1 FSA’s –Defining FSA’s –Computing with FSA’s Defining L(M) –Defining language class...
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1 • FSA’s – Defining FSA’s – Computing with FSA’s • Defining L(M) – Defining language class LFSA – Comparing LFSA to set of solvable languages (REC)
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Transcript of 1 FSA’s –Defining FSA’s –Computing with FSA’s Defining L(M) –Defining language class...
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• FSA’s– Defining FSA’s– Computing with FSA’s
• Defining L(M)
– Defining language class LFSA– Comparing LFSA to set of solvable languages
(REC)
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Finite State Automata
New Computational Model
3
Tape
• We assume that you have already seen FSA’s in CSE 260– If not, review material in reference textbook
• Only data structure is a tape– Input appears on tape followed by a B character
marking the end of the input– Tape is scanned by a tape head that starts at
leftmost cell and always scans to the right
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Data type/States
• The only data type for an FSA is char
• The instructions in an FSA are referred to as states
• Each instruction can be thought of as a switch statement with several cases based on the char being scanned by the tape head
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Example program
1 switch(current tape cell) {case a: goto 2case b: goto 2case B: return yes
}2 switch (current tape cell) {
case a: goto 1case b: goto 1case B: return no;
}
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New model of computation
• FSA M=(Q,,q0,A,)– Q = set of states = {1,2}
– = character set = {a,b}• don’t need B as we see below
– q0 = initial state = 1
– A = set of accepting (final) states = {1}
• A is the set of states where we return yes on B
• Q-A is set of states that return no on B
– = state transition function
1 switch(current tape cell) {case a: goto 2
case b: goto 2
case B: return yes
}
2 switch (current tape cell) {case a: goto 1
case b: goto 1
case B: return no;
}
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Textual representations of *
1 switch(current tape cell) {case a: goto 2
case b: goto 2
case B: return yes
}
2 switch (current tape cell) {case a: goto 1
case b: goto 1
case B: return no;
}
1
2
a b
1 1
2 2
(1,a) = 2, (1,b)=2, (2,a)=1, (2,b) = 1
{(1,a,2), (1,b,2), (2,a,1), (2,b,1)}
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Transition diagrams
1 2
a,b
a,b1 switch(current tape cell) {case a: goto 2
case b: goto 2
case B: return yes
}
2 switch (current tape cell) {case a: goto 1
case b: goto 1
case B: return no;
}
Note, this transition diagramrepresents all 5 components of an FSA, not just the transition function
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Exercise
• FSA M = (Q, , q0, A, )
– Q = {1, 2, 3}– = {a, b}
– q0 = 1
– A = {2,3}– : {(1,a) = 1, (1,b) = 2, (2,a)= 2, (2,b) = 3,
(3,a) = 3, (3,b) = 1}
• Draw this FSA as a transition diagram
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Transition Diagram
1
2
3
a
a
a
b
b
b
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Computing with FSA’s
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Computation Example *
1
2
3
a
a
a
b
b
b
Input: aabbaa
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Computation of FSA’s in detail
• A computation of an FSA M on an input x is a complete sequence of configurations
• We need to define– Initial configuration of the computation– How to determine the next configuration given
the current configuration– Halting or final configurations of the
computation
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• Given an FSA M and an input string x, what is the initial configuration of the computation of M on x?– (q0,x)
– Examples• x = aabbaa
• (1, aabbaa)
• x = abab
• (1, abab)
• x = • (1, )
Initial Configuration
1
2
3
a
a
a
b
b
b
FSA M
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• (1, aabbaa) ├M (1, abbaa)– config 1 “yields” config 2 in
one step using FSA M• (1,aabbaa) ├ M (2, baa)
– config 1 “yields” config 2 in 3 steps using FSA M
• (1, aabbaa) ├ M (2, baa)– config 1 “yields” config 2 in 0
or more steps using FSA M• Comment:
– ├ M determined by transition function
– There must always be one and only one next configuration
• If not, M is not an FSA
Definition of ├M
1
2
3
a
a
a
b
b
b
3
FSA M
*
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• Halting configuration– (q, )
– Examples• (1, )
• (3, )
• Accepting Configuration– State in halting configuration
is in A
• Rejecting Configuration– State in halting configuration
is not in A
Halting Configurations *
1
2
3
a
a
a
b
b
b
FSA M
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• Two possibilities for M running on x– M accepts x
• M accepts x iff the computation of M on x ends up in an accepting configuration
• (q0, x) ├M (q, ) where q is in A
– M rejects x• M rejects x iff the computation of M on x ends up in a
rejecting configuration• (q0, x) ├M (q, ) where q is not in A
– M does not loop or crash on x• Why?
FSA M on xb
bb
aa
a
FSA M
*
*
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– For the following input strings, does M accept or reject?• • aa• aabba• aab• babbb
Examplesb
bb
aa
a
FSA M
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• Notation from the book
• (q, c) = p
• k(q, x) = p– k is the length of x
• *(q, x) = p
• Examples– (1, a) = 1
– (1, b) = 2
– 4(1, abbb) = 1
– *(1, abbb) = 1
– (2, baaaaa) = 3
Definition of *(q, x)
1
2
3
a
a
a
b
b
b
FSA M
