Minimization of Finite State Machines 1

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1 FORMALIZED MATHEMATICS Volume 5, Number 2, 1996 Warsa University - Bia lystok Minimization of Finite State Machines 1 Miroslava Kaloper University of Alberta Department of Computing Science Piotr Rudnicki University of Alberta Department of Computing Science Summary. We have formalized deterministic finite state machines closely folloing the textbook [9], pp up to the minimization theorem. In places, e have changed the approach presented in the book as it turned out to be too specific and inconvenient. Our ork also revealed several minor mistakes in the book. After defining a structure for an outputless finite state machine, e have derived the structures for the transition assigned output machine (Mealy) and state assigned output machine (Moore). The machines are then proved similar, in the sense that for any Mealy (Moore) machine there exists a Moore (Mealy) machine producing essentially the same response for the same input. The rest of ork is then done for Mealy machines. Next, e define equivalence of machines, equivalence and k-equivalence of states, and characterize a process of constructing for a given Mealy machine, the machine equivalent to it in hich no to states are equivalent. The final, minimization theorem states: Theorem 4.5: Let M 1 and M 2 be reduced, connected finite-state machines. Then the state graphs of M 1 and M 2 are isomorphic if and only if M 1 and M 2 are equivalent. and it is the last theorem in this article. MML Identifier: FSM 1. The papers [19], [23], [10], [2], [21], [13], [16], [8], [20], [18], [24], [5], [6], [7], [22], [3], [4], [1], [14], [17], [12], [11], and [15] provide the terminology and notation for this paper. 1 This ork as partially supported by NSERC Grant OGP c 1996 Warsa University - Bia lystok ISSN

2 d d 174 miroslava kaloper and piotr rudnicki 1. Preliminaries For simplicity e adopt the folloing convention: m, n, i, k ill denote natural numbers, D ill denote a non empty set, d ill denote an element of D, and d 1, d 2 ill denote finite sequences of elements of D. Next e state several propositions: (1) If m < n, then there exists a natural number p such that n = m + p and 1 p. (2) If i Seg n, then i + m Seg(n + m). (3) If i > 0 and i + m Seg(n + m), then i Seg n and i Seg(n + m). (4) If k < i, then there exists a natural number j such that j = i k and 1 j. (5) If 1 len d 1, then there exist d, d 2 such that d = d 1 (1) and d 1 = d (6) If i dom d 1, then ( d 1 )(i + 1) = d 1 (i). For simplicity e adopt the folloing rules: S is a set, D 1, D 2 are non empty sets, f 1 is a function from S into D 1, and f 2 is a function from D 1 into D 2. One can prove the folloing propositions: (7) If f 1 is bijective and f 2 is bijective, then f 2 f 1 is bijective. (8) For every set Y and for all equivalence relations E 1, E 2 of Y such that Classes E 1 = Classes E 2 holds E 1 = E 2. (9) For every non empty set W holds every partition of W is non empty. (10) For every finite set Z holds every partition of Z is finite. Let W be a non empty set. Note that every partition of W is non empty. Let Z be a finite set. Note that every partition of Z is finite. Let X be a non empty finite set. Observe that there exists a partition of X hich is non empty and finite. We adopt the folloing rules: X, A ill be non empty finite sets, P 1 ill be a partition of X, and P 2, P 3 ill be partitions of A. We no state several propositions: (11) For every set P 4 such that P 4 P 1 there exists an element x of X such that x P 4. (12) card P 1 card X. (13) If P 2 is finer than P 3, then card P 3 card P 2. (14) If P 2 is finer than P 3, then for every element p 2 of P 3 there exists an element p 1 of P 2 such that p 1 p 2. (15) If P 2 is finer than P 3 and card P 2 = card P 3, then P 2 = P 3. 2.

