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Finite State Machines. Chapter 5. Languages and Machines. Regular Languages. Regular Language. L. Regular Expression. Accepts. Finite State Machine. Finite State Machines. An FSM to accept $.50 in change:. Definition of a DFSM. M = ( K , , , s , A ), where:

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## Finite State Machines

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**Finite State Machines**Chapter 5**Regular Languages**Regular Language L Regular Expression Accepts Finite State Machine**Finite State Machines**An FSM to accept $.50 in change:**Definition of a DFSM**M = (K, , , s, A), where: K is a finite set of states is an alphabet sK is the initial state AK is the set of accepting states, and is the transition function from (K) to K**Accepting by a DFSM**Informally, M accepts a string w iff M ends up in some element of A (an accept state) when it has finished reading w. The language accepted by M, denoted L(M), is the set of all strings accepted by M.**Configurations of DFSMs**A configuration of a DFSM M is an element of: K* It captures the two things that can make a difference to M’s future behavior: • its current state • the input that is still left to read. The initial configuration of a DFSM M, on input w, is: (sM, w)**The Yields Relations**The yields-in-one-step relation |-M: (q, w) |-M (q', w') iff • w = aw' for some symbol a, and • (q, a) = q' |-M * is the reflexive, transitive closure of |-M.**Computations Using FSMs**A computation by M is a finite sequence of configurations C0, C1, …, Cn for some n 0 such that: • C0 is an initial configuration, • Cn is of the form (q, ), for some state qKM, • C0 |-MC1 |-MC2 |-M … |-MCn.**Accepting and Rejecting**A DFSM Maccepts a string w iff: (s, w) |-M * (q, ), for some qA. A DFSM Mrejects a string w iff: (s, w) |-M* (q, ), for some qAM. The language accepted byM, denoted L(M), is the set of all strings accepted by M. Theorem: Every DFSM M, on input s, halts in |s| steps.**An Example Computation**An FSM to accept odd integers: even odd even q0q1 odd On input 235, the configurations are: (q0, 235) |-M (q0, 35) |-M |-M Thus (q0, 235) |-M* (q1, )**Regular Languages**A language is regular iff it is accepted by some FSM.**A Very Simple Example**L = {w {a, b}* : every a is immediately followed by a b}.**A Very Simple Example**L = {w {a, b}* : every a is immediately followed by a b}.**Parity Checking**L = {w {0, 1}* : w has odd parity}.**Parity Checking**L = {w {0, 1}* : w has odd parity}.**No More Than One b**L = {w {a, b}* : every a region in w is of even length}.**No More Than One b**L = {w {a, b}* : w has no more than 1 b}.**Checking Consecutive Characters**L = {w {a, b}* : no two consecutive characters are the same}.**Checking Consecutive Characters**L = {w {a, b}* : no two consecutive characters are the same}. Error in Book, p.61, d should not be an accept state.**Dead States**L = {w {a, b}* : every a region in w is of even length}**Dead States**L = {w {a, b}* : every a region in w is of even length}**Dead States**L = {w {a, b}* : every b in w is surrounded by a’s}**The Language of Floating Point Numbers is Regular**Example strings: +3.0, 3.0, 0.3E1, 0.3E+1, -0.3E+1, -3E8 The language is accepted by the DFSM:**Programming FSMs**Cluster strings that share a “future”. Let L = {w {a, b}* : w contains an even number of a’s and an odd number of b’s}**Vowels in Alphabetical Order**L = {w {a - z}* : all five vowels, a, e, i, o, and u, occur in w in alphabetical order}.**Vowels in Alphabetical Order**L = {w {a - z}* : all five vowels, a, e, i, o, and u, occur in w in alphabetical order}.**Programming FSMs**L = {w {a, b}* : w does not contain the substring aab}.**Programming FSMs**L = {w {a, b}* : w does not contain the substring aab}. Start with a machine for L: How must it be changed?**A Building Security System**L = {event sequences such that the alarm should sound}**Finite State Representations in Software Engineering**A high-level state chart model of a digital watch.**The Missing Letter Language**Let = {a, b, c, d}. Let LMissing= {w : there is a symbol ai not appearing in w}. Try to make a DFSM for LMissing:**Definition of an NDFSM**• M = (K, , , s, A), where: • K is a finite set of states • is an alphabet • sK is the initial state • AK is the set of accepting states, and is the transition relation. It is a finite subset of • (K ( {})) K**Accepting by an NDFSM**M accepts a string w iff there exists some path along which w drives M to some element of A. The language accepted by M, denoted L(M), is the set of all strings accepted by M.**Analyzing Nondeterministic FSMs**Two approaches: • Explore a search tree: • Follow all paths in parallel**Optional Substrings**L = {w {a, b}* : w is made up of an optional a followed by aa followed by zero or more b’s}.**Multiple Sublanguages**L = {w {a, b}* : w = aba or |w| is even}.**The Missing Letter Language**Let = {a, b, c, d}. Let LMissing= {w : there is a symbol ai not appearing in w}**Pattern Matching**L = {w {a, b, c}* : x, y {a, b, c}* (w = xabcabby)}. A DFSM:**Pattern Matching**L = {w {a, b, c}* : x, y {a, b, c}* (w = xabcabby)}. An NDFSM:**Pattern Matching with NDFSMs**L = {w {a, b}* : x, y {a, b}* : w = xaabbby or w = xabbaby }**Multiple Keywords**L = {w {a, b}* : x, y {a, b}* ((w = xabbaay) (w = xbabay))}.**Checking from the End**L = {w {a, b}* : the fourth to the last character is a}**Checking from the End**L = {w {a, b}* : the fourth to the last character is a}**Another Pattern Matching Example**L = {w {0, 1}* : w is the binary encoding of a positive integer that is divisible by 16 or is odd}

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