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Sequential Machine Theory. Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 1 http://cpe.gmu.edu/~khintz. Adaptation to this class and additional comments by Marek Perkowski. Why Sequential Machine Theory (SMT)?. Sequential Machine Theory – SMT

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Consecutive Machine Theory Prof. K. J. Hintz Department of Electrical and Computer Engineering Lecture 1 http://cpe.gmu.edu/~khintz Adaptation to this class and extra remarks by Marek Perkowski

Why Sequential Machine Theory (SMT)? Consecutive Machine Theory – SMT Some Things Cannot be Parallelized Theory Leads to New Ways of Doing Things, has likewise commonsense applications in programming and equipment (compiler outline, controllers plan, and so on.) Understand Fundamental FSM Limits Minimize FSM Complexity and Size Find the " Essence " of a Machine, what does it imply that there is a machine for certain undertaking?

Why Sequential Machine Theory? Examine FSM properties that are unhampered by Implementation Issues: Software Hardware FPGA/ASIC/Memory, and so on. Innovation is Changing Rapidly, the center of the hypothesis remains perpetually . Hypothesis is a Framework inside which to Understand and Integrate Practical Considerations

Hardware/Software There Is an Equivalence Relation Between Hardware and Software Anything that should be possible in one should be possible in alternate… maybe speedier/slower System outline now done in equipment portrayal dialects (VHDL, Verilog, higher) without respect for acknowledgment technique Hardware/programming/split choice conceded until later stage in configuration

Hardware/Software Hardware/Software comparability reaches out to formal dialects Different classes of computational machines are identified with various classes of formal dialects Finite State Machines (FSM) can be identically spoken to by one class of dialects

Formal Languages Unambiguous Can Be Finite or Infinite Give some straightforward illustrations Can Be Rule-based or Enumerated Various Classes With Different Properties

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 type of state machine. FSM Costs Hardware: More States More Hardware Time: More States, Slower Operation Technology subordinate : what number of CPLD chips?

Goal of this arrangement of addresses Develop comprehension 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 diminishment is interesting

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

Combinational Logic Feedforward Output Is Only a Function of Input No Feedback No memory No transient reliance Two-Valued Function Minimization Techniques Well-known Minimization Techniques Multi-esteemed Function Minimization Well-known Heuristics

Finite State Machine Feedback Behavior Depends Both on Present State and Present Input State Minimization Well-known With Guaranteed Minimum Realization Minimization Unsolved issue of Digital Design Technology related, combinational outline related

Computer Design: Turing Machines Defined by Turing Computability Can figure anything that is "processable" Some things are not calculable Assumed Infinite Memory State Dependent Behavior Elements: Control Unit is determined and actualized as FSM Tape boundless Head developments Show case of an extremely straightforward Turing machine now: x- - > x+1

Intelligent Machines Some machines show a capacity to figure out How a machine can learn? A few issues are perhaps not processable What issues? Why not processable? Something must be interminable ?

Automata, otherwise known as FSM Concepts of Machines: Mechanical Counters, adders Computer programs Political Towns, interstates, social gatherings, parties, and so forth Biological Tissues, cells, hereditary, neural, social orders Abstract numerical Functions, relations, diagrams You ought to have the capacity to utilize FSM ideas in different zones like apply autonomy

FSM - Abstract scientific idea of numerous sorts of conduct Discrete Continuous framework can be discretized to any level of determination Finite State: limited letter sets for information sources, yields and states. Input/Output Some cause , some outcome

Set Theoretic Formulation of Finite State Machine S: Finite set of conceivable states I: Finite set of conceivable information sources O: Finite set of conceivable yields : Rule characterizing state change : Rule deciding yields

Types of FSMs Moore FSM Output is an element of state just Mealy FSM Output is a component of both the present state and the present info Discuss timing contrasts, show cases and charts, talk about quick flagging and PLD acknowledgment

Types of FSMs Finite State Acceptors , Language Recognizers Start in a solitary, indicated state End specifically state(s) Pushdown Automata Not a FSM Assumed unbounded stack with access just to highest component

Computer Turing Machine Assumed vast read/compose tape FSM controls read/compose/tape movement Definition of processable capacity Universal Turing Machine peruses FSM conduct from tape

Review of Set Theory Element: " a ", a solitary article with no exceptional property Set: " A ", a gathering of components, i.e. , Enumerated Set: Finite Set :

Sets Infinite Set of sets

Subsets All components of B are components of An and there might be one or more components of A that is not a component of B A 3 Larry, Curly, Moe A 6 whole numbers A 7

Proper Subset All components of B are components of An and there is no less than one component of A that is not a component of B

Set Equality Set An is equivalent to set B

Sets Null Set A set without any components , Every set is a subset of itself Every set contains the invalid set

Operations on Sets Intersection Union Logical AND Logical OR

Operations on Sets Set Difference Cartesian Product , Direct Product

Special Sets Powerset : set of all subsets of A * no supports around the invalid set since the image speaks to the set

Special Sets Disjoint sets: An and B are disjoint if Cover: We know set covering issue from 572. It was characterized as a grid issue

Properties of Operations on Sets Commutative , Abelian Associative Distributive Left hand distributive

Partition of a Set Properties p i are called "pi-squares" or " -pieces " of PI

Relations Between Sets If An and B are sets, then the connection from A to B , is a subset of the Cartesian result of An and B , i.e. , R - related :

Domain of a Relation Domain of R B an A b

Range of a Relation Range of R b An a B

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

Partial Function , Mapping A solitary esteemed connection to such an extent that R a b " a " * A B * can be numerous to one

Partial Function Also called the Image of an under R Only one component of B for every component of A Single-esteemed Can be a numerous to-one mapping

Function A halfway capacity with A b compares to each a , however one and only b for each a Possibly numerous to-one : various a \'s could guide to the same b

Function Example 1 2 3 4 u v w Unique , one picture for every component of An and not any more Defined for every component of A , so a capacity , not incomplete Not coordinated since 2 components of A guide to v

Surjective (called additionally Onto) relations Range of the connection is B At slightest one an is identified with every b Does not suggest single-esteemed balanced R B Not mapped An a 1 234 s1s2s3

Injective, or One-to-One relations "A connection between 2 sets to such an extent that sets can be expelled , one part from every set , until both sets have been all the while depleted ."

Bijective A capacity which is both Injective and Surjective is Bijective . Additionally called "coordinated" and "onto" A bijective capacity has a backwards , R - 1, and it is novel

B A B an a " A Function Examples Monotonically expanding if injective Not balanced, however single-esteemed

b " b "" an A Function Examples Multivalued, yet balanced There are no two a\'s which would have the same b, so it is balanced