 0, y = zip 2 (x + 3y, y) Lemma A system of linear zip-sdes has a unique solution.

34 Stream Specification Streams specified in linear BDE format: x(0) = 1, x 1 = 2x + y, x 2 = 3x y(0) = 0, y 1 = x + 3y, y 2 = y equivalent with linear even-odd specification: x(0) = 1, even(x ) = 2x + y, odd(x ) = 3x y(0) = 0, even(y ) = x + 3y, odd(y ) = y equivalent with linear zip-stream differential equations: x(0) = 1, x = zip 2 (2x + y, 3x) y(0) = 0, y = zip 2 (x + 3y, y) Lemma A system of linear zip-sdes has a unique solution.

35 Stream Specification Streams specified in linear BDE format: x(0) = 1, x 1 = 2x + y, x 2 = 3x y(0) = 0, y 1 = x + 3y, y 2 = y equivalent with linear even-odd specification: x(0) = 1, even(x ) = 2x + y, odd(x ) = 3x y(0) = 0, even(y ) = x + 3y, odd(y ) = y equivalent with linear zip-stream differential equations: x(0) = 1, x = zip 2 (2x + y, 3x) y(0) = 0, y = zip 2 (x + 3y, y) Lemma A system of linear zip-sdes has a unique solution.

36 Automatic and Regular Sequences Def. Let σ S ω and k N. The k-kernel of σ is the set of streams obtained by closing {σ} under unzip j,k for all j < k.

37 k-automatic Sequences (cf. Automatic Sequences, J.P. Allouche and J. Shallit, CUP 2003) A sequence σ S ω is k-automatic if one of the following equivalent conditions hold: σ is generated by a finite k-automaton, i.e., n N : σ(n) = o(d (x)(base k (n))) where base k (n) is base k expansion of n. E.g. base 2 (6) = 110. the k-kernel of σ is finite, i.e., there is Σ = {τ 1,..., τ n } S ω such that σ Σ and unzip j,k (τ i ) Σ i n j < k solution to a finite systems of zip-equations: τ 1 = zip k (τ 1,1,..., τ 1,k ).. τ n = zip k (τ n,1,..., τ n,k ) where τ i,j S ω.

38 k-automatic Sequences (cf. Automatic Sequences, J.P. Allouche and J. Shallit, CUP 2003) A sequence σ S ω is k-automatic if one of the following equivalent conditions hold: σ is generated by a finite k-automaton, i.e., n N : σ(n) = o(d (x)(base k (n))) where base k (n) is base k expansion of n. E.g. base 2 (6) = 110. the k-kernel of σ is finite, i.e., there is Σ = {τ 1,..., τ n } S ω such that σ Σ and unzip j,k (τ i ) Σ i n j < k solution to a finite systems of zip-equations: τ 1 = zip k (τ 1,1,..., τ 1,k ).. τ n = zip k (τ n,1,..., τ n,k ) where τ i,j S ω.

39 Example Thue-Morse is 2-automatic σ = (0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0,...) As 2-automaton with output in 2={0,1}: 1 0 x 0 y As system of zip-equations 2-kernel has size 2. x = zip 2 (x, y) y = zip 2 (y, x)

40 k-regular Sequences Def. (cf. Allouche-Shallit) A sequence σ S ω is k-regular if one of the following equivalent conditions hold: the k-kernel of σ is contained in a finitely generated S-semimodule, i.e., there is Σ = {τ 1,..., τ n } S ω such that σ Σ and for all i = 1,..., n unzip j,k (τ i ) S(Σ) j < k (S(Σ) = S-linear combinations over τ 1,..., τ n ) σ is solution to finite system of linear zip-equations: τ 1 = zip k (γ 1,1,..., γ 1,k ).. τ n = zip k (γ n,1,..., γ n,k ) where γ i,j S({τ 1,..., τ n }), τ 1,..., τ n S ω.

41 Example of k-regular Sequence The sequence of numbers whose base 3 representation does not contain the digit 2 is solution to Hence is 2-regular. σ = (0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31,...) σ = zip 2 (3σ, 3σ + ones) ones = zip 2 (ones, ones)

42 Characterisation Thm. TFAE A sequence σ S ω is k-regular σ is the solution to a finite system of linear zip k -SDEs σ is generated by a finite linear k-automaton. h(σ) is a rational power series. Thm. TFAE A sequence σ S ω is k-automatic σ is the solution to a finite system of zip k -SDEs σ is generated by a finite k-automaton. (cf. [Endrullis-Grabmayer-Hendriks-Klop-Moss 11])

43 Characterisation Proof sketch: Transform linear zip-equations into linear zip-sdes. E.g., for earlier system: σ = zip(3σ + ones, 3σ ) σ = zip(3σ, 3σ + ones ) ones = zip(ones, ones ) ones = zip(ones, ones ) introduce variables and add initial values: w(0) = 3 σ(0), w = zip(3w + y, 3x) x(0) = 3 σ(0) + 1, x = zip(3x, 3x + z) y(0) = 1, y = zip(y, z) z(0) = 1, z = zip(z, z)

44 Characterisation Proof sketch: Transform linear zip-equations into linear zip-sdes. E.g., for earlier system: σ = zip(3σ + ones, 3σ ) σ = zip(3σ, 3σ + ones ) ones = zip(ones, ones ) ones = zip(ones, ones ) introduce variables and add initial values: w(0) = 3 σ(0), w = zip(3w + y, 3x) x(0) = 3 σ(0) + 1, x = zip(3x, 3x + z) y(0) = 1, y = zip(y, z) z(0) = 1, z = zip(z, z)

45 Application: Divide and Conquer Recurrences On the Online Encyclopedia of Integer Sequences (OEIS), some formats for divide and conquer recurrences are given. E.g. a(2n) = Ca(n) + Ca(n 1) + P(n) a(2n + 1) = 2Ca(n) + Q(n) where C Z, P and Q are expressible by a rational g.f. Question (asked on oeis.org/somedcgf.html): An open question would be whether all sequences here discussed are 2-regular. Answer: if you replace the condition expressible by a rational g.f. by 2-regular then yes (includes all their examples), otherwise no.

46 Application: Divide and Conquer Recurrences On the Online Encyclopedia of Integer Sequences (OEIS), some formats for divide and conquer recurrences are given. E.g. a(2n) = Ca(n) + Ca(n 1) + P(n) a(2n + 1) = 2Ca(n) + Q(n) where C Z, P and Q are expressible by a rational g.f. Question (asked on oeis.org/somedcgf.html): An open question would be whether all sequences here discussed are 2-regular. Answer: if you replace the condition expressible by a rational g.f. by 2-regular then yes (includes all their examples), otherwise no.

47 Conclusion Summary: k-regular weighted automata = k-automatic determ. automata = rational streams event.per. streams Isomorphism between final k-automaton of languages and final k-automaton of streams (via bijective k-adic numeration). Bijective correspondence between rational power series and k-regular sequences. Characterisation of k-regular sequences as solutions to finite systems of linear zip k -SDEs.

48 Conclusion Future work: (constructively) algebraic power series (in BDEs, replace linear comb s with polynomials). Connection: algebraic power series and context-free sequences? other numeration systems?