LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN 510.6-1 ho. 2I2-220 Digitized by the Internet Archive in 2013 http://archive.org/details/weakcongruencere213yehr M6aS y?ijj^L Report No. 213 COO-IU69-OO38 P WEAK CONGRUENCE RELATIONS ON GRAPHS by Raymond Tzuu-Yau Yen September 1, 1966 DEPARTMENT OF COMPUTER SCIENCE • UNIVERSITY OF ILLINOIS • URBANA, ILLINOIS Report No, 213 WEAK CONGRUENCE RELATIONS ON GRAPHS* by Raymond Tzuu-Yau Yeh September 1, I966 Department of Computer Science University of Illinois Urbana^ Illinois 61803 This work was supported in part by Contract No„ AT(ll-l)-lU69 of the Atomic Energy Commission and submitted in partial fulfillment of the degree of Doctor of Philosophy in Mathematics. Ill ACKNOWLEDGMENT The author would like to express his most sincere gratitude to Professor David E. Muller for his constant encouragement, indis- pensable guidence and constructive criticisms in the preparation of this thesis. IV TABLE OF CONTENTS Page I. INTRODUCTION 1 II. ELEMENTARY PROPERTIES OF GRAPHS 3 III. STRUCTURE OF WEAK CONGRUENCE RELATIONS 10 IV. CANONICAL AND FINITELY REDUCIBLE COVERS 30 V. BIBLIOGRAPHY 39 APPENDIX 1+0 VITA 1+3 I . INTRODUCTION State reduction has "been a central problem in the theory of sequential machines. For the complete specified machines, the problem is relatively simple since the states of the reduced machine are congruence classes on the set of states of the original machine. The problem becomes more complicated in the incomplete case because the states of any reduced machine correspond to a cover with substitution property on the states of the original machines as have been shown by many authors [^>5>7>9]« Recently Hartmanis and Stearns pointed out the danger of state reduction in connection with the realizability of a machine by smaller machines. Their investigation [7] showed that sometimes it is desirable to perform "state splitting" before reduction. The concept of "state splitting" seems to tie in with Yoeli's [13] consideration of multi-valued homomorphism quite naturally. The purpose of this thesis is to investigate the concepts of homomorphic relations (multi-value homomorphisms ) , weak congruence relations and covers with substitution property on generalized graphs. As the thesis will show these concepts are closely related. In fact, homomorphic relations and weak congruence relations on graphs bear the same natural relations as that of homomorphisms and congruence relations on algebra. More precisely, each weak congruence relation 9 on a graph G determines a unique homomorphic relation H on G, and each homomorphic relation H between a graph G and a productive graph G' determines a unique weak congruence relation 9 on G such that n T 9 = H • H . Theorems similar to isomorphism theorems on algebra are n obtained. An important theorem in section IV gives conditions under which a finite subcollection of an infinite cover with substitution property on a graph is again a cover with substitution property on the same graph. It is well known that every sequential machine can be represented by a directed, labelled graph. It is hoped that through the investigation reported here, new insight will be gained in state reduction and realizability of non-deterministic, incomplete sequential machines . II. ELEMENTARY PROPERTIES OF GRAPHS Definition 1. Let a and 5 be two relations on a set X, and X. a 1 subset of X. We define the compositions a • 6 of a and 5, X. ' 6 of X . and S, and a -X. of a and X. by the following rules: a • 8 = ((x,z)|3y e X, (x,y) e a ~ (y,z) e 5} X ± -5= {y|3 xe X., (x,y) e 6} a "X j .= {x|H y e X ± , (x,y) e o] Definition 2. Let X be a set and c X x X. Then I x = [(x,x)|x e X} and T = ((x,y)|(y,x) e 0} . is reflexive iff I c: 9 , 9 is weakly reflexive iff ~~ — ^— — — — — _x_ — — ■ — — — — 9 c (8 n 1^) -e. Remark 1. Let a, b and y be relations on a set X. Then (1) a '• (6 • T ) = (a • 6) • r (2) a • (5 u r) = (a • S) u (a ■ r) (3) (a U o) • r = (a • r) u (5 • r) (4) a . (& n r) e ( a * & ) n ( a • r) (5) (a n &) • r c (a • r) n (6 • r) (6) (a ■ o) T = 5 T • a T (7) a c (a fl 3^) • a £ y x ,y € X, (x,y) e a - (x,x) e Definition 3. Let X and X ! be two sets and 9 c X x X' , then is productive iff V x e X, there exists at least one x' e X' such that (x,x r ) € 0. 