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

Free SemiGroup

String or Word

Concatenation

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

Types of Relations Partial, Binary, Single-Valued System Groupoid SemiGroup Monoid 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

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

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.

Group Monoid With an Inverse

‘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

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. R=DR=D ‘morphism’

SemiGroup Homomorphism

SmGp. HmMphsm. Example* *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 g 4 correspond to the same symbol, 0

SmGp. HmMphsm. Example* Do the results of operations correspond? same

Monoid Homomorphism

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:A1 = { positive real numbers } * 1 = multiplication = * SG2:A2 = { positive real numbers } * 2 = addition = + *Ginzberg, pg 10

Isomorphism Example

SemiGroup Isomorphism

Machine Isomorphisms 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

Machine Isomorphisms

Interpret

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 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

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

Identity Machine Isomorphism

Inverse Machine Isomorphism

Machine Equivalence

Machine Homomorphism

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

Behavioral Equivalence