Scalar Ambiguity and Freeness in Matrix Semigroups over Bounded Languages

Transcription

1 in Matrix Semigroups over Bounded Languages Paul C. Bell [1], Shang Chen [1], Lisa Jackson [2] Dept of Computer Science [1], Dept of Aeronautical and Automotive Engineering [2], Loughborough University, UK Language and Automata Theory and Applications (LATA 2016) Czech Technical University

2 Outline of the talk Introduction and notation Matrix freeness problems Scalar ambiguity and freeness Problems on bounded languages Probabilistic finite automata Conclusion

3 Notations We denote an n-dimensional matrix over a semiring F by F n n Given a set of matrices G = {M 1, M 2,..., M k } K n n (where K {Z, Q, R, C, H}), we denote by S = G the semigroup generated by G

4 Decision Problems for Matrix Semigroups There has been much study on properties of finitely generated matrix semigroups Given a matrix semigroup S generated by a finite set G = {M 1, M 2,..., M k } K n n (where K {Z, Q, R, C, H}): Decide whether the semigroup S contains the zero matrix (Mortality Problem) contains the identity matrix (Identity Problem) is free (Freeness Problem) is bounded, finite, etc. Vector reachability problems: Given two vectors x and y. Decide whether the semigroup S contains a matrix M such that Mx = y Variants of such problems are important for probabilistic and quantum automata models

5 Early Reachability Results Problem (The Mortality Problem) Given a matrix semigroup S generated by a finite set G = {M 1, M 2,..., M k } Z n n, determine if 0 G, where 0 is the n-dimensional zero matrix. The Mortality Problem was one of the earliest undecidability results of reachability for matrix semigroups Theorem ([Paterson 70]) The Mortality Problem is undecidable over Z 3 3 holds even when the semigroup is generated by just 6 matrices over Z 3 3, or for 2 matrices over Z [Cassaigne et al., 14]

6 The Identity Problem Problem (The Identity Problem) Given a matrix semigroup S generated by a finite set G = {M 1, M 2,..., M k } Z n n, determine if I n G, where I n is the n-dimensional multiplicative identity matrix. Theorem ([B., Potapov 11]) The Identity Problem is undecidable over Z 4 4 The problem is decidable over Z 2 2 [Choffrut, Karhumäki 05]

7 Semigroup Freeness Definition (Code) Let S be a semigroup and G a subset of S. We call G a code if the property u 1 u 2 u m = v 1 v 2 v n for u i, v i G, implies that m = n and u i = v i for each 1 i n. Definition (Semigroup freeness) A semigroup S is called free if there exists a code G S such that S = G +. For example, consider the semigroup {0, 1} + under concatenation. Then the set {00, 01, 10, 11} is a code, but {01, 10, 0} is not (since 0 10 = 01 0 for example)

8 Matrix Freeness Problem (Matrix semigroup freeness) Semigroup freeness problem - Given a finite set of matrices G Z n n generating a semigroup S, does every element M S have a single, unique factorisation over G? Alternatively, is G a code? Theorem (Klarner, Birget and Satterfield, 91) The semigroup freeness problem is undecidable over N 3 3 Undecidability holds even over N 3 3 uptr [Cassaigne, Harju and Karhumäki, 99]

9 Let G = {G 1, G 2,..., G k } F n n, S = G and ρ, τ F n. Define Λ(G) = {λ : λ = ρ T Mτ M S}. For example, λ = (1, 0, 0, 0)

10 If for λ Λ(G) there exists a unique matrix M S such that λ = ρ T Mτ, then λ is unambiguous with respect to G, ρ, τ. Λ(G) is unambiguous if every λ Λ(G) is unambiguous. If for λ Λ(G) there exists a unique product G i1 G i2 G im S, where G il G such that λ = ρ T G i1 G i2 G im τ, then λ is free with respect to G, ρ, τ. Λ(G) is called free if every λ Λ(G) is free.

11 Scalar Ambiguity Example {( ) ( )} Given G =,, and ρ = τ = (1, 0) T, then G is a free semigroup, but ( ) T ( ) ( ) = = ( 1 0 ) T ( ) ( ) 1 0 1, thus 1 is ambiguous with respect to G, ρ, τ and thus Λ(G) is ambiguous even though G is free.

12 Example 2 {( ) ( )} Given G =,, and ρ = (1, 0) T, τ = (0, 1) T, then each scalar k N has a unique representation ( ) T ( ) ( ) 1 1 k 0 k = = Thus Λ(G) is unambiguous. However, ( ) T ( ) ( ) k ( ) k = = ( 1 0 ( 1 0 ) T ( ) k ( ) ) T ( ) k ( ) ( ) , thus scalar k has two factorizations implying Λ(G) is not free.

13 Properties of Lemma Let G = {G 1, G 2,..., G k } F n n, S = G and ρ, τ F n, then: (1) If Λ(G) is ambiguous, then Λ(G) is not free. (2) if Λ(G) is free, then S is free.

14 Problems related to Scalar Ambiguity/Freeness Let G = {G 1, G 2,..., G k } F n n, S = G and ρ, τ F n, then we may consider the following decision problems. Problem (Scalar Ambiguity) Is Λ(G) unambiguous with respect to G, ρ, τ? Problem (Scalar Freeness) Is Λ(G) free with respect to G, ρ, τ?