6 is single-valued iff V x e X, there is exactJy one x' £ X' such that (x,x ! ) e 0. Remark 2. Let Q c X x X' , then (1) 9 is productive iff ■ (2) 9 is single-valued iff • T 3 I . - X ?h ,T c: I , and A. Definition k. A graph G is a pair , where X is an arbitrary set and A is a finite set of relational symbols such that for each a € A, there corresponds a unique relation R c X x X. a — Definition 5. Two graphs G = and G' = are similar iff A = A' . Definition 6. Let G = and G 1 = be two similar graphs. The direct product G x G 1 of G and G' is defined by the pair such that V a e A, the corresponding relation R " is defined by the rule: a J V (x,x'),(y,y') in x x x', ((x,x'),(y,y')) e r; 1 z (x,y) e R a - (x',y') e R^ . When G = G* , denote G x G by G 2 . Definition 7. Let G = and G' = be two similar graphs G' is a subgraph of G iff X' c X and V a g A, R* = R fl (X' x X' ). a a v ' Remark 3„ Let B = be a graph, then any nonempty subset X' of X determines a unique subgraph of G. Definition 8. Let G = be a graph. A relation on G satisfies the substitution property (S.P.) iff V a e A, R T • . R c 0. When satisfies S.P,. we call a a - ' a S.P. relation . Remark k. Let G = be a graph, a relation on G. (1) satisfies S.P. iff V x,y e X, (x,y) e - (V a e A)[V x'e x • R , 3. V y ' e y . R d (x',y') e 0] (2) satisfies S.P. iff <0,A> is a subgraph of G 2 = . Let G - be any graph and S = {R } .. Let S' be the semi-group a a eA generated by S under composition, and let S(g) = S 1 U L V OL e S(g), we define 0(a) to be the number of elements of S contained in the minimum representation of OL, Let be any relation on G, then for each a e S(g), we define a new relation. = a T • • a a T Remark 5. Let G = be a graph such that VaeA, R • R = L.. a a a Then V OL e S(G), V c X x X, V i, j > 0, we have (e a }1 = {Q±) a > Where 6± = d± ' 1 ' 9 ' i > 1 - G ° = h° Definition 9- Let 9 be a relation on a set X. Define the complement 9 of 9 by the rule : V x,y € X, (x,y) e iff (x,y) £ 9. Remark 6. Let a and 5 be two relations on a set X. Then (1) (2) 0=0 o c S -» 5" c a (3) a U & = a n 6 (4) a n 5 = a u & Definition 10. Let G = be a graph. G is onto iff V a e A, T R is productive. G is productive iff V a e A, 3. R is productive. Remark 7. Let G = be a graph, and S = {a.}. a family of relations on G. Then V a e S(g), (l) o . a o . -> (a. ) c (a.V (3) if G is productive, then (o . • o .) a (o .) '(a.) lei iel (5) ( n ff.V c n (a.) iel iel Theorem 1. Let G = be a graph such that S(G) is abelian, Then V 9 c X x X, 9 satisfies S.P. iff V a e S(g), satisfies S.P. a Proof: Assume S(g) is abelian and 9 satisfies S.P. Then V a g A, V a e S(G), R T • • R = R T • a T ' ^ ' ' a OL a a a - R a T T = a • R . e • r -a a a c: a • -a = a The converse is clear. Q.E.D. Lemma 1 Let G = be a graph, and OcXxL Then satisfies S.P. iff V a e S(g), q <=■ Proof: If V a e S(g), 9 c 0. Then in particular, T V a e A, cz -> R ' a — a R cz a — Suppose satisfies S.P. V a e S(G), 0(a) < 1 - c 0. Assume the lemma holds for all a such that 0(a) < n, Let e S(G), 0(p) = n + 1. Then =j a e A, a e S(G) such that 3 = a • R and 0(a) < n. 3. = P T T 3 = R ■ a ^ a a • R -- R • ' R a a a T c R x — a R e a — Q.E.D. Theorem 2, Let G = be productive and a S.P* relation on G. Then V a e S(g), c c a Proof; V a e S(G), c by lemma 1 G is productive -> V a e S(g), ..T c a ' a T T ■ a • a = a • -a a c a T a = a So, c c m . ' a — - T a Q.E.D. Definition 11, 8 Let G = be a graph and c X x X. satisfies the dual substitution property (D.S.P.) iff V a e A, R • i T R c: a — 8 is regular if satisfies both S.P. and D.S.P. Theorem 3. Let G = be a graph and c X x X. Then satisfies S.P. iff 6 satisfies D.S.P. Proof: Suppose R • • R c 0. a a — V x,y e X, (x,y) e 2 (x,y) i V a e A, V x' e x • R , Vy'eyR^ (x',y') { - V a e A, R • • R T c . v 'a a — Converse is proved similarly. Q.E.D. Definition 12. Let G = be a graph, define the dual graph G of G by the rule: ,T Remark 8. G = such that V a e A, R = R ' ' a a Let G = be a graph. Then (1) G = G (2) G is productive (onto) iff G is onto (productive) (3) c X x X satisfies S.P. (D.S.P.) on G iff satisfies D.S.P. (S.P.) on G. 9 Corollary: Let G = "be an arbitrary graph. Then for each theorem on G involving S.P., there is a corresponding theorem on G involving D.S.P. Definition 13. A graph G is totally onto iff G is both productive and onto. Theorem 4. Proof Let G = be a totally onto graph, c: x x X is regular iff V a e S(g), (■ a T a is regular ;VaeS(G), 6 (=0ee rp ^0 m c=ee0 a a' Q.E.D. 10 III. STRUCTURE OF WEAK CONGRUENCE RELATIONS Definition Ik. Let G = "be a graph and a relation on G. Then (1) is a weak congruence relation of type I (WCRI) iff (a) = T (b) c (0 n L^ • (c) V a e A ; R T • e • R e e - a a — (2) is a weak congruence relation of type II (WCRII) iff (b) is replaced by (b 1 ) L. c 0. A (3) is a weak congruence relation of type III (WCRIII) iff is WCRI and (d) V a e A, • R • X c: R -X. a — a (k) is a weak congruence relation of type IV (WCRIV) iff is WCRII and satisfies (d). We note that when G is productive, then WCRI = WCRIII and WCRII = WCRIV. Definition 15 . Let G = be a graph. A family of nonempty distinct subsets C = (X . } . _ of X is a cover 1'iel — on G iff U X. = X. C is a cover with S.P. on G iel X iff C is a cover on G, and V a e A, V X. e C, nX. £ C, X. • R c X. . Cisan irredundant cover j ' 1 a - j on G iff V X..X. 6 C, X. c X. -* X. = X. . 