The Zeta Function of a Cyclic Language with Connections to Elliptic Curves and Chip-Firing Games

Transcription

1 The Zeta Function of a Cyclic Language with Connections to Elliptic Curves and Chip-Firing Games University of California, San Diego March 14, 2007 University of California, San Diego Slide 1

2 OUTLINE I. Introduction II. Elliptic Curves III. A Combinatorial Interpretation of N k IV. Determinantal Formula V. Chip-Firing Games VI. Critical Groups VII. Connection to Cyclic Languages University of California, San Diego Slide 2

3 I. INTRODUCTION University of California, San Diego Slide 3

4 The zeta function of a Formal Language L is defined as [Berstel and Reutenauer, 1990] ( ζ(l) = exp n=1 a n T n n where a n = L A n is the number of words of length n in L. ) University of California, San Diego Slide 4

5 The zeta function of a Formal Language L is defined as [Berstel and Reutenauer, 1990] ( ζ(l) = exp n=1 a n T n n where a n = L A n is the number of words of length n in L. Furthermore, a Language is cyclic if it is closed under conjugation and powers (i.e. uv L if and only if vu in L and w in L implies that w m L for all m.) ) University of California, San Diego Slide 5

6 The zeta function of a Formal Language L is defined as [Berstel and Reutenauer, 1990] ( ζ(l) = exp n=1 a n T n n where a n = L A n is the number of words of length n in L. Furthermore, a Language is cyclic if it is closed under conjugation and powers (i.e. uv L if and only if vu in L and w in L implies that w m L for all m.) Theorem 1. If L is a cyclic language which is recognizable by a finite automaton, then its zeta function is rational. ) University of California, San Diego Slide 6

7 The zeta function of a Formal Language L is defined as [Berstel and Reutenauer, 1990] ( ζ(l) = exp n=1 a n T n n where a n = L A n is the number of words of length n in L. Furthermore, a Language is cyclic if it is closed under conjugation and powers (i.e. uv L if and only if vu in L and w in L implies that w m L for all m.) Theorem 1. If L is a cyclic language which is recognizable by a finite automaton, then its zeta function is rational. We compare with the theory of zeta functions for algebraic varieties. ) University of California, San Diego Slide 7

8 We let K be F q, a finite field containing q elements, where q is a power of a prime. We can also let K be a field extension of F q, such as F q k, or even the algebraic closure F q. University of California, San Diego Slide 8

9 We let K be F q, a finite field containing q elements, where q is a power of a prime. We can also let K be a field extension of F q, such as F q k, or even the algebraic closure F q. A model for a non-singular projective curve (with a rational point) defined over K is the zero locus of an equation f(x, y) with coefficients in K, plus a single point at infinity. We denote such a curve as C. University of California, San Diego Slide 9

10 We let K be F q, a finite field containing q elements, where q is a power of a prime. We can also let K be a field extension of F q, such as F q k, or even the algebraic closure F q. A model for a non-singular projective curve (with a rational point) defined over K is the zero locus of an equation f(x, y) with coefficients in K, plus a single point at infinity. We denote such a curve as C. C(F q ), C(F q k), or C(F q ) will denote the curve C over these fields, respectively. (This means that solutions (x, y) to equation f have coordinates in F q, F q k and F q, respectively.) C(F q ) C(F q k 1 ) C(F q k 2 ) C(F q ) for any sequence of natural numbers 1 k 1 k University of California, San Diego Slide 10

11 Curve C over field K has defining equation f(x, y) = 0 with coefficients in K. Such a curve consists of a single point at infinity, P, and affine points expressed as a pair of coordinates over K. The Frobenius map π acts on curve C over finite field F q via π(a, b) = (a q, b q ) and π(p ) = P. Fact 1. For point P C(F q ), Fact 2. For point P C(F q k), π(p) C(F q ). π k (P) = P. University of California, San Diego Slide 11

12 Let N m be the number of points on curve C, over finite field F q m. Alternatively, N m counts the number of points in C(F q ) which are fixed by the mth power of the Frobenius map, π m. Using this sequence, we define the zeta function of an algebraic variety, which can be written several different ways, including as an exponential generating function. ( T m ) Z(C, T) = exp N m m = p m=1 1 1 T deg p where p is a prime ideal ζ(s) = p prime integer 1 1 p s University of California, San Diego Slide 12

