Coalgebra, Lecture 15: Equations for Deterministic Automata

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1 Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined nd show the generl ide nd concepts for this purpose. 1 Deterministic utomt We fix (not necessrily finite) set A clled n lphet. A deterministic utomton on A is pir (X, α) where X is set of sttes nd α X A X is function clled its trnsition function. In other words, deterministic utomton on A is n F lger for the functor F Set Set given y F X = A X. By considering deterministic utomt s F lgers we get the notion of homomorphism of utomt, which is the one of eing n F lger morphism. Tht is, n F lger morphism h (X, α) (Y, β) is function h X Y such tht the following digrm commutes: A X α X id A h h A Y Y β which mens tht h(α(, x)) = β(, h(x)) for every A nd x X. In the cse tht A nd X re finite sets, we cn drw the digrm of the utomton (X, α) which is the digrm with nodes in X nd rrows from node x 1 to node x 2 with lel s in x 2 for every A nd x 1, x 2 X such tht α(, x 1 ) = x 2. x 1 Exmple 1. If A = {, } nd X = {x 1, x 2, x 3 }, then the following digrm, x 1 x 2 x 3 represents the utomton (X, α) such tht α(, x 1 ) = α(, x 1 ) = x 2, α(, x 2 ) = x 2, α(, x 2 ) = x 3, α(, x 3 ) = x 3 nd α(, x 3 ) = x 2. 1

2 2 Exercise 1. Let A = {, }, X = {x 1, x 2, x 3 } nd Y = {y 1, y 2, y 3 }. Consider the utomt (X, α) nd (Y, β) whose digrms re given y:, x 1 x 2 x 3 y 1 y 2 y 3 i) Is there ny homomorphism of utomt h (X, α) (Y, β) such tht h(x 1 ) = y 1? Find such n h or, if it is not possile, explin why. ii) How mny homomorphisms of utomt h (X, α) (Y, β) re there? Justify your nswer. iii) How mny homomorphisms of utomt h (Y, β) (X, α) re there? Justify your nswer. We denote y A the set of ll words with symols in A. Tht is, every element w A is of the form w = 1 n, n N, where ech i A, 1 i n. In the prticulr cse tht n = 0, we otin the empty word which we denote y ɛ. Nottion. Given deterministic utomton (X, α) on A, w A nd x X, we define the stte w(x) X y induction s follows: x if w = ɛ, w(x) = α(, u(x)) if w = u, u A, A thus w(x) is the stte we rech from x y processing the word w from right to left. Exmple 2 (Exmple 1 continued). If we consider the utomton (X, α) given in Exmple 1, then we hve the following: (x 1 ) = x 2, (x 1 ) = x 3, 8 (x 3 ) = x 2, ɛ(x 2 ) = x 2, (x 1 ) = x 3. An esy wy to rememer how to do the previous clcultions is y introducing prenthesis for ech symol in A. For exmple: (x 1 ) = ((x 1 )) = (x 2 ) = ((x 2 )) = (x 2 ) = x 2. Remrk. Using the previous nottion, homomorphism of utomt h (X, α) (Y, β) is function h X Y such tht for every A nd x X we hve h((x)) = (h(x)). Exercise 2. Let A = {, } nd X = {x 1, x 2 }. Find the numer of functions α A X X such tht the utomton (X, α) stisfies the following two conditions: i) (x 1 ) = x 1. ii) For ll x X we hve tht (x) = (x). (Condition ii) sys tht the eqution = is stisfied y the utomton (X, α))

3 3 2 Equtions for deterministic utomt Now we turn our ttention to the study of equtions for deterministic utomt (cf. Exercise 2 ii)) which, informlly, re pirs of words (u, v) A A such tht the utomton cnnot distinguish etween processing the word u nd processing the word v from ny given stte. This is defined s follows. Definition 3. Let A e n lphet, n eqution on A is pir (u, v) A A. We sy tht the utomton (X, α) on A stisfies the eqution (u, v), denoted s (X, α) (u, v) if for every x X we hve u(x) = v(x). We denote y Eq(X, α) the set of equtions tht (X, α) stisfies, tht is Eq(X, α) = {(u, v) A A (X, α) (u, v)}. As (X, α) (u, v) if nd only if (X, α) (v, u), we cn denote the eqution (u, v) s u = v nd then we hve: (X, α) u = v x X u(x) = v(x). Exmple 4. Let A = {, }, X = {x, y} nd consider the utomton (X, α) on A given y the following digrm: x y Then (X, α) stisfies equtions such s =, = nd =, ut it does t stisfies the eqution = nor the eqution ɛ =, since (x) = y x = (x) nd ɛ(y) = y x = (y). Now, how cn we find ll the equtions tht (X, α) stisfies? This is in generl nontrivil tsk since there could e infinitely mny of them. For exmple, since (X, α) stisfies = then it lso stisfies ny eqution of the form = 2n+1 for n 1, ll of them re otined from = y replcing y, since =, s follows: = = = = 7 = 9 = 11 = In this sense, ech eqution = 2n+1, n 1, is generted (i.e., cn e deduced y using sustitution) y the single eqution =. In the cses tht A nd X re finite sets we cn find finite set of equtions tht genertes ll the equtions stisfied y (X, α). We will illustrte how to otin genertor set of equtions for the utomton given ove nd then formlize the concepts in Section 4. By Definition 3 ove we hve tht (X, α) u = v iff x X u(x) = v(x), so we re going to consider ll the sttes of (X, α) t the sme time nd mke trnsitions for ll the symols in A to see when the stte we rech with two different words is the sme. Tht is, we put ll the sttes of (X, α) in the tuple (x, y) nd strt to mke trnsitions, ccording to (X, α), for ech symol in A. For exmple, if we mke n trnsition nd trnsition from the tuple (x, y) we get the tuples (y, y) nd (x, x), respectively, which is illustrted in the following picture:

4 4 (x, y) (y, y) (x, x) Until now, there re no different words, strting from (x, y), tht will tke us to the sme tuple, hence we hve not found ny equtions yet. But, s we hve new tuples, nmely (y, y) nd (x, x), we need to do the trnsitions from those sttes for every symol in A. We cn strt with (y, y) y mking ll the trnsitions for every symol in A to otin the following: (x, y) (y, y) (x, x) In ech of those trnsitions we found two different words tht will tke us to the sme tuple. In fct, i) The words nd tke us from (x, y) to the sme tuple (y, y), which mens tht the eqution = is stisfied y (X, α). ii) The words nd tke us from (x, y) to the sme tuple (x, x), which mens tht the eqution = is stisfied y (X, α). Note tht in i) nd ii) we lwys strt from the tuple (x, y) which is the tuple tht represents ll the sttes of the utomton. Also, the equtions otined in i) nd ii) ove re given y the two shortest pths tht tke us to the sme tuple, which in some sense is the minimum informtion we wnt to cpture in n eqution. For exmple, the words nd will tke us to the sme tuple, i.e., the utomton stisfies the eqution =, ut = cn e deduced from = nd =. Therefore, the equtions in i) nd ii) ove re enough to deduce every eqution tht comes from the previous digrm. Now, we still hve to do the trnsitions from the tuple (x, x) to find new equtions nd/or new tuples. By mking ll the trnsitions from the tuple (x, x) we otin the following: (x, y) (y, y) (x, x) Agin, in ech of those trnsitions we found two different words tht will tke us to the sme tuple. In fct, iii) The words nd tke us from (x, y) to the sme tuple (y, y), which mens tht the eqution = is stisfied (X, α).

5 5 iv) The words nd tke us from (x, y) to the sme tuple (y, y), which mens tht the eqution = is stisfied (X, α). Finlly, s every tuple in the previous digrm hs ll the trnsitions for ech symol in A, we hve finished the process for finding generting set for the equtions tht the utomton (X, α) stisfies. Hence, generting set for Eq(X, α) is given y the equtions: =, =, =, = Tht is, every eqution in Eq(X, α) cn e deduced from the four equtions ove. Exercise 3. Find generting set for the equtions of the utomt (X, α) nd (Y, β) given in Exercise 1. 3 Monoids In this section, we study some sic concepts nd fcts out monoids tht will e used in the next section. We strt y defining the lgeric structure of monoid. Definition 5. A monoid is tuple M = (M,, e) such tht M is set, is inry opertion on M, clled multipliction, nd e is element in M, clled the identity element, nd they stisfy the following xioms: i) Associtivity: For every x, y, z M we hve tht (x y) z = x (y z). ii) e is the identity: for every x M we hve tht x e = e x = x. Exmple 6. The following re some exmples of monoids: i) The monoid (N, +, 0) of nturl numers with ddition nd identity element 0. ii) The monoid (N,, 1) of nturl numers with multipliction nd identity element 1. iii) The monoid (R,, 1) of rel numers with multipliction nd identity element 1. iv) The monoid (M 2 2 (R),, I 2 ) of 2 2 mtrices on the rel numers with mtrix multipliction nd identity element I 2 given y: I 2 = [ ] v) For ny given set A the monoid A = (A,, ɛ) of words on A with multipliction given y conctention nd identity element the empty word ɛ. This monoid plys n importnt role in (universl) lger nd lnguge theory nd it is clled the free monoid on A. vi) For ny given set X the monoid X X = (X X,, id X ) of functions from X to X with composition of functions nd identity element given y the identity function id X on X.