1 J 1 - J 1 J Let G = be a graph. Denote the family of all covers on G by C(G). Define an "ordering" "<" on C(G) such that if C . = {X . ) . T € C(g), i = 1,2., where J n and he a graph. (1) "<" is reflexive and transitive but not anti- symmetric on C(G). However, "<" is a partial order on the class of all irredundant covers on G. (2) Every cover (cover with S.P.) C = {X.}. on G defines a unique reflexive and symmetric (WCRIl) relation on G by the rule: V x,y e X, (x,y) ee^^.eC, x,y e X.. (3) Every reflexive and symmetric (WCRIl) relation 9 on T defines a unique cover (cover with S.P.) C fl on G by the rule : If X. is an arbitrary subset of X, then X. e C Z (i) V x,y e X., (x,y) e 9, and (ii) (Vx e X)[Vy e X., (x,y) € 9] -» x eX.. 1 We note that if 9 is not reflexive, then CL is simply a family of subsets of X. (h) There is in general more than one cover (cover with S.P.) on G which determines the same reflexive and symmetric ( WCRIl) relation on G. (5) If C ± ,C e C(G), then 12 (6) If a and & are reflexive and symmetric relations on G, then a c 6 £ C < C ft - a - o (7) If 9 is a reflexive and symmetric relation on G, and C £ C(g), then = 0, but C fl ^C in general. L 6 C (8) Mutual refinement of covers is an equivalence relation. Theorem 5 Let G = be a graph. Then the partly ordered sets of weak congruence relations of type II and type IV, ordered by refinement, each forms a complete, distributive lattice which is a sublattice of the lattice of all reflexive and symmetric relations on X. Proof: Let S be any set of weak congruence relations a on G, then a = inf S and p = sup S are defined by the rules (1) a = n a aeS (2) p = U a aeS It is clear that both a and p are reflexive and symmetric. V a e A, we have T T R • a • R = R a a a ,T P • R = R a a n a ■ R e n R T • a oeS aeS c fl a = a " 0£S U a • R = u R • a aeS a oeS a R R = p a K 13 ( PL a) ' R • X c n (a • R • X) c R - X v ogS a - aeS a ' — a ( U a) • R ■ X = U (a • R • X) c R • X aeS a aeS a — a Distributivity follows from the fact that the lattice of all reflexive and symmetric relations on X is a sublattice of the lattice of all subsets of X which is distributive. E D Notation. Let L(G,Il) and L(G,IV) be lattices of WCRII and WCRIV respectively on G. Corollary. Let G = be a graph, and 9 a WCRII on G. Let L Q (G,II) = (o|eco a 6 l(G,II)), and a L (G,II) = [a\a cG-^a e L(G,Il)}. a Then both L„(G,Il) and L (G.Il) are complete a sublattices of L(G,Il) such that 9 is the least element of L~(G,Il) and the greatest element of L (G,II). Similar results hold if 9 is WCRIV. Definition 16. Let G = be a graph, S a family of nonempty subsets of X. The quotient graph G/ S of G with respect to S is defined as follows: G/S = such that V a e A, the corresponding relation R' is defined a such that V X.,X. eS, (X.,X.) eR' £ (X. • R D £ 0WX. ■ R c X.). l j i j a ^ v l a ' r ' l a — j If 9 is a weakly reflexive and symmetric relation, we define the quotient graph G/Q of G with respect to 9 by letting G/0 = G/C fl . Ik Definition 17. Let G = and G' = be two similar graphs. Then a relation H c X x X' is a homomorphio relation between G and G' iff V a e A, R^ • H • R' c_ H. H is an endomorphic relation on G iff G = G' . Remark 10. Let G = , G' = and G" = and G' = be two similar graphs, and H c X x X' . Then V a e A, R T • H • R" c H Z R ! c H T • R • H . 7 a a — a — a Proof: R T • H • R' C H 2 T(R T • H • R' • H 1 ) = a a — v a a /-T T ^ T(H • B • H • B') = x a a' -T T H • R a T T H e R ,T H • R ■ HcR' a — a B 1 a H • B • H a — a Q.E.D. Theorem 6, Let G = be a graph and X' an arbitrary set. If Ha x X', then there is a graph G' = such that H is a homomorphic relation between G and G' . Proof V a e A, define R' c H ' a — from lemma 2. R • H . Theorem follows a Lemma 3 • Let G = be a graph. Then every family of sub- sets S - {X.}. of X determines a unique homomorphic relation H a between G and G/S. o 16 Proof Define H "by the relation: V x e X, V X. e S, (x,X. ) e H Q Z x e X . . (x,X.) € R^ • H s • P/ Z^y e X, 3 X^ € S, (y,x) e R a , (y,X.) e K s , (X.,X.) e R ; ->(y £ X.HX. • R ex.). w j J a - i' Since xey«R,sox€X.o This implies (x,X.) € H_. Thus, R^ ■ H Q • R ' c H a S a - S Q.E.D. Theorem 7' Proof: Let G = and G 1 = be two similar graphs and H c X x X' is a homomorphic relation between G and G' . If G' is productive, then H determines a <] unique weak congruence relation of type I 6 n = H • H n on Go Conversely, every WCRI on G determines a unique homomorphic relation H~ between G and g/0. Suppose H is a homomorphic relation on G, and G' T is productive. Define 9 = H • H . Then n (i) e = e T K J H H (2) (x, y ) e e H - ^c (e H n i x ) ■ e H (3) V a g A, R T • H • H T • R cz R T • H • R* • R T • H T • R w/ 'a a— a aa a c H • H Thus is a WCRI on G. ri The converse follows from lemma 3' Q.E.D. 17 Corollary. Let G and G' be two similar graphs , and H a homo- morphic relation between G and G 1 . If both G' and H are productive, then H determines a unique WCR1I. Definition 19 • Let G = be a graph and a,y two reflexive and symmetric relations on G such that a c y . Then (1) y/a is reflexive and symmetric. (2) x/o = H T • y • H . (3) a cz y c= 5 -* y/a c b/o . {k) If G is productive and o ,x are both WCRII, then x/o is a WCRII on G/a. Definition 20. Let G = and G' = be two similar graphs, a a reflexive and symmetric relation on G' , and H a homomorphic relation between G and G' , we define o on G by the rule : V x,y e X, (x,y) e a Z 3 x 1 £ x • H, 3 y ' e y • H, (x',y' ) e a . 