13 Theorem 2 (Rationality - Weil 1948). Z(C, T) = (1 α 1T)(1 α 2 T) (1 α 2g 1 T)(1 α 2g T) (1 T)(1 qt) for complex numbers α i s, where g is the genus of the curve C. Furthermore, the numerator of Z(C, T), which we will denote as L(C, T), has integer coefficients. Theorem 3 (Functional Equation - Weil 1948). Z(C, T) = q g 1 T 2g 2 Z(C, 1/qT) University of California, San Diego Slide 13

14 Theorem 2 (Rationality - Weil 1948). Z(C, T) = (1 α 1T)(1 α 2 T) (1 α 2g 1 T)(1 α 2g T) (1 T)(1 qt) for complex numbers α i s, where g is the genus of the curve C. Furthermore, the numerator of Z(C, T), which we will denote as L(C, T), has integer coefficients. Theorem 3 (Functional Equation - Weil 1948). Z(C, T) = q g 1 T 2g 2 Z(C, 1/qT) In particular, the zeta function for a cyclic language and the zeta function for an algebraic curve are both rational. University of California, San Diego Slide 14

15 Theorem 2 (Rationality - Weil 1948). Z(C, T) = (1 α 1T)(1 α 2 T) (1 α 2g 1 T)(1 α 2g T) (1 T)(1 qt) for complex numbers α i s, where g is the genus of the curve C. Furthermore, the numerator of Z(C, T), which we will denote as L(C, T), has integer coefficients. Theorem 3 (Functional Equation - Weil 1948). Z(C, T) = q g 1 T 2g 2 Z(C, 1/qT) In particular, the zeta function for a cyclic language and the zeta function for an algebraic curve are both rational. Question: Is there a cyclic language whose zeta function agrees with the zeta function for an elliptic curve? University of California, San Diego Slide 15

16 II. ELLIPTIC CURVES University of California, San Diego Slide 16

17 Specializing to the case of an elliptic curve E, or a genus one curve, a lot more is known and there is additional structure. Fact 3. E can be represented as the zero locus in P 2 of the equation for A, B F q. (if p 2, 3) y 2 = x 3 + Ax + B Fact 4. E has a group structure where two points on E can be added to yield another point on the curve. Fact 5. The Frobenius map is compatible with the group structure: π(p Q) = π(p) π(q). University of California, San Diego Slide 17

18 Draw Chord/Tangent Line and then reflect about horizontal axis Q P P + Q = R R University of California, San Diego Slide 18

19 If P 1 = (x 1, y 1 ), P 2 = (x 2, y 2 ), then 1) If x 1 x 2 then P 1 P 2 = P 3 = (x 3, y 3 ) where x 3 = m 2 x 1 x 2 and y 3 = m(x 1 x 3 ) y 1 with m = y 2 y 1 x 2 x 1. 2) If x 1 = x 2 but (y 1 y 2, or y 1 = 0 = y 2 ) then P 3 = P. 3) If P 1 = P 2 and y 1 0, then x 3 = m 2 2x 1 and y 3 = m(x 1 x 3 ) y 1 with m = 3x2 1 + A 2y 1. 4) P acts as the identity element in this addition. University of California, San Diego Slide 19

20 Rationality (Weil 1948, Hasse 1933) Z(E, T) = (1 α 1T)(1 α 2 T) (1 T)(1 qt) = 1 (1 + q N 1)T + qt 2 (1 T)(1 qt) for complex numbers α 1 and α 2. (In fact α 1 = α 2 = q.) Functional Equation (Weil 1948) Z(E, 1/qT) = Z(E, T). N k = p k [1 + q α 1 α 2 ] = 1 + q k α k 1 α k 2 and the Functional Equation implies α 1 α 2 = q. Thus the entire sequence of N k s, for elliptic curve E, only depends on q and N 1. University of California, San Diego Slide 20