6 6 Definition 7. Let M = (M,, e) nd N = (N,, e ) e monoids. A monoid homomorphism h M N from M to N is function h M N tht preserves the monoid opertions, tht is: i) For ll m 1, m 2 M we hve tht h(m 1 m 2 ) = h(m 1 ) h(m 2 ), nd ii) h(e) = e. Exmple 8. Consider the function h N N defined y h(n) = 2 n. Then h is monoid homomorphism from the monoid (N, +, 0) of nturl numers with ddition to the monoid (N,, 1) of nturl numers with multipliction. In fct, for every n, m N we hve h(n + m) = 2 n+m = 2 n 2 m = h(n) h(m) nd h(0) = 2 0 = 1. Exercise 4. Define the determinnt function det M 2 2 (R) R s: det ([ c ]) = d c d i) Show tht det is monoid homomorphism from the monoid (M 2 2 (R),, I 2 ) of 2 2 mtrices on the rel numers with mtrix multipliction to the monoid (R,, 1) of rel numers with multipliction. ii) Is det monoid homomorphism from the monoid (M 2 2 (R), +, ) of 2 2 mtrices on the rel numers with mtrix ddition to the monoid (R, +, 0) of rel numers with ddition? Justify your nswer. For ny given set A we defined the free monoid (A,, ɛ) where A is the set of words with symols on A nd is given y the conctention of words. As every symol in A is n element in A, nmely the word w with only one symol given y w =, we hve function η A A A defined s η A () =. The monoid (A,, ɛ) is clled the free monoid on A since it stisfies the following (universl) property: (UP) For ny monoid (M,, e) nd ny function f A M there exists unique monoid homomorphism f from (A,, ɛ) to (M,, ɛ) such tht the following digrm commutes: η A A A f f M The monoid homomorphism f is cnoniclly defined for ny w = 1 n, n 1, i A s: f (w) = f ( 1 n ) = f( 1 ) f( n ) nd, clerly, f (ɛ) = e. The homomorphism f is clled the extension of f, since they oth gree on A, i.e., f () = f() for every A. The previous universl property sys tht in order to get monoid homomorphism from (A,, ɛ) to (M,, e) it is enough to define function f A M.

7 7 Exmple 9. Let A = {,, c} nd consider the monoid (Z 7, +, 0) of integers modulo 7 with ddition. Let f A Z 7 e the function such tht: f() = 2, f() = 5, f(c) = 3 Then, y the universl property (UP) ove, we cn find the vlue of the monoid homomorphism f for ny element w A. For exmple, we hve the following: f ( 2 c 3 ) = f() + f() + f(c) + f(c) + f(c) = 2f() + 3f(c) = = 6 f ( 4 2 cc 4 ) = 4f() + 2f() + f(c) + f() + 4f(c) = = 3. Definition 10. Let h M N e homomorphism of monoids etween the monoid (M,, e) nd the monoid (N,, e ). Define the kernel ker(h) of h nd the imge Im(h) of h s follows: ker(h) = {(m 1, m 2 ) M M h(m 1 ) = h(m 2 )} Im(h) = {n N m M h(m) = n} The imge of homomorphism hs cnonicl monoid structure s follows. Proposition 11. Let h M N e homomorphism of monoids etween the monoid (M,, e) nd the monoid (N,, e ). Then, Im(h) = (Im(h),, e ) is monoid nd it is sumonoid of (N,, e ). Proof. Clerly e Im(h) since h(e) = e. Associtivity follows from the fct tht h is homomorphism of monoids. Definition 12. Let M = (M,, e) e monoid. A congruence of M is n equivlence reltion θ on M such tht for every (m, n), (x, y) θ we hve tht (m x, n y) θ. Given set of pirs {(m i, n i )} i I in M M, we denote y {(m i, n i )} i I the lest congruence of M tht contins ech (m i, n i ), i I. {(m i, n i )} i I is clled the congruence generted y {(m i, n i )} i I. In cse tht I is finite nd {(m i, n i )} i I = {(m 1, n 1 ),..., (m k, n k )} we denote {(m i, n i )} i I s (m 1, n 1 ),..., (m k, n k ). Proposition 13. Let M = (M,, e) e monoid nd let θ e congruence of M. Let M/θ the set of ll equivlence clsses nd denote y [m] θ the equivlence clss of m M with respect to θ. Define the opertion on M/θ s [m] θ [n] θ = [m n] θ. Then M/θ = (M/θ,, [e] θ ) is monoid. Proof. Note tht is well defined since θ is congruence. Associtivity of nd the fct tht [e] θ is the identity of M/θ follow from the definition of nd the fct tht M is monoid. Exercise 5. i) Prove tht the kernel of monoid homomorphism is congruence. ii) Prove tht n equivlence reltion θ on A is congruence of the monoid (A,, ɛ) if nd only if is stisfies the following property: - For every A, (w, v) θ implies (w, v), (w, v) θ.