18 Remark 13. Let H be a homomorphic relation between graphs G and G' , and a, 6 two reflexive and symmetric relations on G, then (1) a is symmetric (2) a is reflexive iff H is productive (3) c r - o c r \y rp (k) a = H • a • H (5) If both H and G 1 are productive, then a is a WCRII on G' implies a is a WCRII on G. Remark Ik. Let G = be a productive graph, o,r,& WCRII on G such that a c y c 5. Let 9, be WCRII on G/a, then (1) T E (r/a) (2) e c 9/0 (3) (T U b)/a - r/o U S/a (*0 (r n &)/a 5 r / a n &/a (5) (e u ^) = e u x (6) (en\)cen\ Theorem 8. Let G = be a graph* Then to every relation T on G there corresponds a unique maximum WCRII (WCRIV) &(r), and a unique minimum WCRII (WCRIV) \(x) satisfying S(y) c y c: \(y) such that 6(y)/a C &(r/a), \(r/a) c \(r)/a if a c r. Furthermore, if S is a set of relations y on G, then 19 (1) 5 (inf S) (2) 5(sup S) (3) \(inf S) (h) \(sup S) = inf [5(y)} res = sup fs(r)) res = inf {\(r)3 res = sup (>,(r)) yes Proof : Let C be a family of WCRII (WCRIV) a on G such that a c y . Then define S(y) = sup C. Similarly;, define \(y) = inf C , where C is the family of all WCRII (WCRIV) a' on G such that r^ a'. Then a c 5(r) c r c \( r ) -» o(y)/a c y/a c fc(y)/a - S(y)/a c S(r/a) and \(j / o) cz \(r)/o. The rest follows from properties of sets. Q.E.D. Definition 21. Two relations a and y on a set X are commutative with each other iff o • y = y . a. Definition 22, Theorem 9- Let G = "be a graph. A family S of pairwise commutative weak congruence relations o of type k is maximal iff for any weak congruence relation 9 of type k on G, 9 -o=c • 6 -» G € S. Where k = I, II, III, or IV. Let G = be a productive graph. Then every maximal family of weak congruence relation of type 20 k, k = I, II, III, and IV is closed under union and composition. Proof: Let S = fa . } . T "be a maximal family of weak 1 i j iel congruence relations. Then (1) [(va k = V o.Wa..o k = eye.)] -. [(o..o.) ■ a^ (2) [a. c (a. fl l x ) ■ a ± , i = 1,2] - [(a-^) c [(a 1 -a 2 ) fl I x ] • (a 1 -a 2 )} (3) [a. = I, i = 1,2.] - [(o^) 3 I] (*0 [^ = a*, i = 1,2.] - [(a 1 -a 2 ) T = a^ • a* = a^] (5) Va e A, R T -0.-a.-R cR T -a..R -R T -a.-R K ' ' a l j a — aiaaja <= a. 'a . - i J (6) G is productive -*Va e A, Vi e I, o. • R • X c R -X l a — a (7) (a. U a.) • a. = a. -a. U a. -a. = a. • (a. u a.) l j k lkjk k v i j' Q.E.D. Theorem 10. Let G = be a totally onto graph, and S(g) ahelian. Then 9 is a weak congruence relation of type k on G iff Va e S(G), 9 is a weak congruence relation of type k, where k = I, II, III, or IV. Proof: In view of theorm 1, it is suffice to show that (1) V a g S(G), (a T -e-a) T = a T -e-a - 9^ = 9 a a 21 (2) e c (eni^) -e -> V a e s(g), e^ = a T • • a c a T «(eni v )-e-a T T c: a • (yni )-a-a -0-a A. E (e a n V • e a T (3) i c - V a e S(G), (a • • a) m m => cr ■ I • a = a ■ a ^ I (4) G is productive -* V a e A, • R • X c: R -X a — a Q.E.D. Definition 23- Let G = be a graph and any relation on G. The closure of is defined by the rule: U 9. * aeS(G) a Theorem 11, Let G = be a graph and 9 ,o ,b be relations on G. Then (1) is the smallest S.P. relation which contains (2) Voe S(G), q c 9 (3) o c 5 -» o ^5 y . \ . */V 1 "A" "A" * Proof : V a e A, R T -* R T • U ° R aeS(G) U aeS(G) T U R • a. • aeS(G) a c 5 - Va e S(G), a„ <= o„ -» a* c 5 a — a c ) c (0 ) a -R c a — 22 V a e A, R J Thus, (0*) \ E 6 "*Va 6 s(G), (e*) a E 9 ' -.(e) c e = 0' Suppose 7 is a S.P. relation on G such that c 7. We see 957 for 0(a) - 0. Assume c 7, ya e S(G) such that 0(a) < n. Let p e S(G), 0(p) = n + 1. Then = a • R for some s. a e A, and some a e S(g) such that 0(a) < n. 9 o = 6 T -0-p = R T -a T -0-a-R c R T -7-R c 7 -* 0* c 7. P K a a— a a— — Q.E.D. Definition 2k. Let G = be a graph and a relation on G. Define the reflexive extension R(0) and symmetric extension S(0) of by the rule : R(0) = U I, S(0) = U T . Thus, we see RS(0) = SR(0) - u ^ U L Theorem 12. Let G = be a graph and a relation on G. Then \(0) = RS(0)*. Proof: If a is any WCRII on G such that c o , then RS(0) c \(0) c a. By theorem 11, RS(0)* c a. In particular, RS(0)*c= \(0). However, RS(0) is a WCRII containing 0. So, \(0) C RS(0)*. Thus, \(0) = RS(0)*. Q.E.D. 23 Definition 25. Let G = be a graph and 9 a relation on G. 9 is a dual weak congruence relation of type k iff satisfies all the conditions, except S.P., for a weak congruence relation of type k and D.S.P., where k = I, II, III, or IV. Remark 15 Let G - be a graph and 9 a relation on