21 Theorem 4 (Garsia 2004). For an elliptic curve, we can write N k as a polynomial in terms of N 1 and q such that N k = k ( 1) i 1 P k,i (q)n1 i i=1 where each P k,i is a polynomial in q with positive integer coefficients. University of California, San Diego Slide 21

22 Theorem 4 (Garsia 2004). For an elliptic curve, we can write N k as a polynomial in terms of N 1 and q such that N k = k ( 1) i 1 P k,i (q)n1 i i=1 where each P k,i is a polynomial in q with positive integer coefficients. N 2 = (2 + 2q)N 1 N 2 1 N 3 = (3 + 3q + 3q 2 )N 1 (3 + 3q)N N 3 1 N 4 = (4 + 4q + 4q 2 + 4q 3 )N 1 (6 + 8q + 6q 2 )N (4 + 4q)N 3 1 N 4 1 N 5 = (5 + 5q + 5q 2 + 5q 3 + 5q 4 )N 1 ( q + 15q q 3 )N ( q + 10q 2 )N 3 1 (5 + 5q)N N 5 1 Question 1. What is a combinatorial interpretation of these expressions, i.e. of the P k,i s? University of California, San Diego Slide 22

23 III. A COMBINATORIAL INTERPRETATION OF N k. University of California, San Diego Slide 23

24 Fibonacci Numbers F n = F n 1 + F n 2 F 0 = 1, F 1 = 1 1, 1, 2, 3, 5, 8, 13, 21, Counts the number of subsets of {1, 2,..., n 1} with no two elements consecutive e.g. F 5 = 8 : { }, {1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4} University of California, San Diego Slide 24

25 Lucas Numbers L n = L n 1 + L n 2 L 1 = 1, L 2 = 3 1, 3, 4, 7, 11, 18, 29, 47,... Counts the number of subsets of {1, 2,...,n} with no two elements circularly consecutive e.g. L 4 = 7 : { }, {1}, {2}, {3}, {4}, {1, 3}, {2, 4} By Convention and Recurrence: L 0 = 2 University of California, San Diego Slide 25

26 Definition 1. We define the (q,t) Lucas numbers to be a sequence of polynomials in variables q and t such that L n (q, t) is defined as L n (q, t) = S q # even elements in S t n 2 #S where the sum is over subsets S of {1, 2,...,n} such that no two numbers are circularly consecutive. Theorem 5. L 2k (q, t) = 1 + q k N k N1 = t The L 2k (q, t) s satisfy recurrence relation L 2k+2 (q, t) = (1 + q + t)l 2k (q, t) ql 2k 2 (q, t). University of California, San Diego Slide 26

27 Symmetric Function Aside: We can also think of this plethystically as L 2k (q, N 1 ) = 1 + q k p k [1 + q α 1 α 2 ]. University of California, San Diego Slide 27

28 Symmetric Function Aside: We can also think of this plethystically as L 2k (q, N 1 ) = 1 + q k p k [1 + q α 1 α 2 ]. Symmetric Functions give rise to other identities including: # Positive Divisors of degree k = h k [1 + q α 1 α 2 ] and ( 1) k F 2k 1 (q, N 1 ) = e k [1 + q α 1 α 2 ] for suitably defined bivariate Fibonacci polynomials. University of California, San Diego Slide 28

29 Question 2. Is there a generating function equal to N k directly? University of California, San Diego Slide 29

30 Question 2. Is there a generating function equal to N k directly? We can come close. University of California, San Diego Slide 30

31 We let W n denote the wheel graph which consists of n vertices on a circle and a central vertex which is adjacent to every other vertex. University of California, San Diego Slide 31

32 We let W n denote the wheel graph which consists of n vertices on a circle and a central vertex which is adjacent to every other vertex. We note that a spanning tree will consist of arcs on the rim and spokes. We orient the arcs clockwise and designate the head of each arc. University of California, San Diego Slide 32