3 minimization of finite state machines Definitions and Terminology Let I 1 be a non empty set. We consider FSM over I 1 as systems states, a Tran, a InitS, here the states constitute a finite non empty set, the Tran is a function from [:the states, I 1 :] into the states, and the InitS is an element of the states. Let I 1 be a non empty set and let f 3 be a FSM over I 1. A state of f 3 is an element of the states of f 3. For simplicity e follo a convention: I 1, O 1 are non empty sets, f 3 is a FSM over I 1, s is an element of I 1,, 1, 2 are finite sequences of elements of I 1, q, q, q 1, q 2 are states of f 3, and q 3 is a finite sequence of elements of the states of f 3. Let us consider I 1, f 3, s, q. The functor s-succ(q) yielding a state of f 3 is defined by: (Def.1) s-succ(q) = (the Tran of f 3 )( q, s ). Let us consider I 1, f 3, q,. The functor (q,)-admissible yields a finite sequence of elements of the states of f 3 and is defined by the conditions (Def.2). (Def.2) (i) (q, )-admissible(1) = q, (ii) len((q, )-admissible) = len + 1, and (iii) for every i such that 1 i and i len there exists an element 3 of I 1 and there exist states q 4, q 5 of f 3 such that 3 = (i) and q 4 = (q,)-admissible(i) and q 5 = (q,)-admissible(i + 1) and 3 -succ(q 4 ) = q 5. The folloing proposition is true (16) (q,ε (I1 ))-admissible = q. Let us consider I 1, f 3,, q 1, q 2. The predicate q 1 q2 is defined as follos: (Def.3) (q 1,)-admissible(len + 1) = q 2. We no state the proposition (17) q ε (I 1 ) q. Let us consider I 1, f 3,, q 3. We say that q 3 is admissible for if and only if: (Def.4) There exists q 1 such that q 1 = q 3 (1) and (q 1,)-admissible = q 3. We no state the proposition (18) q is admissible for ε (I1 ). Let us consider I 1, f 3, q,. The functor -succ(q) yields a state of f 3 and is defined by: (Def.5) q -succ(q). One can prove the folloing propositions: (19) (q,)-admissible(len((q,)-admissible)) = q iff q q. (20) For every k such that 1 k and k len 1 holds (q 1, 1 2 )-admissible(k) = (q 1, 1 )-admissible(k).

4 176 miroslava kaloper and piotr rudnicki (21) If q 1 1 q2, then (q 1, 1 2 )-admissible(len 1 + 1) = q 2. (22) If q 1 1 q2, then for every k such that 1 k and k len holds (q 1, 1 2 )-admissible(len 1 + k) = (q 2, 2 )-admissible(k). (23) If q 1 1 q2, then (q 1, 1 2 )-admissible = ((q 1, 1 )-admissible len 1 +1) (q 2, 2 )-admissible. 3. Mealy and Moore Machines Let I 1, O 1 be non empty sets. We consider Mealy-FSM over I 1, O 1 as extensions of FSM over I 1 as systems states, a Tran, a OFun, a InitS, here the states constitute a finite non empty set, the Tran is a function from [:the states, I 1 :] into the states, the OFun is a function from [:the states, I 1 :] into O 1, and the InitS is an element of the states. We introduce Moore-FSM over I 1, O 1 hich are extensions of FSM over I 1 and are systems states, a Tran, a OFun, a InitS, here the states constitute a finite non empty set, the Tran is a function from [:the states, I 1 :] into the states, the OFun is a function from the states into O 1, and the InitS is an element of the states. For simplicity e adopt the folloing convention: t 1, t 2, t 3, t 4 ill denote Mealy-FSM over I 1, O 1, s 1 ill denote a Moore-FSM over I 1, O 1, q 6 ill denote a state of s 1, q, q 1, q 2, q 7, q 8, q 9, q 10, q 1, q 11, q 12, q 13 ill denote states of t 1, q 14, q 15 ill denote states of t 2, and q 21, q 22 ill denote states of t 3. Let us consider I 1, O 1, t 1, q 11,. The functor (q 11,)-response yields a finite sequence of elements of O 1 and is defined as follos: (Def.6) len((q 11,)-response) = len and for every i such that i dom holds (q 11,)-response(i) = (the OFun of t 1 )( (q 11,)-admissible(i), (i) ). The folloing proposition is true (24) (q 11,ε (I1 ))-response = ε (O1 ). Let us consider I 1, O 1, s 1, q 6,. The functor (q 6,)-response yields a finite sequence of elements of O 1 and is defined by: (Def.7) len((q 6,)-response) = len + 1 and for every i such that i Seg(len + 1) holds (q 6,)-response(i) = (the OFun of s 1 )((q 6,)-admissible(i)). One can prove the folloing propositions: (25) (q 6,)-response(1) = (the OFun of s 1 )(q 6 ). (26) If q 1 12 q13, then (q 12, 1 (q 13, 2 )-response. 2 )-response = (q 12, 1 )-response (27) If q 1 14 q15 and q 1 21 q22 and (q 15, 2 )-response (q 22, 2 )-response, then (q 14, 1 2 )-response (q 21, 1 2 )-response.

