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**Sequential Machine Theory**

Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 1 Adaptation to this class and additional comments by Marek Perkowski

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**Why Sequential Machine Theory (SMT)?**

Some Things Cannot be Parallelized Theory Leads to New Ways of Doing Things, has also practical applications in software and hardware (compiler design, controllers design, etc.) Understand Fundamental FSM Limits Minimize FSM Complexity and Size Find the “Essence” of a Machine, what does it mean that there is a machine for certain task?

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**Why Sequential Machine Theory?**

Discuss FSM properties that are unencumbered by Implementation Issues: Software Hardware FPGA/ASIC/Memory, etc. Technology is Changing Rapidly, the core of the theory remains forever. Theory is a Framework within which to Understand and Integrate Practical Considerations

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Hardware/Software There Is an Equivalence Relation Between Hardware and Software Anything that can be done in one can be done in the other…perhaps faster/slower System design now done in hardware description languages (VHDL, Verilog, higher) without regard for realization method Hardware/software/split decision deferred until later stage in design

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Hardware/Software Hardware/Software equivalence extends to formal languages Different classes of computational machines are related to different classes of formal languages Finite State Machines (FSM) can be equivalently represented by one class of languages

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**Formal Languages Unambiguous Can Be Finite or Infinite**

Give some simple examples Can Be Rule-based or Enumerated Various Classes With Different Properties

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**Finite State Machines FSMs are Equivalent to One Class of Languages**

Prototypical Sequence Controller Generator acceptor controller Many Processes Have Temporal Dependencies and Cannot Be Parallelized, the need some form of state machine. FSM Costs Hardware: More States More Hardware Time: More States, Slower Operation Technology dependent: how many CPLD chips?

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**Goal of this set of lectures**

Develop understanding of Hardware/Software/Language Equivalence Understand Properties of FSM Develop Ability to Convert FSM Specification Into Set-theoretic Formulation Develop Ability to Partition Large Machine Into Greatest Number of Smallest Machines This reduction is unique

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**Machine/Mathematics Hierarchy**

AI Theory Intelligent Machines Computer Theory Computer Design Automata Theory Finite State Machine Boolean Algebra Combinational Logic

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**Combinational Logic Feedforward Output Is Only a Function of Input**

No Feedback No memory No temporal dependency Two-Valued Function Minimization Techniques Well-known Minimization Techniques Multi-valued Function Minimization Well-known Heuristics

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**Finite State Machine Feedback**

Behavior Depends Both on Present State and Present Input State Minimization Well-known With Guaranteed Minimum Realization Minimization Unsolved problem of Digital Design Technology related, combinational design related

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**Computer Design: Turing Machines**

Defined by Turing Computability Can compute anything that is “computable” Some things are not computable Assumed Infinite Memory State Dependent Behavior Elements: Control Unit is specified and implemented as FSM Tape infinite Head Head movements Show example of a very simple Turing machine now: x--> x+1

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**Intelligent Machines Some machines display an ability to learn**

How a machine can learn? Some problems are possibly not computable What problems? Why not computable? Something must be infinite?

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**concepts in other areas**

Automata, aka FSM Concepts of Machines: Mechanical Counters, adders Computer programs Political Towns, highways, social groups, parties, etc Biological Tissues, cells, genetic, neural, societies Abstract mathematical Functions, relations, graphs You should be able to use FSM concepts in other areas like robotics

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**FSM - Abstract mathematical concept of many types of behavior**

Discrete Continuous system can be discretized to any degree of resolution Finite State: finite alphabets for inputs, outputs and states. Input/Output Some cause, some result

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**Set Theoretic Formulation of Finite State Machine**

S: Finite set of possible states I: Finite set of possible inputs O: Finite set of possible outputs : Rule defining state change : Rule determining outputs

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**Types of FSMs Moore FSM Mealy FSM Output is a function of state only**

Output is a function of both the present state and the present input Discuss timing differences, show examples and diagrams, discuss fast signaling and PLD realization

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**Types of FSMs Finite State Acceptors, Language Recognizers**

Start in a single, specified state End in particular state(s) Pushdown Automata Not an FSM Assumed infinite stack with access only to topmost element

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**Computer Turing Machine Assumed infinite read/write tape**

FSM controls read/write/tape motion Definition of computable function Universal Turing Machine reads FSM behavior from tape

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Review of Set Theory Element: “a”, a single object with no special property Set: “A”, a collection of elements, i.e., Enumerated Set: Finite Set:

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Sets Infinite set Set of sets

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Subsets All elements of B are elements of A and there may be one or more elements of A that is not an element of B A3 Larry, Curly, Moe A6 integers A7

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Proper Subset All elements of B are elements of A and there is at least one element of A that is not an element of B

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Set Equality Set A is equal to set B

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**Sets Null Set Every set is a subset of itself**

A set with no elements, Every set is a subset of itself Every set contains the null set

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Operations on Sets Intersection Union Logical AND Logical OR

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Operations on Sets Set Difference Cartesian Product, Direct Product

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**Special Sets Powerset: set of all subsets of A**

*no braces around the null set since the symbol represents the set

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**Special Sets Disjoint sets: A and B are disjoint if Cover:**

We know set covering problem from 572. It was defined as a matrix problem

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**Properties of Operations on Sets**

Commutative, Abelian Associative Distributive Left hand distributive

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**Partition of a Set Properties**

pi are called “pi-blocks” or “-blocks” of PI

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**Relations Between Sets**

If A and B are sets, then the relation from A to B, is a subset of the Cartesian product of A and B, i.e., R-related:

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Domain of a Relation Domain of R R B a A b

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Range of a Relation Range of R R b A a B

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Inverse Relation, R-1 R-1 B a A b

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**Partial Function, Mapping**

A single-valued relation such that R a b b’ a’ * A B * can be many to one

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**Partial Function Also called the Image of a under R**

Only one element of B for each element of A Single-valued Can be a many-to-one mapping

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**Function A partial function with**

A b corresponds to each a, but only one b for each a Possibly many-to-one: multiple a’s could map to the same b

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Function Example u v w Unique, one image for each element of A and no more Defined for each element of A, so a function, not partial Not one-to-one since 2 elements of A map to v

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**Surjective (called also Onto) relations**

Range of the relation is B At least one a is related to each b Does not imply single-valued one-to-one R B Not mapped A a 1234 s1s2s3

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**Injective, or One-to-One relations**

“A relation between 2 sets such that pairs can be removed, one member from each set, until both sets have been simultaneously exhausted.”

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**Injective, One-to-One a could map to b’ also if it were not at**

least a partial function which implies single-valued a a’ = R b 12 abc

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Bijective A function which is both Injective and Surjective is Bijective. Also called “one-to-one” and “onto” A bijective function has an inverse, R-1, and it is unique

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**Function Examples Monotonically increasing if injective**

Not one-to-one, but single-valued A B B A b a a’

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**Function Examples Multivalued, but one-to-one b**

There are no two a’s which would have the same b, so it is one-to-one

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