33 Definition 2. W k (q, t) = spanning trees of W k q total dist from spokes to tails t # spokes. Theorem 6. k W k (q, t) = N N1 k = t = P k,i (q) t i for all k 1. i=1 dist = 0 dist = 1 dist = 0 dist = 1 q 2 t 3 dist = 1 q 3 t 3 dist = 2 The proof uses combinatorial facts from [Eğeciouğlu-Remmel 1990] and [Benjamin-Yerger 2004]. University of California, San Diego Slide 33

34 IV. A DETERMINANTAL FORMULA FOR N k University of California, San Diego Slide 34

35 Let M 1 = [ N 1 ], M 2 = 1 + q N 1 1 q, and for k 3, 1 q 1 + q N 1 let M k be the k-by-k three-line circulant matrix 1 + q N q q 1 + q N q 1 + q N q 1 + q N q 1 + q N 1 Theorem 7. The sequence of integers N k = #E(F q k) satisfies the relation N k = detm k for all k 1. Analogously, W k (q, t) = detm k N1 = t. University of California, San Diego Slide 35

36 dist = 0 dist = 1 q 2 t 3 dist = 1 q = 3 t = 2 University of California, San Diego Slide 36

37 Proof by the Matrix-Tree Theorem: The Laplacian Matrix for W k (q, t) is 1 + q + t q t q 1 + q + t t t L = 0... q 1 + q + t 1 0 t q 1 + q + t 1 t q 1 + q + t t t t t... t t kt The last row and column correspond to hub vertex, the root. By the Matrix-Tree theorem, the number of directed rooted spanning trees is detl 0 where L 0 is matrix L with the last row and last column deleted. University of California, San Diego Slide 37

38 V. CHIP-FIRING GAMES University of California, San Diego Slide 38

39 Let G be a finite loopless directed multi-graph. That is G = (V, E) where V is a finite set {v 1, v 2,...,v n } and E is a multiset whose elements are pairs from V V. For every vertex v i let C i be a nonnegative integer representing the number of chips on vertex v i. University of California, San Diego Slide 39

40 Let G be a finite loopless directed multi-graph. That is G = (V, E) where V is a finite set {v 1, v 2,...,v n } and E is a multiset whose elements are pairs from V V. For every vertex v i let C i be a nonnegative integer representing the number of chips on vertex v i. If there is an edge e = (v i, v j ) in E, we say that v i and v j are adjacent, and that edge e is directed from v i to v j. The outdegree of a vertex v i, d i is the number of edges in E with first coordinate v i. We call vertex v j a neighbor of v i if edge (v i, v j ) E. Finally, we let d ij be the number of edges (v i, v j ) in E. University of California, San Diego Slide 40

41 Chip-Firing: (Björner, Lovász, Shor) 1. Start with vertex v If C i, the number of chips on v i, is greater than the outdegree of v i, then vertex v i fires. Otherwise move on to v i If vertex v i fires, then we take d i chips off of v i and distribute them to v i s neighbors. 4. Now C i := C i d i and C j := C j + d i,j if v j is a neighbor of v i. 5. We continue until we get to v n. 6. We then start over with v 1 and repeat. 7. We continue forever or terminate when all C i < d i. University of California, San Diego Slide 41

42 We consider a variant due to Norman Biggs known as the Dollar Game: 1. We designate one vertex v 0 to be the bank, and allow C 0 to be negative. All the other C i s still must be nonnegative. 2. To limit extraneous configurations, we presume that the sum #V 1 i=0 C i = 0. (Thus in particular, C 0 will be non-positive.) 3. The bank, i.e. vertex v 0, is only allowed to fire if no other vertex can fire. Note that since we now allow C 0 to be negative, v 0 is allowed to fire even when it is smaller than its outdegree. University of California, San Diego Slide 42

43 A configuration is stable if v 0 is the only vertex that can fire A configuration C is recurrent if there is firing sequence which will lead back to C. (Note that this will necessarily require the use of v 0 firing.) We call a configuration critical if it is both stable and recurrent. University of California, San Diego Slide 43

44 A configuration is stable if v 0 is the only vertex that can fire A configuration C is recurrent if there is firing sequence which will lead back to C. (Note that this will necessarily require the use of v 0 firing.) We call a configuration critical if it is both stable and recurrent. Theorem 8 (Biggs 1999). For any initial configuration C with k i=0 C i = 0 and C i 0 for all 1 i k, there exists a unique critical configuration that can be reached by an allowable firing sequence. University of California, San Diego Slide 44