8 8 iii) Let (X, α) e n utomton on A. Show tht Eq(X, α) is congruence of the monoid (A,, ɛ). iv) Let M = (M,, e) nd N = (N,, e ) e monoids. We sy tht M nd N re isomorphic if there exist ijective monoid homomorphisms ϕ M N. Show tht: ) The inverse ϕ 1 of ijective monoid homomorphism ϕ is monoid homomorphism. ) For ny homomorphism h M N from M to N the monoids M/ker(h) nd Im(h) re isomorphic. v) Let θ e congruence on (A,, ɛ). Construct n utomton (X, α) on A such tht Eq(X, α) = θ. 4...putting everything together We finish y showing the connections etween the concepts we previously defined, this will give us cler ide on how we cn use lgeric nd ctegoricl techniques to study equtions for deterministic utomt. Given n utomton (X, α) on A, which is completely determined y its trnsition function α, we define the function α A X X s α()(x) = α(, x), A nd x X. We hve tht oth kind of functions re in one to one correspondence y the identity α()(x) = α(, x), i.e., we cn recover ny of them if we know the other one nd this correspondence is ijective. This is summrized s: α A X X α A X X α()(x) = α(, x) Now, given such n α A X X, y the universl property (UP) given in the previous section, there exists unique monoid homomorphism α from the monoid (A,, ɛ) to the monoid (X X,, id X ) such tht α η A = α. Lemm 14. Let (X, α) e deterministic utomton on A. Then for every x X nd w A we hve tht α (w)(x) = w(x). Proof. We proof this y induction on the length of w. In fct, i) If w = ɛ, then α (ɛ) = id X, which implies tht α (ɛ)(x) = id X (x) = x = ɛ(x). ii) If w = u with A nd u A, then we hve: α (u)(x) = ( α () α (u)) (x) = α () ( α (u)(x)) = α()(u(x)) = α(, u(x)) = u(x).

9 9 where the first equlity follows from the fct tht α is monoid homomorphism, the second one from definition of composition, the third one from the induction hypothesis nd the fct tht A, the fourth one from the definition of α, nd the lst one is the nottion we defined. Corollry 15. Let (X, α) e deterministic utomton on A. Then Eq(X, α) = ker( α ). Definition 16. Let (X, α) e deterministic utomton on A. The trnsition monoid trns(x, α) of (X, α) is the monoid defined s: trns(x, α) = Im( α ). By Exercise 5 nd the previous corollry, we hve tht trns(x, α) is isomorphic to (A,, ɛ)/eq(x, α). Exmple 17 (Exmple 4 continued). For A = {, } nd X = {x, y} we considered the utomton (X, α) on A given y the digrm: x In order to get generting set for the equtions of (X, α) we constructed the digrm y (x, y) (y, y) (x, x) From this digrm we cn otin ll the functions α (w) for ny w A, which is the function tht mps ech element in the tuple (x, y), which is the tuple contining ll the different elements in X, to its corresponding element in the tuple we rech from (x, y) y processing the word w from right to left. For exmple, if we consider the word then we rech the tuple (y, y) from the tuple (x, y), this mens tht the function α () mps ech element in (x, y) to its corresponding element in (y, y), i.e., α ()(x) = α ()(y) = y. According to this, since trns(x, α) = Im( α ) nd it is isomorphic to (A,, ɛ)/eq(x, α), we otin the equtions Eq(X, α) of (X, α) y looking t the pirs of words (u, v) such tht from the tuple (x, y) we get the sme tuple y processing the word u nd y processing the word v from right to left. This is wht we did in Exmple 4 ut we restricted our ttention to the equtions tht generted the congruence Eq(X, α). In this cse, we hve tht Eq(X, α) = (, ), (, ), (, ), (, ). Exercise 6. Let A e n lphet nd let E e set of equtions on A. Given n utomton (X, α) on A, we sy tht (X, α) stisfies E, denoted s (X, α) E, if (X, α) (u, v) for every (u, v) E. Let θ e congruence of (A,, ɛ) nd define the function τ θ A A /θ s τ θ (w) = [w] θ, then we hve tht τ θ is monoid homomorphism from (A,, ɛ) to (A /θ,, [ɛ] θ ) (why?). Show tht (X, α) θ if nd only if there exists monoid homomorphism g from (A /θ,, [ɛ] θ ) to (X X,, id X ) such tht α = g τ θ.