Morphisms of State Machines Sequential Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 7 Updated and adapted by Marek Perkowski

Notation

Free SemiGroup

String or Word

Concatenation

Partition of a Set Properties p i are called “pi-blocks” of a partition,  (A)

Types of Relations 11. Partial, Binary, Single-Valued System 22. Groupoid 33. SemiGroup 44. Monoid 55. Group

Partial Binary Single-Valued

Groupoid Closed Binary Operation Partial, Binary, Single-Valued System with It is defined on all elements of S x S Not necessarily surjective a a b a b c a b c c a c a b ca(ba) = ac = a (ab) a = ba = c arguments value Surjective Surjective: each y in the R has at least one x in the D Also:

SemiGroup An Associative Groupoid –Binary operation, e.g., multiplication –Closure –Associative Can be defined for various operations, so sometimes written as a a b c b b c a c c a b a b c

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. a a b c b b c a c c a b a b c

Group Monoid With an Inverse a a b c b b c a c c a b a b c a a b c b b c a c c a b a b c Operation is modulo addition. Check that this is a group

‘Morphisms’‘Morphisms’ Homomorphism “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 continued “If operations such as multiplication, addition, or multiplication by scalars are defined for D and R, it is required that these correspond...” and...

Example: Homomorphism of groups “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 Note that homomorphism can map many elements to one. But homomorphic properties must be preserved in the range

Homomorphism preserves correspondence Correspondence must be –Single-valued –Single-valued: therefore at least a partial function –Surjective –Surjective: each y in the R has at least one x in the D –Non-Injective: not one-to-one else isomorphism

Endomorphism Question: What is endomorphism? Answer: An endomorphism is a ‘morphism’ which maps back onto itself The range, R, is the same set as the domain, D, e.g., the real numbers. R=DR=D ‘morphism’

SemiGroup Homomorphism Operation in range Operation in domain

Graphical Explanation of Homomorphism of Semi-Groups Operation in domain Operation in range

Homomorphism of Semi-Groups. Example* *Larsen, Intro to Modern Algebraic Concepts, p. 53 Ask a student to draw operations in domain and range and then show this homomorphism graphically

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 g 4 correspond to the same symbol, 0 Homomorphism of Semi-Groups. Example*

Do the results of operations correspond? same Homomorphism of Semi-Groups. Example*

\Homomorphism of Monoids

IsomorphismIsomorphism Homomorphism Which Is InjectiveAn 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

Graphical illustration of Isomorphism of Semi-Groups Injective Homomorphism

Example of function Log being Isomorphism of two semi-groups Define two groupoids –non-associative semigroups –groups without an inverse or identity element SG1SG1 :A1 = { positive real numbers } * 1 = multiplication = * SG2SG2 :A2 = { positive real numbers } * 2 = addition = + *Ginzberg, pg 10

Isomorphism Example Example of function Log being Isomorphism of two semi-groups (continued)

Graphical illustration of this SemiGroup Isomorphism

Machine Isomorphisms Formally, it should be called Machine Input- output isomorphism, but usually abbreviated to just isomorphism An I/O isomorphism exists between two machines, M 1 and M 2 if there exists a triple alpha

Machine Isomorphisms (cont) alpha iota

Interpret Machine Isomorphisms (cont) Two machine isomorphisms should be introduced, for states and for outputs delta Machine state isomorphism Machine output isomorphism

State Machine State Isomorphism

Output 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 I1I1 I2I2 O2O2 O1O1 M2M2 M1M1 I1I1 O1O1

which is the same as the preceding state diagram and block diagram definitions therefore M 1 and M 2 are Isomorphic to each other

Information in Isomorphic Machines 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 Output strings of one machine are equivalent to output strings of other machine

Identity Machine Isomorphism Al three are identity functions

Inverse Machine Isomorphism

Machine Equivalence Remember: machine isomorphism is an equivalence relation defined on M

Machine Homomorphism

If alpha is injective, then have isomorphism –“State Behavior” assignment, –“Realization” of M 1 If alpha not injective –“Reduction Homomorphism”

Behavioral Equivalence of two State Machines

Behavioral Equivalence

Homework Problem Take an arbitrary machine M and minimize it to machine M2 which has less states. Next specify the homomorphism between Machine M and Machine M2 that corresponds to the relation of combining compatible states. To specify this homomorphism use the formalisms and notations from this lecture.