45 For example, consider the following two wheels with chip distributions as given. These are both critical configurations. We do not label the number of chips on the hub vertex since forced If we add these together pointwise we obtain University of California, San Diego Slide 45

46 This is not a critical configuration, but by the theorem, reduces to a unqiue critical configuration. University of California, San Diego Slide 46

47 University of California, San Diego Slide 47

48 University of California, San Diego Slide 48

49 This last one is critical. University of California, San Diego Slide 49

50 The critical group of graph G, with respect to vertex v 0, to be the set of critical configurations, with addition given by C 1 C 2 = C 1 + C 2. Here + signifies the usual pointwise vector addition and C represents the unique critical configuration reachable from C. When v 0 is understood, we will abbreviate this group as the critical group of graph G, and denote it as C(G). University of California, San Diego Slide 50

51 The critical group of graph G, with respect to vertex v 0, to be the set of critical configurations, with addition given by C 1 C 2 = C 1 + C 2. Here + signifies the usual pointwise vector addition and C represents the unique critical configuration reachable from C. When v 0 is understood, we will abbreviate this group as the critical group of graph G, and denote it as C(G). Theorem 9 (Biggs 1999). C(G) is an abelian (associative) group. University of California, San Diego Slide 51

52 The critical group of graph G, with respect to vertex v 0, to be the set of critical configurations, with addition given by C 1 C 2 = C 1 + C 2. Here + signifies the usual pointwise vector addition and C represents the unique critical configuration reachable from C. When v 0 is understood, we will abbreviate this group as the critical group of graph G, and denote it as C(G). Theorem 9 (Biggs 1999). C(G) is an abelian (associative) group. Alternative definition: / C(G) = Z V (G) 1 V (G) 1 Im L 0 Z where L 0 is the Laplacian matrix of graph G with the last row and last column deleted. University of California, San Diego Slide 52

53 We get in particular that C(G) = #Spanning Trees in Graph G (using the Matrix-Tree Theorem again.) University of California, San Diego Slide 53

54 We get in particular that C(G) = #Spanning Trees in Graph G (using the Matrix-Tree Theorem again.) Proposition 1. There exists a natural bijection between rooted spanning trees of the directed (q, t)-wheel multi-graph on k rim vertices, and critical configurations of the same graph. (Multi-graph analogue of Biggs-Winkler 1997 for this special case.) Note: We abbreviate the configuration vector as [C 1, C 2,...,C #V (G) 1 ], leaving off coefficient C 0, which is forced by the relation #V (G) 1 i=0 C i = 0. University of California, San Diego Slide 54

55 VI. CRITICAL GROUPS OF (q, t)-wheel GRAPHS University of California, San Diego Slide 55

56 Understanding the sequence of Critical Groups: The set C(W 1 (q, t)), C(W 2 (q, t)), C(W 3 (q, t)),... { } Elements of the critical group C(W k (q, t)) is a subset of the set of length k words in alphabet {0, 1, 2,..., q + t}. Proposition 2. The map ψ : w is an injective group homomorphism between C(W k1 (q, t)) and C(W k2 (q, t)) whenever k 1 k 2. Here map ψ replaces w with k 2 /k 1 copies of w. University of California, San Diego Slide 56

57 Example: [2, 4, 2] [0, 4, 1] [1, 0, 4] in W 3 (q = 3, t = 2) versus = 1 4 [2, 4, 2, 2, 4, 2] [0, 4, 1, 0, 4, 1] [1, 0, 4, 1, 0, 4] in W 6 (q = 3, t = 2) = Chip-firing is a local process. University of California, San Diego Slide 57