5 )-response minimization of finite state machines 177 In the sequel O 2 is a finite non empty set, t 5 is a Mealy-FSM over I 1, O 2, and s 2 is a Moore-FSM over I 1, O 2. Let us consider I 1, O 1, t 1, s 1. We say that t 1 is similar to s 1 if and only if the condition (Def.8) is satisfied. (Def.8) Let t 6 be a finite sequence of elements of I 1. Then (the OFun of s 1 )(the InitS of s 1 ) (the InitS of t 1, t 6 )-response = (the InitS of s 1, t 6 )-response. The folloing propositions are true: (28) There exists t 1 hich is similar to s 1. (29) There exists s 2 such that t 5 is similar to s Equivalence of States and Machines Let us consider I 1, O 1, t 2, t 3. We say that t 2 and t 3 are equivalent if and only if: (Def.9) For every holds (the InitS of t 2, )-response = (the InitS of t 3, )-response. Let us observe that the predicate introduced above is reflexive and symmetric. We no state the proposition (30) If t 2 and t 3 are equivalent and t 3 and t 4 are equivalent, then t 2 and t 4 are equivalent. Let us consider I 1, O 1, t 1, q 8, q 9. We say that q 8 and q 9 are equivalent if and only if: (Def.10) For every holds (q 8,)-response = (q 9,)-response. We no state several propositions: (31) q and q are equivalent. (32) If q 1 and q 2 are equivalent, then q 2 and q 1 are equivalent. (33) If q 1 and q 2 are equivalent and q 2 and q 7 are equivalent, then q 1 and q 7 are equivalent. (34) If q 1 = (the Tran of t 1)( q 8, s ), then for every i such that i Seg(len +1) holds (q 8, s )-admissible(i+1) = (q 1,)-admissible(i). (35) If q 1 = (the Tran of t 1)( q 8, s ), then (q 8, s = (the OFun of t 1 )( q 8, s ) (q 1,)-response. (36) q 8 and q 9 are equivalent if and only if for every s holds (the OFun of t 1 )( q 8, s ) = (the OFun of t 1 )( q 9, s ) and (the Tran of t 1 )( q 8, s ) and (the Tran of t 1 )( q 9, s ) are equivalent. (37) Suppose q 8 and q 9 are equivalent. Given, i. Suppose i dom. Then there exist states q 16, q 17 of t 1 such that q 16 = (q 8,)-admissible(i) and q 17 = (q 9,)-admissible(i) and q 16 and q 17 are equivalent. Let us consider I 1, O 1, t 1, q 8, q 9, k. We say that q 8 and q 9 are k-equivalent if and only if:

6 178 miroslava kaloper and piotr rudnicki (Def.11) For every such that len k holds (q 8,)-response = (q 9,)-response. One can prove the folloing propositions: (38) q 8 and q 8 are k-equivalent. (39) If q 8 and q 9 are k-equivalent, then q 9 and q 8 are k-equivalent. (40) If q 8 and q 9 are k-equivalent and q 9 and q 10 are k-equivalent, then q 8 and q 10 are k-equivalent. (41) If q 8 and q 9 are equivalent, then q 8 and q 9 are k-equivalent. (42) q 8 and q 9 are 0-equivalent. (43) If q 8 and q 9 are k + m-equivalent, then q 8 and q 9 are k-equivalent. (44) Suppose 1 k. Then q 8 and q 9 are k-equivalent if and only if the folloing conditions are satisfied: (i) q 8 and q 9 are 1-equivalent, and (ii) for every element s of I 1 and for every natural number k 1 such that k 1 = k 1 holds (the Tran of t 1 )( q 8, s ) and (the Tran of t 1 )( q 9, s ) are k 1 -equivalent. Let us consider I 1, O 1, t 1, i. The functor i-eqs-rel(t 1 ) yielding an equivalence relation of the states of t 1 is defined as follos: (Def.12) For all q 8, q 9 holds q 8, q 9 i-eqs-rel(t 1 ) iff q 8 and q 9 are i-equivalent. Let us consider I 1, O 1, t 1, i. The