G. Let Sn(9) and X (0) Le the unique maximum dual weak congruence relation of type II and unique minimum dual weak congruence relation of type II on G satisfying & d (0) c 9c ^(9). If we let =8 U?UI, then we have (1) 9 is reflexive and symmetric -+9=9 (2) a c 6 -» 6 c a (3) a is WCRII and a c implies that a is a dual weak congruence relation of type II such that 9 c a. Theorem 13. Let G = be a graph and 9 a relation on G. Then 6(0) = ^ d (0)- Proof: X, (0) is reflexive and symmetric by definition. \ Ad) satisfies D.S.P. implies \ (0 ) satisfies S.P. Thus \,(0) is a WCRII. d^ e c \. (e) -» \_@) c e = 0. — d v ' d ' — X, (0) c a for all dual weak congruence relation a of d^ ' — D type II such that c a. Thus, a c \ (0), Since 2k 6(e) c e and 5(e) = 5(e), so 6(e) c >- d (e). But k (e) c e -> \ (0) c &(e). so, 6(e) = x d (ej. Q.E.D. Definition 26. Let G = and G' = "be two similar graphs. Then H C X x X' is a strong homomorphic relation between G and G ! iff V a e A, (i) R T • H • R c H, v ' a a — ' (ii) H T • R ■ X c R 1 • X' , K ' a — a ' (iii) H • R' • X' c R • X . v ' a — a Remark 16. (l) H is a strong homomorphic relation between G and T G iff H is a strong homomorphic relation between G' and G. (2) Let G = , G' = ^X',A> and G" = be similar graphs. If H c X x X' and H' c X' x X" are strong homomorphic relations between G and G' , and between G' and G" respectively, then H • H' <= X x X" is a strong homomorphic relation between G and G" . Theorem Ik. Let G = be a graph and eeXxX. is a weak congruence relation of type III on G iff there exist a graph G' = and a strong homomorphic relation H between G and G' such that = H • H T . 25 Proof: Suppose H is a strong homomorphic relation between G and G ! , and = H • H T . .It is clear that = T , and 0c(fiflL) -9. — A V a e A. H T -R -X c R' -X' - H'H T -R -X c H-R' -X' c R -X. ' a— a a— a— a I.e., 9-R •XcR -x. ' a — a V a e A, V x,y e X, (x,y) e R^-0'R - (x,y) e R^H«H T .R -^weX, ^w'eX'j^zeX, xR T wHw' H T zR y a a -» 3 lo. ' e X ' , x R T w Hw ' R ' ll ' R ' T w ' H T z R y -* ^ 'a a ' a a ^ since H T • R • X c R 1 • X 1 a — a - (x,y) e R T ' H • R' • R ,T • H T • R c H • H T = ^ '* ' a a a a — So, V a e A, R T • • R c 6 . ; 'a a — We have shown that is a weak congruence relation of type III. We now suppose that is a weak congruence relation of type III on G. Define G' = by the rules: (i) X ! = Cg, and (ii) V a € A, V X.,X. e C Q , (X.,X.) e R' l j l j a z (x.-R, / 0) - (x. • R a EXj). Define H 5 X x X' by the rule: V x e X, V X. e C Q , (x.X.) e H Z x £ X.. It is clear that = H • H T . 26 VaeA, (x,X.)el£..H-R' ;:jyeX, d X € C fll lad J x R T y EX. R' X. . a J j a 1 (y,X.) eH^yeX-Vy' e y-R . y' e X, , J o a i since (X,,X. ) e R' . - x c X. -» (x.X.) 6 H - V aeA, R T -H-R' c H. i ■ ' i' 'a a — X. e H T -R -X Z3 x e X, (x,X.) e H and x-R f ^ X. °R / -* =1 X. e C Q , X. • R ex., l a ' r j 0' l a — j' since C fl satisfies S.P. - V a e A, H T • R • X <=■ R> . X' . ' a — a x e H-R 1 -X' 23X. eX', (x.X,) e H and X. -R 1 / a i * v ' i' i a ^ r -* 3 y e X , y • R / X d ■* x • R I 0, since (x,y) e Q , ct and 0-R -X c R -X. a — a - V a e A, H • R' • X' c R -X. ' a — a We have shown that there exist a graph G' and a strong homomorphic relation H between G and G' such that T e = h • h . Q.E.D. Definition 27. Let G = and G' = be two similar graphs. H C X x X 1 is an isomorphic relation between G and G 1 iff H is a strong homomorphic relation between G and T G' such that both H and H are productive. 27 Notation We will use the symbols G ~ G' to denote the fact that there is an isomorphic relation between G and G' . Theorem 15 Proof Let H be a strong homomorphic relation between two graphs G = and G' = . Then G/d„ ~ . n T 9 = H • H is a weak congruence relation of type III H by theorem 1^. By the same theorem, there exists a strong homomorphic relation H' between and T G/0„ such that H' and H' are both productive. Define H C x(X • H) by H = H ,T • H. Then H is a strong H ATT homomorphic relation. Moreover , both H and H are productive. So, G/e R ~ . Q.E.D. Definition 28. Let G = be a graph. G' = is a semi-productive subgraph of G iff G' is a subgraph Theorem 16. Proof: of G, and V x e X' , x • R f 3. x • R 1 £ a ' Let G 1 = be a graph, G = a semi- productive subgraph of G, , and a weak congruence relation of type III on G, , then G 2 /e n (x 2 x x 2 ) ~ /e. determines a strong homomorphic relation H between G-, and G-./9 by theorem 1^. Let H be the relation H /\ restricted to X„. We see then H is a strong homomorphic relation between G p and /Q defined 28 by the rule : V x e X 2 , x • H = (x-9)/e . Both H and H are productive, and H • H = 6 fi (X xX ), Thus, by theorem 15, we have Gje n (xxxj ~ /e . 22 2 Q.E.D. Theorem 17 • Let G = be a productive graph, and a ,b two weak congruence relations of type II on G such that o c 6. Then (G/a)/(b/o) ~ G/&. Proof: Let C = {X.