58 Proposition 2. The map ψ : w is an injective group homomorphism between C(W k1 (q, t)) and C(W k2 (q, t)) whenever k 1 k 2. Here map ψ replaces w with k 2 /k 1 copies of w. Define ρ to be the rotation map on C(W k (q, t)). If we consider elements of the critical group to be configuration vectors, then we mean clockwise rotation of elements to the right. Equivalently, ρ acts by rotating the rim vertices of W k clockwise if we view elements of C(W k (q, t)) as spanning trees. Proposition 3. The kernel of (1 ρ k 1 ) acting on C(W k2 (q, t)) is isomorphic to the subgroup C(W k1 (q, t)) whenever k 1 k 2. University of California, San Diego Slide 58

59 Proposition 3. The kernel of (1 ρ k 1 ) acting on C(W k2 (q, t)) is isomorphic to the subgroup C(W k1 (q, t)) whenever k 1 k 2. We therefore can define a direct limit C(W(q, t)) = C(W k (q, t)) k=1 where ρ provides the transition maps. Another view of C(W(q, t)): The set of bi-infinite words which are (1) periodic, and (2) have fundamental subword, i.e. pattern, equal to a configuration vector in C(W k (q, t)) for some k 1. In this interpretation, map ρ acts on C(W(q, t)) as the shift map. University of California, San Diego Slide 59

60 In particular we obtain C(W k (q, t)) = Ker(1 ρ k ) : C(W(q, t)) C(W(q, t)). We now can describe a combinatorial interpretation for the factorizations of W k (q, t) = C(W k (q, t)) into irreducible integral polynomials. University of California, San Diego Slide 60

61 In particular we obtain C(W k (q, t)) = Ker(1 ρ k ) : C(W(q, t)) C(W(q, t)). We now can describe a combinatorial interpretation for the factorizations of W k (q, t) = C(W k (q, t)) into irreducible integral polynomials. Let ρ denote ( the shift map, Cyc d (x) be the dth cyclotomic polynomial x k 1 = ) d k Cyc d(x), and C(W(q, t)) be the direct limit of the sequence {C(W k (q, t))} k=1. Theorem 10. W k (q, t) = d k WCyc d (q, t) and WCyc d = ) (Cyc Ker d (ρ) : C(W(q, t)) C(W(q, t)). University of California, San Diego Slide 61

62 Shift map ρ is the wheel graph-analogue of the Frobenius map π on elliptic curves. 1. We have an analogous family of bivariate integral polynomials and factorizations N k (q, t) = d k ECyc d (q, t) and ECyc d = ) (Cyc Ker d (π) : E(F q ) E(F q ) where for d 2, ECyc d (q, N 1 ) = WCyc d (q, t) t= N1. 2. C(W k (q, t)) = Ker(1 ρ k ) : C(W(q, t)) C(W(q, t)) just as E(F q k) = Ker(1 π k ) : E(F q ) E(F q ). University of California, San Diego Slide 62

63 3. We get the equation ρ 2 (1 + q + t)ρ + q = 0 on C(W(q, t)). This can be read off from matrix 1 + q N q q 1 + q N M k = 0... q 1 + q N q 1 + q N q 1 + q N 1 and the configuration vectors images under clockwise and counter-clockwise rotation. This is a direct analogue of the characteristic equation π 2 (1 + q N 1 )π + q = 0 on E(F q ). University of California, San Diego Slide 63

64 VII. CONNECTION TO CYCLIC LANGUAGES. University of California, San Diego Slide 64

65 Spanning trees of wheel graphs have cyclic symmetry, and Consist of disconnected arcs on the rim (one such piece for each spoke) University of California, San Diego Slide 65

66 Spanning trees of wheel graphs have cyclic symmetry, and Consist of disconnected arcs on the rim (one such piece for each spoke). We can characterize (conjugates of) critical configurations of the wheel graph (q, t)-w k as a concatonation of blocks with form with the properties 1. B {q + 1,...,q + t}, 2. M i {0, 1,...,q}, and B, M 1,...,M r 3. if M j = 0, then M j+1 = = M r = q. University of California, San Diego Slide 66