15 Main Theorem 1 Theorem The Scalar Freeness Problem is undecidable over Z 3 3 and the Scalar Ambiguity Problem is undecidable over Z 4 4. The proof uses a reduction from the Mixed Modification Post s Correspondence Problem (MMPCP)

16 Mixed Modification Post s Correspondence Problem (MMPCP) Problem (Mixed Modification PCP (MMPCP)) Given alphabet Σ, binary alphabet, and morphisms h, g : Σ, does there exist w = x 1... x k Σ + ; x i Σ s.t. h 1 (x 1 )h 2 (x 2 )... h k (x k ) = g 1 (x 1 )g 2 (x 2 )... g k (x k ), where h i, g i {h, g}, and there exists at least one j such that h j g j. Theorem (Cassaigne, Karhumäki, Harju 96) The Mixed Modification PCP is undecidable.

17 From Words to Rationals Let Σ = {x 1, x 2,..., x n 2 } and = {x n 1, x n } be alphabets and h, g : Σ be an instance of MMPCP. We define two morphisms α, β : (Σ ) Q by: α(x i1 x i2 x im ) = Σ m j=1 i j(n + 1) m j N, β(x i1 x i2 x im ) = Σ m j=1 i j(n + 1) j m 1 (0, 1) Q, and α(ε) = β(ε) = 0. w 1, w 2 (Σ ), (n + 1) w 2 α(w 1 ) + α(w 2 ) = α(w 1 w 2 ) and (n + 1) w 2 β(w 1 ) + β(w 2 ) = β(w 1 w 2 ).

18 From Pairs of Words to 3 3 Rational Matrices Define γ : (Σ ) (Σ ) Q 3 3 by (n + 1) u 0 α(u) γ (u, v) = 0 (n + 1) v β(v) It is easy to verify that γ (u 1, v 1 )γ (u 2, v 2 ) = γ (u 1 u 2, v 1 v 2 ), i.e., γ is a homomorphism.

19 From Pairs of Words to 3 3 Rational Matrices Define two more matrices T and T 1 : T = 0 1 0, T 1 = and let γ : (Σ ) (Σ ) Q 3 3 be defined as γ(u, v) = (n + 1) u (n + 1) v (n + 1) u α(u) + β(v) T γ (u, v)t 1 = 0 (n + 1) v β(v) We can now verify that, γ(u 1, v 1 )γ(u 2, v 2 ) = T γ (u 1, v 1 )TT 1 γ (u 2, v 2 )T 1 = T γ (u 1 u 2, v 1 v 2 )T 1 = γ(u 1 u 2, v 1 v 2 ).

20 From Pairs of Words to 3 3 Rational Matrices Let G = {γ(x i, g(x i )), γ(x i, h(x i )) x i Σ, 1 i n 2}, S = G, ρ = (1, 0, 0) T and τ = (0, 0, 1) T. Assume that there exists M 1 = G i1 G i2 G it G and M 2 = G j1 G j2 G jt G such that t t or else at least one G ip G jp where 1 p t and λ = ρ T M 1 τ = ρ T M 2 τ. We see that: ρ T M 1 τ = (M 1 ) [1,3] = α(x i1 x i2 x it ) + β(f 1 (x i1 )f 2 (x i2 ) f t (x it )), ρ T M 2 τ = (M 2 ) [1,3] = α(x j1 x j2 x jt ) + β(f 1 (x j 1 )f 2 (x j 2 ) f t (x jt )), where each f i, f i {g, h}.

21 From Pairs of Words to 3 3 Matrices Since α : (Σ ) N and β : (Σ ) (0, 1) Q are homomorphisms, then which implies that x i1 x i2 x it = x j1 x j2 x jt, f 1 (x i1 )f 2 (x i2 ) f t (x it ) = f 1(x i1 )f 2(x i2 ) f t (x i t ) is a solution to the instance of MMPCP.

22 Bounded Languages Theorem The Scalar Freeness Problem over a bounded language is undecidable. In other words, given k matrices M 1, M 2,..., M k Q n n, generating bounded language M = M 1 M 2 M k, and two vectors ρ, τ Zn, it is undecidable to decide if there exist l 1, l 2,..., l k, r 1, r 2,..., r k N such that ρ T M l 1 1 M l M l k k τ = ρ T M r 1 1 Mr Mr k k τ, where l j r j for at least one j.

23 PFA on Bounded Languages Definition (Probabilistic Finite Automaton) A Probabilistic Finite Automaton (PFA) over an alphabet A is a triplet (u, ϕ, v). For a given PFA denoted R = (u, ϕ, v) and a word w A, we can define a function f R : A [0, 1], where: f R (w) = u T ϕ(w)v [0, 1] ; w A. This is the probability of R being in a final state after reading word w A.

24 PFA on Bounded Languages Problem (PFA Ambiguity Problem) Given a PFA R = (u, ϕ, v) over a bounded language L A, do there exist two different words w 1, w 2 L such that u T ϕ(w 1 )v = u T ϕ(w 2 )v? Theorem Ambiguity for PFA over a bounded language is undecidable.

25 Conclusion Two new notions of freeness problems for matrix semigroups We studied the problems on arbitrary semigroups and bounded languages Links to problems in Probabilistic Finite Automata