Morphisms of State Machines Sequential Machine Theory Prof
- Slides: 50
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 Semi. Group
String or Word
Concatenation
Partition of a Set • Properties • pi are called “pi-blocks” of a partition, (A)
Types of Relations • • • 1. Partial, Binary, Single-Valued System 2. Groupoid 3. Semi. Group 4. Monoid 5. 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 arguments a a a b c c c Also: b c b a a b a c a(ba) = ac = a (ab) a = ba = c value Surjective: Surjective each y in the R has at least one x in the D
Semi. Group • An Associative Groupoid – Binary operation, e. g. , multiplication – Closure – Associative • Can be defined for various operations, so sometimes written as a a a b b c c a a b
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 a b b c c a a b
Group Monoid With an Inverse 0 1 2 a a a b b c c b c b c c a a b Operation is modulo addition. Check that this is a group
‘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=D ‘morphism’
Semi. Group 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
Homomorphism of Semi-Groups. 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
Homomorphism of Semi-Groups. Example* Do the results of operations correspond? same
Homomorphism of Monoids
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
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 • SG 1: A 1 = { positive real numbers } *1 = multiplication = * • SG 2: A 2 = { positive real numbers } *2 = addition = + *Ginzberg, pg 10
Example of function Log being Isomorphism of two semi-groups Isomorphism Example (continued)
Graphical illustration of this Semi. Group Isomorphism
Machine Isomorphisms • Formally, it should be called Machine Inputoutput 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
Machine Isomorphisms (cont) delta Interpret Machine state isomorphism Machine output isomorphism Two machine isomorphisms should be introduced, for states and for outputs
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 I 1 I 2 I 1 M 2 M 1 O 2 O 1
Block Diagram Isomorphism
Block Diagram Isomorphism 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
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 M 2 which has less states. • Next specify the homomorphism between Machine M and Machine M 2 that corresponds to the relation of combining compatible states. • To specify this homomorphism use the formalisms and notations from this lecture.
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