67 Spanning trees of wheel graphs have cyclic symmetry, and Consist of disconnected arcs on the rim (one such piece for each spoke). We can characterize (conjugates of) critical configurations of the wheel graph (q, t)-w k as a concatonation of blocks with form with the properties B, M 1,...,M r 1. B {q + 1,...,q + t}, (Need at least one element > q 2. M i {0, 1,...,q} for config to be recurrent.) 3. if M j = 0, then M j+1 = = M r = q. (Recurrent also forces.) Considering these as elements of C(W k (q, t)) C(W(q, t)), we identity C 1,...,C k with periodic string...c k, C 1, C 2,...C k 1, C k, C 1,... University of California, San Diego Slide 67

68 A {1+q, 2+q,..., q+t} {1+q, 2+ q,..., q+t} {0} {1, 2,..., q} {1+q, 2+q,..., q+t} B {1, 2,..., q} C {q} {0} t q 1 t q 1 t 0 1 Critical Configurations correspond to Closed Paths in this DFA, D, which go through state A. University of California, San Diego Slide 68

69 A {1+q, 2+q,..., q+t} {1+q, 2+ q,..., q+t} {0} {1, 2,..., q} {1+q, 2+q,..., q+t} B {1, 2,..., q} C {q} {0} t q 1 t q 1 t 0 1 Critical Configurations correspond to Closed Paths in this DFA, D, which go through state A. Thus we disallow cycles containing only state B and cycles containing only state C. University of California, San Diego Slide 69

70 Recall the zeta function of a Cyclic Language L is ( T k ) ζ(l, T) = exp W k k k=1 where W k is the number of words of length k. University of California, San Diego Slide 70

71 Recall the zeta function of a Cyclic Language L is ( T k ) ζ(l, T) = exp W k k k=1 where W k is the number of words of length k. Also can write ( ζ(l, T) = exp allowed closed paths P # words admissible by path P T k ). University of California, San Diego Slide 71

72 Recall the zeta function of a Cyclic Language L is ( T k ) ζ(l, T) = exp W k k k=1 where W k is the number of words of length k. Also can write ( ζ(l, T) = exp allowed closed paths P # words admissible by path P T k ). The trace of an automaton A is the language of words generated by closed paths in A, and satisfies ζ(trace(a)) = 1 det(i M T), where M encodes the number of directed edges between state i and state j in A. University of California, San Diego Slide 72

73 Let L(W(q, t)) be the language of patterns for set C((W)(q, t)). Language L(W(q, t)) also equals the set k=1 C(W k(q, t)). University of California, San Diego Slide 73

74 Let L(W(q, t)) be the language of patterns for set C((W)(q, t)). Language L(W(q, t)) also equals the set k=1 C(W k(q, t)). L(W(q, t)) = trace(d) trace(b) trace(c) where we let B (resp. C) signify the DFA we get by taking D and removing states A and C (resp. A and B). University of California, San Diego Slide 74

75 A {1+q, 2+q,..., q+t} {1+q, 2+ q,..., q+t} {0} {1, 2,..., q} {1+q, 2+q,..., q+t} B {1, 2,..., q} C {q} {0} t q 1 t q 1 t 0 1 Critical Configurations correspond to Closed Paths in this DFA, D, which go through state A. Thus we disallow cycles containing only state B and cycles containing only state C. det(i M T) = 1 (1 + q + t)t qt 2. University of California, San Diego Slide 75

76 Let L(W(q, t)) be the language of patterns for set C((W)(q, t)). Language L(W(q, t)) also equals the set k=1 C(W k(q, t)). L(W(q, t)) = trace(d) trace(b) trace(c) where we let B (resp. C) signify the DFA we get by taking D and removing states A and C (resp. A and B). Theorem 11. ζ(l(w(q, t))) = (1 T)(1 qt) 1 (1 + q + t)t qt 2. University of California, San Diego Slide 76

77 Let L(W(q, t)) be the language of patterns for set C((W)(q, t)). Language L(W(q, t)) also equals the set k=1 C(W k(q, t)). L(W(q, t)) = trace(d) trace(b) trace(c) where we let B (resp. C) signify the DFA we get by taking D and removing states A and C (resp. A and B). Theorem 11. Compare with: ζ(l(w(q, t))) = (1 T)(1 qt) 1 (1 + q + t)t qt 2. Z(E, T) = 1 (1 + q N 1)T qt 2 (1 T)(1 qt). University of California, San Diego Slide 77