functor i-eqs-part(t 1 ) yields a non empty finite partition of the states of t 1 and is defined by: (Def.13) i-eqs-part(t 1 ) = Classes(i-EqS-Rel(t 1 )). One can prove the folloing propositions: (45) (k + 1)-EqS-Part(t 1 ) is finer than k-eqs-part(t 1 ). (46) If Classes(k-EqS-Rel(t 1 )) = Classes((k+1)-EqS-Rel(t 1 )), then for every m holds Classes((k + m)-eqs-rel(t 1 )) = Classes(k-EqS-Rel(t 1 )). (47) If k-eqs-part(t 1 ) = (k + 1)-EqS-Part(t 1 ), then for every m holds (k + m)-eqs-part(t 1 ) = k-eqs-part(t 1 ). (48) If (k + 1)-EqS-Part(t 1 ) k-eqs-part(t 1 ), then for every i such that i k holds (i + 1)-EqS-Part(t 1 ) i-eqs-part(t 1 ). (49) k-eqs-part(t 1 ) = (k + 1)-EqS-Part(t 1 ) or card(k-eqs-part(t 1 )) < card((k + 1)-EqS-Part(t 1 )). (50) [q] 0-EqS-Rel(t 1 ) = the states of t 1. (51) 0-EqS-Part(t 1 ) = {the states of t 1 }. (52) If n + 1 = card (the states of t 1 ), then (n + 1)-EqS-Part(t 1 ) = n-eqs-part(t 1 ). Let us consider I 1, O 1, t 1. A partition of the states of t 1 is final if: (Def.14) For all q 8, q 9 holds q 8 and q 9 are equivalent iff there exists an element X of it such that q 8 X and q 9 X. Next e state three propositions: (53) If k-eqs-part(t 1 ) is final, then (k + 1)-EqS-Rel(t 1 ) = k-eqs-rel(t 1 ).

7 minimization of finite state machines 179 (54) k-eqs-part(t 1 ) = (k + 1)-EqS-Part(t 1 ) iff k-eqs-part(t 1 ) is final. (55) If n + 1 = card (the states of t 1 ), then there exists a natural number k such that k n and k-eqs-part(t 1 ) is final. Let us consider I 1, O 1, t 1. The functor final-partition(t 1 ) yields a partition of the states of t 1 and is defined by: (Def.15) final-partition(t 1 ) is final. We no state the proposition (56) If n + 1 = card (the states of t 1 ), then final-partition(t 1 ) = n-eqs-part(t 1 ). 5. The Reduction of a Mealy Machine In the sequel r 1 ill be a Mealy-FSM over I 1, O 1, q 18 ill be a state of r 1, and q 19 ill be an element of final-partition(t 1 ). Let us consider I 1, O 1, t 1, q 19, s. The functor (s,q 19 )-C-succ yields an element of final-partition(t 1 ) and is defined by: (Def.16) There exist q, n such that q q 19 and n + 1 = card (the states of t 1 ) and (s,q 19 )-C-succ = [(the Tran of t 1 )( q, s )] n-eqs-rel(t 1 ). Let us consider I 1, O 1, t 1, q 19, s. The functor (q 19,s)-C-response yielding an element of O 1 is defined by: (Def.17) There exists q such that q q 19 and (q 19,s)-C-response = (the OFun of t 1 )( q, s ). Let us consider I 1, O 1, t 1. The reduction of t 1 yielding a strict Mealy-FSM over I 1, O 1 is defined by the conditions (Def.18). (Def.18) (i) The states of the reduction of t 1 = final-partition(t 1 ), (ii) for every state Q of the reduction of t 1 and for all s, q such that q Q holds (the Tran of t 1 )( q, s ) (the Tran of the reduction of t 1 )( Q, s ) and (the OFun of t 1 )( q, s ) = (the OFun of the reduction of t 1 )( Q, s ), and (iii) the InitS of t 1 the InitS of the reduction of t 1. The folloing to propositions are true: (57) If r 1 = the reduction of t 1 and q q 18, then for every k such that k Seg(len + 1) holds (q,)-admissible(k) (q 18,)-admissible(k). (58) t 1 and the reduction of t 1 are equivalent. 6. Machine Isomorphism In the sequel q 20, q 23 ill denote states of r 1 and T 1 ill denote a function from the states of t 2 into the states of t 3.