}. _ and C R = (Y.). T . a i J i€l & L J jeJ Define H c C x C c by the rule : — a o J V X. eC , VY. e C R , (X.Y.) eHjX, fl Y. / . V a e A, let R' and R" be the corresponding relations a a in G/a and G/'S respectively. V a e A, VX. e C a , VY. e C & , X. R ,T -H-R" Y. ?3xi, 3y'., X.R ,T X! H Y'.R" Y. i a aj i> y i a l jaj -» X ! fl Y '. =0 i J ^ G is productive -» G/a and G/6 are productive. so, v x" e x; • R' V Y" e Y' • R" XV n Y'! / ^ i i a JJ ai J In particular, X. fl Y. £ 0. So (X.,Y.) e H Thus, V a e A, R' T . H • R" c H. ' 'a a — G/a and G/S are both productive imply that V a e A, H T • R 1 • C C R" • n and H • R" • C R <= R> • C a a— a o a o — a a 29 Therefore, H is a strong homomorphi^ relation between G/a and G/&. By theorem 16, (g/o)/9, - g/B, where 9 C = H ■ H T . n We must show that = b/a . n V X i ,X. e C , (X.,X.) £9„:X. • H fl X. • H f * 3 Y k eC a> x . n y, p <~ x . n y, ^0 ^3 xeX 1 , 9 yeX^, (x,y)e5 Z (X.,X.) € &/c -9 H - B/a. Therefore, (G/a)/(&/a) - G/B 3.E.D. 30 IV. CANONICAL AND FINITELY REDUCIBLE COVERS Definition 29- A cover with S.P. C = {X.). -r on a graph G = is canonical iff VaeA, VX. e C, X. ■ R ^ -> X. • R e C. ' l ' l a ' r l a Theorem 18. Let G = "be a graph . Then to every cover with S.P. C = {X.}. _ on G, there corresponds a /\ canonical cover with S.P. C on G such that 6 = G C <> Proof: Let C be a given cover with S.P. Define C A = C Q = (X °'}. . Aft er C = (X: n 'l . _ has been defined, n > 0, n J J jeJ - we define C , = -jxi n '-R n+1 t j a n 1 aeA, jeJ , X 1 • R ^0^ n j a j = ix .(n+l)l JeJ. n+1 /\ ( (k) t °° Define C = -\X\ J \j e U J., k > i=0 It is clear that C is a canonical cover such that Q.E.D. Corollary: If C . , i = 1,2 are two covers with S.P. on a graph G, then e Ci . e^ , o x . c g 31 Theorem 19. Let G = be an infinite graph such that A = {a} Then a finite canonical cover with 3. P. on X exists iff 3 N > 0, X • R +± = X • R N - 1 — > a a Where X • R° = X n+1 n and X • R = X • R" • R . a a a Proof: Suppose 3 N > 0, X ■ R N+1 = X • R N ■ — a a Then C = iX-B a iN i=0 is a finite canonical cover with S.P. on Gc Suppose now that C = -JX. r is a finite canonical N 4 el cover with S.P. on G. Then X • R = ( U X.) • B = U X. • R a x . _ i y a 1 a lei lei Since C is canoni ical, V X. eC.X.-R £6 -» X. -B e C ' 1 ' 1 a ' r 1 a Therefore, X • R = U/oX,, where 1^ ' c: 1. a . (1) i' - iel X • R n = X > R n_1 ° R = U X . ° R = U X., a a a /.via / \ i iel iel Since I is a finite set, there must exist an N > such that 1 oid) = 1(2)3 => I (N) = j (N+1) Since X ■ R N = U X. a ■ T (N) X iel So, El N > 0, X ■ R N+1 = X • R N ' — ' a a Q.E.D. 32 Definition 30. Let G = be a graph. An element x n e X is a sequential vertex for a subset S of X iff there is an infinite sequence of distinct elements {X.}. such that [Vi>0, 3 aeA, (^ ± ^ i+1 ) e R J ~[2 N>0, V n>N, x n eS] Definition 31. Let G = be a graph and S a subset of X. S is closed if it contains all the sequential vertices for itself. Lemma k . Let G = be a graph. Then Proof: (1) all finite subsets of X are closed. (2) the intersection of an arbitrary family of closed subsets of X is closed. (1) The set of sequential vertices for any finite subset of X is empty. (2) Let (X.}. _ be an arbitrary family of closed subsets of X. Let S = fl X.. If x is a Iel X sequential vertex for S, then x is a sequential vertex for X., iel. This implies that x e X., 1' 1 iel. I.e. , x e S. So, S is closed. Q.E.D. Definition 32. A graph G = is sequential if every infinite subset X. of X has a sequential vertex in X. 33 Remark 17. A graph G = is sequential iff every infinite subgraph of G is sequential. Lemma 5 - Let {X, ), _, be a countably infinite collection of nonempty subsets of X of a graph G = such that (1) X. . c X. , Vk>0, and • ; k+1 k ' (2) each X, is closed and X is sequential, Then h X. is closed and nonempty. k=l k Proof : WW n X, is closed by lemma h. k=l k By the condition of the lemma, each X, must be infinite. Therefore, we can form a collection of 00 distinct points S = fx.l. , such that x. e X.. v i J i=l 1 1 S is an infinite subset of X which is sequential, therefore, there exists a sequential vertex y_ for S such that y_ e X, , I.e. there is an infinite 00 sequence of distinct elements (y . ) . such that [V i > 0, 3 a € A, (y. >y . +1 ) e RJ ~ [3 N > 0, Vn >N, y n e S]. Let S' = {y^^^ Since all but a finite number of elements of S belong to X,, V k > 0, S' must also contain an infinite number of elements of X, . I.e., 3 N, > N+l, V m > N, , y g X, . Therefore, y_ is a sequential k' J m k J vertex for X, , V k > 0. Since V k > 0, X, is closed, so y Q e X k , V k > 0. 