Morphisms of State Machines Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 8 Updated and adapted by Marek Perkowski Notation A relation A function A binary operation called multiplication A binary operation called addition Therefore For all There exists Proper subset Subset + Free SemiGroup The free semigroup generated by the set + is the set of all strings (words) from where = symbols This is equivalent to I * String or Word Given = a nonempty set then a string is a finite sequence of elements from e.g ., 1 2 , 2 4 , 1 7 , etc. Concatenation Given , a nonempty set and 1 n , 1' m' then 1 n 1' m' 1 n 1' m' is the concatenat ion of two strings Partition of a Set • Properties A p | p P A and, a) p are disjoint, b) p A, i c) pi • pi are called “pi-blocks” of a partition, (A) Types of Relations • • • • • Partial, Binary, Single-Valued System Groupoid SemiGroup Monoid Group Partial Binary Single-Valued If R :S S S i.e., s1 , s2 , s3 R such that R is a partial function i.e., s3 is unique single - valued and s1 , s2 S S i.e., proper subset partial & D R S Groupoid • Closed Binary Operation • Partial, Binary, Single-Valued System with s1, s2 S i.e., D R S • It is defined on all elements of S x S • Not necessarily surjective SemiGroup • An Associative Groupoid – Binary operation, e.g., multiplication – Closure – Associative a bc ab c a, b, cS • Can be defined for various operations, so sometimes written as S, Closed Binary Operation • Division Is Not a Closed Binary Operation on the Set of Counting Numbers 6/3 = 2 = counting number 2/6 = ? = not a counting number • Division Is Closed Over the Set of Real Numbers. Monoid Semigroup With an Identity Element, e. aA ea a ae a ee e Group Monoid With an Inverse e, a, b A 1 a a e identity element or unit element e must be the same e as defined in monoid i.e., a e a ‘Morphisms’ Homomorphism (J&J) “A correspondence of a set D (the domain) with a set R (the range) such that each element of D determines a unique element of R [single-valued] and each element of R is the correspondent of at least one element of D.“ and... Homomorphism “If operations such as multiplication, addition, or multiplication by scalars are defined for D and R, it is required that these correspond...” and... Homomorphism “If D and R are groups (or semigroups) with the operation denoted by * and x corresponds to x’ and y corresponds to y’ then x * y must correspond to x’ * y’ “ Product of Correspondence = Correspondence of product Homomorphism Homomorphism • Correspondence must be – Single-valued: therefore at least a partial function – Surjective: each y in the R has at least one x in the D – Non-Injective: not one-to-one else isomorphism Endomorphism • A ‘morphism’ which maps back onto itself • The range, R, is the same set as the domain, D, e.g., the real numbers. ‘morphism’ R=D SemiGroup Homomorphism Given semigroups D, and R, + and a function f :D R then f is a semigroup homomorphi sm iff f x f y f x y SemiGroup Homomorphism SmGp. HmMphsm. Example* let D = e, g, g 2 ,, g 7 with multiplica tion, and R = 0,1, 2, 3 with addition, + f (e) = 0 f (g) = 1 f ( g2 ) = 2 f ( g3 ) = 3 f g 4 0 f g 5 1 f g 6 2 f g 7 3 *Larsen, Intro to Modern Algebraic Concepts, p. 53 SmGp. HmMphsm. Example* Is the relation • single-valued? – Each symbol of D maps to only one symbol of R • surjective? – Each symbol of R has a corresponding element in D • not-injective? – e and g4 correspond to the same symbol, 0 SmGp. HmMphsm. Example* Do the results of operations correspond? e. g., let x g2 then x y = g6 f g6 2 and y g4 same f g4 0 f g2 + f g4 = 2 + 0 = 2 f g2 2 Monoid Homomorphism Given semigroups with identity elements D, and R, + and a function f :D R then f is a monoid homomorphi sm iff