8 180 miroslava kaloper and piotr rudnicki Let us consider I 1, O 1, t 2, t 3. We say that t 2 and t 3 are isomorphic if and only if the condition (Def.19) is satisfied. (Def.19) There exists T 1 such that (i) T 1 is bijective, (ii) T 1 (the InitS of t 2 ) = the InitS of t 3, and (iii) for all q 14, s holds T 1 ((the Tran of t 2 )( q 14, s )) = (the Tran of t 3 )( T 1 (q 14 ), s ) and (the OFun of t 2 )( q 14, s ) = (the OFun of t 3 )( T 1 (q 14 ), s ). Let us observe that the predicate introduced above is reflexive and symmetric. We no state four propositions: (59) If t 2 and t 3 are isomorphic and t 3 and t 4 are isomorphic, then t 2 and t 4 are isomorphic. (60) Suppose that for every state q of t 2 and for every s holds T 1 ((the Tran of t 2 )( q, s )) = (the Tran of t 3 )( T 1 (q), s ). Given k. If 1 k and k len + 1, then T 1 ((q 14,)-admissible(k)) = (T 1 (q 14 ),)-admissible(k). (61) Suppose that (i) T 1 (the InitS of t 2 ) = the InitS of t 3, and (ii) for every state q of t 2 and for every s holds T 1 ((the Tran of t 2 )( q, s )) = (the Tran of t 3 )( T 1 (q), s ) and (the OFun of t 2 )( q, s ) = (the OFun of t 3 )( T 1 (q), s ). Then q 14 and q 15 are equivalent if and only if T 1 (q 14 ) and T 1 (q 15 ) are equivalent. (62) If r 1 = the reduction of t 1 and q 20 q 23, then q 20 and q 23 are not equivalent. 7. Reduced and Connected Machines Let I 1, O 1 be non empty sets. A Mealy-FSM over I 1, O 1 is reduced if: (Def.20) For all states q 8, q 9 of it such that q 8 q 9 holds q 8 and q 9 are not equivalent. One can prove the folloing proposition (63) The reduction of t 1 is reduced. Let us consider I 1, O 1. Note that there exists a Mealy-FSM over I 1, O 1 hich is reduced. In the sequel R 1 ill denote a reduced Mealy-FSM over I 1, O 1. Next e state to propositions: (64) R 1 and the reduction of R 1 are isomorphic. (65) t 1 is reduced iff there exists a Mealy-FSM M over I 1, O 1 such that t 1 and the reduction of M are isomorphic. Let us consider I 1, O 1, t 1. A state of t 1 is accessible if: (Def.21) There exists such that the InitS of t 1 it.