00 00 , I.e., y e n X - n X / k=l k k=l k Q.E.D. Definition 33. Let B = be a graph, and C a cover with S.P. on G. C is finitely reducible iff there is a finite subcollection of elements of C which is again a cover with S.P. on G. Definition 3k. Let G = be a graph. G is semi-cyclic iff V x e X, there is a finite sequence a a n(x) of elements of A such that (x,x) e tt R . . . a n(x) 1=1 i Definition 35. Let G = be a graph and C a cover with S.P. on G. C is a semi-cyclic cover on G iff G/C is semi-cyclic . Theorem 20. If C is a semi-cyclic cover with S.P. on a sequential graph G = . Then C is finitely reducible. Proof: If C is finite, nothing to prove. So, assume C = (X. } . . L 1 i=l V m > 1, consider the finite subcollection S = fX.]. , of C. We claim that there exists an m l i J i=l N > such that S is a cover on G. Suppose there does not exist such an N. Now V i > 0, X - X. is a 7 1 closed subset of X. Since if y. e X. and y„ is a ^O 1 J 35 sequential vertex for X - X., then let fy . } . ^ be the infinite sequence determined by y . Because C is semi-cyclic, an infinite number of elements of X. will occur in this sequence. This contradicts the fact that y n is a sequential vertex for X - X.. Thus, X - X. cannot have a sequential vertex outside of l itself and therefore must be closed. Now let Q = X, and V i > 0, Q. = X - U X.. 1 X.eS. J J i So, Q. = fl (X-X.), V i > 0. 1 X.eS. J J i Since X - X. is closed V j > 0, Q. is also closed J ^- V i > 0. Furthermore, V i > 0, Q. c Q. by the hypothesis that S is not a cover of X, V N > 0. Thus, 00 we have a sequence of closed subsets (Q.)._ n satisfying the hypothesis of lemma 5- Hence, by lemma 5, n Q. f 0. However, we note that V x e X, i=0 X x e Q. -* x jt (J X.. Therefore, :.eS. J i X.eS. J 00 xe fl Q. -* x ^ U X,-^x/X-^Cis not a cover on X 1=0 X j=l J We thus arrive at a contradiction. Hence, zt N such N that (J X. = X. Since C is semi-cyclic, V X. e C, i=l X X there is a finite sequence a ,...,a / s of elements i n(X ) of A such that (X.,X.) e ir R 111 k=l a k 36 Thus, we can construct the following sets X l - X 10> X ll' ••"' X l,n(X 1 )-l X N " X N0' X N1' ■•' , X N,n(X N )-l Where V 1 < i < N, V < j < n(X . ) -1, 3a e A, (X. .,X. . _) e R 1 . And (X. , v n - X.) e RJ, v ij' l j+1 a v i n(X„)-l, i V some b € A, Where V a e A, R' is the corresponding a relation on G/C . Let C = |X_. , r 1 < i < N < J < n(X.)-l It is clear that C is a cover with S.P. on Go Furthermore, C is a finite subcollection of C. So, C is finitely reducible. Q.E.D. Definition 36. Let G = be a graph. A path P(x,y) from x to y on G is a finite sequence x = x ,x ,...,x = y of elements of X such that V < i < n-1, "^ a e A, (x.,x. , ) e R . We will denote by S„/ \ the set of v j/ l+l' a J P(x,y) all elements of X occur in the path P(x,y). Definition 37- A graph G = is semi-connected iff for each infinite subset X' of X, there exists an x e X' and an infinite subset X" of X' such that Vy e X", there is a path P(x,y) and S_,/ \ c X' . P(x,y) - 37 Remark 18. A graph G = is semi-connected iff every infinite subgraph of G is semi-connected. Theorem 21. Let G = be an infinite graph. Then G is sequential iff G is semi-connected. Proof: Suppose G is sequential. Let X' be an arbitrary infinite subset of X. Then there is a sequential vertex x„ € X for X' . Let 00 {X . } . _ n be the infinite sequence determined by x n> we see that "=, N > 0, V n > N, x e X' . Let X " = (x i ] i=N+l ■ Then V X i £ X "> 3 ^V*^ a nd S / \ c X. X' arbitrary implies G is semi- connected. Suppose now G is semi-connected. Let X, be an arbitrary infinite subset of X. Then 3 infinite subset X c X , such that V y e X , ^M^IylS 3 !' LetX- =X 2 -x r Then X' is again an infinite subset of X and hence is semi-connected. Therefore, 3 x e X' , 3 infinite subset X c X' such that V y e X , 3 P(x ,y) ^ S / \ c X' . We may assume that P(x ,x ) is a path consisting of distinct elements (by eliminating all loops). Let X' = X, - S p / n . j 3 -^V x -i ? x pJ Then X' is again an infinite subset of X. We then proceed in the same manner to obtain x,- After we 3 have obtained x and the infinite subset X , , n > 1. n n+1' 38 let X' = X , - S„/ \ , where we may assume n+1 n+1 P(x , x ) v n-1, n' S„/ \ consists of distinct elements. X' . is an P(x . ,x ) n+1 n-1 n infinite subset of X implies 3x , € X' , n+1 n+1' =1 infinite subset X _ c X' , such that V y e X n , n+2 - n+1 J n+2' ^ P(x , ,y) and S„/ \ c: X' . Hence we obtain v n+l ,J ' P(x , ,y ) - n+1 n+1 x ., . Proceed in this way, we obtain an infinite n+1 sequence (x. )._- such that V i > 0, =1 P(x.,x_. ) and S p / i el, Now, enlarge this sequence by v l' 1+1 ' adding all the intermediate elements between x. and x. . in the path P(x.,x. . ) V i > 0. Then the l+l * K V l+l ' 00 resulting enlarged sequence [x\ } • _ n has the properties (l) xi = X, } (2) all elements in the sequence are distinct, (3) V i > 0, "3 a e A, (x» x! \ ) e R , — j. i+j. el- and (k) V i > 0, x'. e X, . Thus x, is a sequential vertex for X, . Since X, is arbitrary, G must be sequential. Q.E.D. 39 V. BIBLIOGRAPHY 1. Berge, Claude. The Theory of Graphs and Its Applications . John Wiley, 1962. 2. Birkhoff, G. Lattice Theory . AMS Colloquium Publication, 28, 1948. 