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Deterministic Finite State Machines
Chapter 5
Languages and Machines
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Regular Languages
Regular Language
Regular Expression
Finite State Machine
L
Accepts
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Finite State Machines
Vending Machine: An FSM to accept $.50 in change
Drink is 25 centsNo more than 50 cents can be deposited
X
Other FSM examples?4
Definition of a DFSM
M = (K, , , s, A), where:
K is a finite set of states is an alphabet s K is the initial state A K is the set of accepting states, and is the transition function from
(K ) to Kstate input symbol state
Quintuple
Cartesian product
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Accepting by a DFSM
Informally, M accepts a string w iff M winds up in some element of A when it has finished reading w.
The language accepted by M, denoted L(M), is the set of all strings accepted by M.
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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), where sM is the start state of M 7
The Yields Relations
The yields-in-one-step relation |-M :
(q, w) |-M (q', w') iff
• w = a w' for some symbol a , and • (q, a) = q'
The relation yields, |-M*, is the reflexive, transitive closure of |-M.
C1 |-M* C2 : M can go from C1 to C2 in 0 or more steps.
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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 q KM,
• C0 |-M C1 |-M C2 |-M … |-M Cn.
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Accepting and Rejecting
A DFSM M accepts a string w iff:
(s, w) |-M * (q, ), for some q A.
A DFSM M rejects a string w iff:
(s, w) |-M* (q, ), for some q AM.
The language accepted by M, denoted L(M), is the set of all strings accepted by M.
Theorem: Every DFSM M, on input s, halts in |s| steps.
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An Example Computation
An FSM to accept odd integers: even odd
even
q0 q1
odd
On input 235, the configurations are:
(q0, 235) |-M (q0, 35)|-M
|-M
Thus (q0, 235) |-M* (q1, ) 11
Regular Languages
A language is regular iff it is accepted by some FSM.
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A Very Simple Example
L = {w {a, b}* : every a is immediately followed by a b}.
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A Very Simple Example
L = {w {a, b}* : every a is immediately followed by a b}.
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A Very Simple Example
L = {w {a, b}* : every a is immediately followed by a b}.
M = (K, , , s, A) = ({q0, q1, q2}, {a, b}, , q0, {q0}), where
= {((q0, a), q1), ((q0, b), q0),
((q1, a), q2), ((q1, b), q0),
((q2, a), q2), ((q2, b), q2)}15
Parity Checking
L = {w {0, 1}* : w has odd parity}.
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Parity Checking
L = {w {0, 1}* : w has odd parity}.
M = (K, , , s, A) = ({q0, q1}, {0, 1}, , q0, {q1}), where
= {((q0, 0), q0), ((q0, 1), q1),
((q1, 0), q1), ((q1, 1), q0)}
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No More Than One b
L = {w {a, b}* : w contains no more than one b}.
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No More Than One b
L = {w {a, b}* : w contains no more than one b}.
M = (K, , , s, A)
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Checking Consecutive Characters
L = {w {a, b}* :
no two consecutive characters are the same}.
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Checking Consecutive Characters
L = {w {a, b}* :
no two consecutive characters are the same}.
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Dead States
L = {w {a, b}* : every a region in w is of even length}
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Dead States
L = {w {a, b}* : every a region in w is of even length}
M = (K, , , s, A)
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Dead States
L = {w {a, b}* : every b in w is surrounded by a’s}
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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:
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A Simple Communication Protocol
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Controlling a Soccer-Playing Robot
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A Simple Controller
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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}
What states are needed? How many states are there?
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Even a’s Odd b’s
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Vowels in Alphabetical Order
L = {w {a - z}* : all five vowels, a, e, i, o, and u,
occur in w in alphabetical order}.
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Vowels in Alphabetical Order
L = {w {a - z}* : all five vowels, a, e, i, o, and u,
occur in w in alphabetical order}.
O
u
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Programming FSMs
L = {w {a, b}* : w does not contain the substring aab}.
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Programming FSMs
L = {w {a, b}* : w does not contain the substring aab}.
Start with a machine for L:
How must it be changed?
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Programming FSMs
L = {w {a, b}* : w does not contain the substring aab}.
Start with a machine for L:
How must it be changed?
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A Building Security System
L = {event sequences such that the alarm should sound}
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FSMs Predate Computers
The Prague Orloj, originally built in 1410
The Chinese Abacus,14 century AD
The Jacquard Loom, invented in 1801
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Finite State Representations in Software Engineering
A high-level state chart model of a digital watch.
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Making the Model Hierarchical
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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:
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