f x f y f x y and f e e' Isomorphism • An Isomorphism Is a Homomorphism Which Is Injective • Injective: One-to-One Correspondence – A relation between two sets such that pairs can be removed, one member from each set until both sets have been simultaneously exhausted SemiGroup Isomorphism Injective Homomorphism Isomorphism Example* • Define two groupoids – non-associative semigroups – groups without an inverse or identity element • SG1: • SG2: *Ginzberg, pg 10 A1 = { positive real numbers } *1 = multiplication = * A2 = { positive real numbers } *2 = addition = + Isomorphism Example then log is an isomorphis m since xy x y log xy log x log y SemiGroup Isomorphism Machine Isomorphisms • Input-output isomorphism, but usually abbreviated to just isomorphism • An I/O isomorphism exists between two machines, M1 and M2 if there exists a triple , , where , , and . . . are bijective (required for isom.) Machine Isomorphisms : S1 S2 : I1 I 2 : O2 O1 note the reverse order of the subscripts such that sS1 i1 i2 1 s1 , i1 2 s1 , i1 1 s1 , i1 2 s1 , i1 o1 and Machine Isomorphisms Interpret 1 s1 , i1 1 s1 , i1 x1 1 y1 z1 x2 2 y 2 z 2 therefore 1 with 1 with S I and S is a semigroup S I and O is a semigroup where M1 = S1 , I1 , O1 , 1 , 1 M2 = S 2 , I 2 , O2 , 2 , 2 Machine State Isomorphism Machine Output Isomorphism Homo- vice Iso- Morphism Reduction Homomorphism • Shows behavioral equivalence between machines of different sizes • Allows us to only concern ourselves with minimized machines (not yet decomposed, but fewest states in single machine) • If we can find one, we can make a minimum state machine Homo- vice Iso- Morphism Isomorphism • Shows equivalence of machines of identical, but not necessarily minimal, size • Shows equivalence between machines with different labels for the inputs, states, and/or outputs Block Diagram Isomorphism 2 I1 I2 I1 2 M2 M1 1 1 O2 O1 O1 Block Diagram Isomorphism Block Diagram Isomorphism s1 s2 s1 s2 O1 O2 O1 2 s2 , i2 = 2 s1 , i1 which is the same as the preceding state diagram and block diagram definitions therefore M1 and M2 are Isomorphic to each other Machine Information • Since the Inputs and Outputs Can Be Mapped Through Isomorphisms Which Are Independent of the State Transitions, All of the State Change Information Is Maintained in the Isomorphic Machine • Isomorphic Machines Produce Identical Outputs Output Equivalence Let : M 1 M 2 i.e., an I/O isomorphis m then s S1 and x I1* the output strings of M 1 are equivalent to the output strings of - M 2 - 1* s1 , x * s1 , x which is = * 2* s2 , * x Identity Machine Isomorphism id : M 1 M 2 Let : SS : II : O O i.e., all elements of the isomorphis m are identity functions Inverse Machine Isomorphism Let , , : M 1 M 2 then 1 1 , 1 , 1 : M 2 M 1 For there to exists a 1 , , must be bijective, i.e., injective and surjective Machine Equivalence Let M 1 , M 2 , M 3 be isomorphic machines then, M M reflexive M1 M 2 M 2 M1 symmetric M 1 M 2 and M 2 M 3 M 1 M 3 transitive which we recognize as the properties of an equivalence relationship, i. e., machine isomorphism is an equivalence relationship defined on M Machine Homomorphism Let , , : M1 M2 then is an I/O Homorphis m of M1 into M2 iff : S1 S2 i.e., many to one : I1 I 2 : O2 O1 note reverse order of subscripts are functions sS1 , i I1 , oO2 and Machine Homomorphism 2 s , i 1 s, i 2 s , i 1 s, i • If alpha is injective, then have isomorphism – “State Behavior” assignment, M 1 M 2 – “Realization” of M1 • If alpha not injective – “Reduction Homomorphism” M 1 < M 2 Behavioral Equivalence Two machines, M1 and M2 are behavioral ly equivalent iff I1 I 2 O1 O2 and R S1 S 2 for which Behavioral Equivalence D R S1 R R S2 and if s1 R s2 then x I1 * 1* s1 , x 2* s2 , x