3. Cohn, P. M. Universal Algebra. Harper and Row, 1965- k. G impel, J. F. "A Reduction Technique for Prime Implicant Tables." Switching Circuit Theory and Logical Design , pp. 183-191, Oct. 1964. 5- Ginsburg, S. An Introduction to Mathematical Machine Theory . Addison-Wesley, 1962. 6. Hartmanis, J. and Stearns, R. E. "Some Danger in State Reduction of Sequential Machine," Information and Control, 5> pp. 252-260, Sept. 1962. 7. Hartmanis, J. and Stearns, R. E. "Pair Algebra and Its Applica- tion to Automata Theory," Information and Control, 7 > PP- ^85- 507, Dec. I96J+. 8. Hohn, F. E. and Schissler, L. R. "Boolean Matrices and the Design of Combinational Relay Circuits," Bell System Technical Journal , vol. 3I+, pp. 177-202, Jan. 1955- 9« McCluskey, E. J. Introduction to the Theory of Switching Circuits , McGraw-Hill, 1965 . 10. Muller, D. E. "A Treatment of Sequential Machines," File No. 567, Dept. of Computer Science, University of Illinois 11. Muller, D. E. "Lecture Notes on Automata Theory." University of Illinois, Sept. 1965 • 12. Pickett, H. E. "Direct and Subdirect Product: Structure Theorems for Non-Deterministic Automata." Chevon Research Company, 1966. 13. Wensel, G. "Report on the paper by M. Yoeli: Multi- Valued Homomorphism and Subdirect Covers of Partial Algebras." 1+0 APPENDIX The concepts of weak congruence relations, covers with 5. P., and homomorphic relations on graphs have been shown to be closely re- lated. The intention here is to state briefly how these concepts arise in state reduction and decomposition of sequential machines. Definition A. A sequential machine M is a 5 -tuple such that A, B and Q are finite sets and f c= (Q x A) x Q and g c (Q x A) x B. It is well known that every sequential machine M = ^A.,B,Q,F,g> can be represented by a graph G(M) = such that V a e A, q e (p,a) • f ^ (p>q) € R . We note that M is complete iff G(M) is productive, a and M is deterministic iff V a € A, R is single -valued. When M is a deterministic, we will use the convention that Va € A, Vq e Q,, (q,a) • f = qa. In the following, only incomplete, deterministic sequential machines will be considered. Definition B. Let M = be a sequential machine and A the free semi-group generated by A under concatenation. Then V x = a . . .a e A , V q e Q, (q,x) • g exists iff each state q.a., 1 < i < n and each output ( a -) ' g> 1 ^ i 1 n exist; and when (q,x) • g exists, it is defined to be the sequence (q ,a ) • g, ..., ( 1. kl Definition C. Let M = be a sequential machine. V x e A ; V q e Q, x is applicable to q iff (cb x ) ' S exists. Definition D. Two states p,q in M = is compatible iff V x e A applicable to both p and q, (p,x)-g = (q,x)-g. The state reduction problem of a machine M = is essentially a problem of finding a cover with S.P. C = (X.) on Q such that Vi e I, Vp,q e X., p and q are compatible. We will show in the following that compatibility on Q is a weak congruence relation of type II onM. Define e Q x Q such that Vp,q e Q, (p,q) € 9 iff V a e A applicable to both p and q, (p,a) • g = (q,a) • g. Assume 0, has been defined V k > 1. Define 0. , on Q, by the rule: k - k+1 J V p,q e Q, (p,q) e © k+1 Z (p,q) e ^ A V a e A, (p-a,q-a) e R . If |q| = n, then define such that n(n-l) 2 It can be shown readily that = 6(0 ) and that V p,q e Q, (p,q) € ~Z p and q are compatible . Now, if C is any irredundant cover with So P. on Q such that 0=0. Then M/C is a reduced machine of M. Definition E. Let M = be a machine and C = {X.}. a cover with S.P. on Q. Then V q e Q, the degree of q with respect to C, d(q), is defined by d(q) = | {X. |X. e C ~ q € X.} k2 In studying the realizability of a given machine by smaller machines, the concept of "state splitting" arises when a given machine is indecomposable or does not have economical realizations. It was found that enlarging the given machine by adding redundant states (state splitting) may lead to desirable realization. More specifically, the state splitting operation on a machine M = is a multi- valued mapping H: Q -* Q' such that V q e Q, q • H = {q,q, , • • . ,q, / \ -, } • (i.e., embedding G(M) into a larger graph G(M')). Then a cover with S.P. C on M induces a partition C on M' = <^,B,Q' ,f ' ,g'> such that elements of C and C correspond one to one. It can be shown that with suitable definition of f ' , H is indeed a (strong) homomorphic relation. We note that, however, f is not unique. h3 VITA The author, Raymond Tzuu-yau Yeh, was born in Hunan, China on November 5, 1937* He received his B»S. degree in Electrical Engineering in June, I96I, and his M.A. degree in Mathematics in February, 1963 from the University of Illinois. From September I96I to August 1963> ne was a research assistant in Biological Computer group of the Electrical Engineering Research Laboratory of the University of Illinois. Since September, 19&3* ne nas heen a research assistant in the Department of Computer Science. * "\ °-. ^