9 minimization of finite state machines 181 Let us consider I 1, O 1. A Mealy-FSM over I 1, O 1 is connected if: (Def.22) Every state of it is accessible. Let us consider I 1, O 1. One can check that there exists a Mealy-FSM over I 1, O 1 hich is connected. In the sequel C 1, C 2, C 3 ill be connected Mealy-FSM over I 1, O 1. We no state the proposition (66) The reduction of C 1 is connected. Let us consider I 1, O 1. Note that there exists a Mealy-FSM over I 1, O 1 hich is connected and reduced. Let us consider I 1, O 1, t 1. The functor accessible-states(t 1 ) yields a finite non empty set and is defined as follos: (Def.23) accessible-states(t 1 ) = {q : q ranges over states of t 1, q is accessible}. The folloing propositions are true: (67) accessible-states(t 1 ) the states of t 1 and for every q holds q accessible-states(t 1 ) iff q is accessible. (68) (The Tran of t 1 ) [: accessible-states(t 1 ), I 1 :] is a function from [:accessible-states(t 1 ), I 1 :] into accessible-states(t 1 ). (69) Let c 1 be a function from [: accessible-states(t 1 ), I 1 :] into accessible-states(t 1 ), and let c 2 be a function from [:accessible-states(t 1 ), I 1 :] into O 1, and let c 3 be an element of accessible-states(t 1 ). Suppose c 1 = (the Tran of t 1 ) [:accessible-states(t 1 ), I 1 :] and c 2 = (the OFun of t 1 ) [:accessible-states(t 1 ), I 1 :] and c 3 = the InitS of t 1. Then t 1 and Mealy-FSM accessible-states(t 1 ),c 1,c 2,c 3 are equivalent. (70) There exists C 1 such that (i) the Tran of C 1 = (the Tran of t 1 ) [:accessible-states(t 1 ), I 1 :], (ii) the OFun of C 1 = (the OFun of t 1 ) [: accessible-states(t 1 ), I 1 :], (iii) the InitS of C 1 = the InitS of t 1, and (iv) t 1 and C 1 are equivalent. 8. Machine Union Let us consider I 1, O 1, t 2, t 3. The functor Mealy-U(t 2,t 3 ) yields a strict Mealy-FSM over I 1, O 1 and is defined by the conditions (Def.24). (Def.24) (i) The states of Mealy-U(t 2,t 3 ) = (the states of t 2 ) (the states of t 3 ), (ii) the Tran of Mealy-U(t 2,t 3 ) = (the Tran of t 2 ) + (the Tran of t 3 ), (iii) the OFun of Mealy-U(t 2,t 3 ) = (the OFun of t 2 ) + (the OFun of t 3 ), and (iv) the InitS of Mealy-U(t 2,t 3 ) = the InitS of t 2. One can prove the folloing propositions: (71) If t 1 = Mealy-U(t 2,t 3 ) and (the states of t 2 ) (the states of t 3 ) = and q 14 = q, then (q 14,)-admissible = (q,)-admissible.

10 182 miroslava kaloper and piotr rudnicki (72) If t 1 = Mealy-U(t 2,t 3 ) and (the states of t 2 ) (the states of t 3 ) = and q 14 = q, then (q 14,)-response = (q,)-response. (73) If t 1 = Mealy-U(t 2,t 3 ) and (the states of t 2 ) (the states of t 3 ) = and q 21 = q, then (q 21,)-admissible = (q,)-admissible. (74) If t 1 = Mealy-U(t 2,t 3 ) and (the states of t 2 ) (the states of t 3 ) = and q 21 = q, then (q 21,)-response = (q,)-response. In the sequel R 2, R 3 ill be reduced Mealy-FSM over I 1, O 1. The folloing proposition is true (75) Suppose t 1 = Mealy-U(R 2,R 3 ) and (the states of R 2 ) (the states of R 3 ) = and R 2 and R 3 are equivalent. Then there exists a state Q of the reduction of t 1 such that the InitS of R 2 Q and the InitS of R 3 Q and Q = the InitS of the reduction of t 1. For simplicity e follo a convention: C 4, C 5 ill denote connected reduced Mealy-FSM over I 1, O 1, c 11, c 12 ill denote states of C 4, c 21, c 22 ill denote states of C 5, and q 24, q 25 ill denote states of t 1. The folloing propositions are true: (76) Suppose that (i) c 11 = q 24, (ii) c 12 = q 25, (iii) (the states of C 4 ) (the states of C 5 ) =, (iv) C 4 and C 5 are equivalent, (v) t 1 = Mealy-U(C 4,C 5 ), and (vi) c 11 and c 12 are not equivalent. Then q 24 and q 25 are not equivalent. (77) Suppose that (i) c 21 = q 24, (ii) c 22 = q 25, (iii) (the states of C 4 ) (the states of C 5 ) =, (iv) C 4 and C 5 are equivalent, (v) t 1 = Mealy-U(C 4,C 5 ), and (vi) c 21 and c 22 are not equivalent. Then q 24 and q 25 are not equivalent. (78) Suppose (the states of C 4 ) (the states of C 5 ) = and C 4 and C 5 are equivalent and t 1 = Mealy-U(C 4,C 5 ). Let Q be a state of the reduction of t 1. Then there do not exist elements q 1, q 2 of Q such that q 1 the states of C 4 and q 2 the states of C 4 and q 1 q 2. (79) Suppose (the states of C 4 ) (the states of C 5 ) = and C 4 and C 5 are equivalent and t 1 = Mealy-U(C 4,C 5 ). Let Q be a state of the reduction of t 1. Then there do not exist elements q 1, q 2 of Q such that q 1 the states of C 5 and q 2 the states of C 5 and q 1 q 2. (80) Suppose (the states of C 4 ) (the states of C 5 ) = and C 4 and C 5 are equivalent and t 1 = Mealy-U(C 4,C 5 ). Let Q be a state of the reduction of t 1. Then there exist elements q 1, q 2 of Q such that q 1 the states of

11 minimization of finite state machines 183 C 4 and q 2 the states of C 5 and for every element q of Q holds q = q 1 or q = q The Minimization Theorem We no state several propositions: (81) There exist Mealy-FSM f 4, f 5 over I 1, O 1 such that (the states of f 4 ) (the states of f 5 ) = and f 4 and t 2 are isomorphic and f 5 and t 3 are isomorphic. (82) If t 2 and t 3 are isomorphic, then t 2 and t 3 are equivalent. (83) If (the states of C 4 ) (the states of C 5 ) = and C 4 and C 5 are equivalent, then C 4 and C 5 are isomorphic. (84) If C 2 and C 3 are equivalent, then the reduction of C 2 and the reduction of C 3 are isomorphic. (85) Let M 1, M 2 be connected reduced Mealy-FSM over I 1, O 1. Then M 1 and M 2 are isomorphic if and only if M 1 and M 2 are equivalent. References [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2): , [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41 46, [3] Grzegorz Bancerek and Krzysztof Hrynieiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1): , [4] Czes la Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3): , [5] Czes la Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55 65, [6] Czes la Byliński. Functions from a set to a set. Formalized Mathematics, 1(1): , [7] Czes la Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3): , [8] Agata Darmocha l. Finite sets. Formalized Mathematics, 1(1): , [9] P. J. Denning, J. B. Dennis, and J. E. Qualitz. Machines, Languages, and Computation. Prentice-Hall, [10] Krzysztof Hrynieiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35 40, [11] Katarzyna Jankoska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5): , [12] Yatsuka Nakamura and Andrzej Trybulec. A mathematical model of CPU. Formalized Mathematics, 3(2): , [13] Beata Padleska. Families of sets. Formalized Mathematics, 1(1): , [14] Konrad Raczkoski and Pae l Sadoski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3): , [15] Yozo Toda. The formalization of simple graphs. Formalized Mathematics, 5(1): , [16] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2): , [17] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): , 1991.

12 184 miroslava kaloper and piotr rudnicki [18] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): , [19] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9 11, [20] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97 105, [21] Micha l J. Trybulec. Integers. Formalized Mathematics, 1(3): , [22] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67 71, [23] Zinaida Trybulec and Halina Śie czkoska. Boolean properties of sets. Formalized Mathematics, 1(1):17 23, [24] Edmund Woronoicz. Relations and their basic properties. Formalized Mathematics, 1(1):73 83, Received November 18, 1994