Finite State Automata

1
Finite State Automata
2
Finite State Automata

• A very simple and intuitive formalism suitable
  for certain tasks
• A bit like a flow chart, but can be used for both
  recognition and generation
• Transition network
• Unique start point
• Series of states linked by transitions
• Transitions represent input to be accounted for,
  or output to be generated
• Legal exit-point(s) explicitly identified

3
ExampleJurafsky Martin, Figure 2.10

• Loop on q3 means that it can account for infinite
  length strings
• Deterministic because in any state, its
  behaviour is fully predictable

4
Non-deterministic FSAJurafsky Martin, Figure
2.18

• At state q2 with input a there is a choice of
  transitions
• We can also have jump arcs (or empty
  transitions), which also introduce non-determinism

5
Augmented Transition Networks

• ATNs were used for parsing in the 60s and 70s
• For parsing, you need to pass constraints (e.g.
  for agreement) as well as account for input the
  Transition Networks were augmented by having a
  register into/from which such information could
  be put/taken.
• Its easy to write recognizers, but computing
  structure is difficult
• ATNs quickly become very complex one solution
  isto have a cascade of ATNs, where transitions
  can call other networks

6
Augmented Transition Networks
push NP put num
push VP get num
adj
det put num
n put num
pop NP
prep
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Exercises
0,b,1 1,a,2 2,a,3 3,a,3
3,!,end
fsa(0,b,1,1,a,2,2,a,3,3,a,3,3,!,end).
8
NDSFA
0,b,1 1,a,2 2,a,3 3,!,end
3,empty,2
fsa(0,b,1,1,a,2,2,a,3,3,empty,2,3,!,end
).
9
FSA and NDFSA programs
First load (consult) the file, eg 219.pl ?-
help. Options are as follows run - a simple
recognizer on prompt type in string with
space between each element, ending in . or ! or
? run(v) - verbose recognizer gives trace of
transitions gen(X) - generate text will interact
at choice points rec(X,quiet) - to generate text
deterministically. Type to get other
grammatical sequences
?- run.
b a a a a !
Enter your string
yes
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FSA and NDFSA programs
?- run(v).
Enter your string
0-b-1 1-a-2 2-a-3 3-skip-2 2-
a-3 3-skip-2 2-a-3 3-skip-2 3-!-end yes
b a a a a !
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FSA and NDFSA programs
?- gen(X).

• Choice at state 3. Choose state from
• !,end
• (2) empty,2
• Select choice number

2. Choice at state 3.
Choose state from (1) !,end (2)
empty,2 Select choice number
2. Choice at state 3.
Choose state from (1) !,end (2)
empty,2 Select choice number
1. X b,a,a,a,a,! ?
yes
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FSA and NDFSA programs
?- rec(X,quiet). X b,a,a ?
X b,a,a,a ?
X b,a,a,a,a ?
X b,a,a,a,a,a ?
yes
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FSAs and regular expressions

• FSAs have a close relationship with regular
  expressions, a formalism for expressing strings,
  mainly used for searching texts, or stipulating
  patterns of strings
• Regular expressions are defined by combinations
  of literal characters and special operators

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Regular expressions
Character Meaning Examples
alternatives /aeiou/, /maen/ range
/a-z/ not /pbm/,
/oxs/ ? optionality /Kath?mandu/ zero or
more /baa!/ one or more /ba!/ . any
character /cat.aeiou/ , start, end of
line \ not special character
\.\?\ alternate strings /catdog/ (
) substring /cit(yies)/ etc.
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Regular expressions

• A regular expression can be mapped onto an FSA
• Can be a good way of handling morphology
• Especially in connection with Finite State
  Transducers

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Finite State Transducers

• A transducer defines a relationship (a mapping)
  between two things
• Typically used for two-level morphology, but
  can be used for other things
• Like an FSA, but each state transition stipulates
  a pair of symbols, and thus a mapping

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Finite State Transducers

• Three functions
• Recognizer (verification) takes a pair of
  strings and verifies if the FST is able to map
  them onto each other
• Generator (synthesis) can generate a legal pair
  of strings
• Translator (transduction) given one string, can
  generate the corresponding string

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Some conventions

• Transitions are marked by
• A non-changing transition xx can be shown
  simply as x
• Wild-cards are shown as _at_
• Empty string shown as e

19
An exampleJM Fig. 3.9, p.74
f o x c a t d o g
P s
Ne
q4
q1
g o o s e s h e e p m o u s e
S
Ne
q0
q5
q2
q7
S
g oe oe s e s h e e p m oi uesc e
Ne
P
q6
q3
lexicalintermediate
20

• 0 ff oo xx 1 Ne 4 P ss 7
• 0 ff oo xx 1 Ne 4 S 7
• 0 cc aa tt 1 Ne 4 P ss 7
• 0 ss hh ee pp 2 Ne 5 S 7
• 0 gg oo oo ss ee 2 Ne 5 P 7

f o x N P s f o x s f o x N S f o x c
a t N P s c a t s s h e e p N S s h e e
p g o o s e N P g e e s e
f o x c a t d o g
P s
Ne
q4
q1
g o o s e s h e e p m o u s e
S
Ne
q0
q5
q2
q7
S
g oe oe s e s h e e p m oi uesc e
Ne
P
q6
q3
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Lexicalsurface mappingJM Fig. 3.14, p.78
f o x N P s f o x s c a t N P s c a t
s
e ? e / x s z __ s
22
0 ff 0 oo 0 xx 1 e 2 ee 3 ss
4 0 0 cc 0 aa 0 tt 0 e 0
ss 0 0
f o x s f o x e s c a t s c a t s
other
q5
e other
z, s, x
s
e
z, s, x
e
ee
s
q0
q1
q4
q2
q3
, other
z, x

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FST

• Can be generated automatically
• Therefore, slightly different formalism

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FST compiler http//www.xrce.xerox.com/competencie
s/content-analysis/fsCompiler/fsinput.html d o g
N P .x. d o g s c a t N P .x. c a t s
f o x N P .x. f o x e s g o o s e N P .x.
g e e s e  
s0 c -gt s1, d -gt s2, f -gt s3, g -gt s4. s1 a
-gt s5. s2 o -gt s6. s3 o -gt s7. s4 ltoegt
-gt s8. s5 t -gt s9. s6 g -gt s9. s7 x -gt
s10. s8 ltoegt -gt s11. s9 ltNsgt -gt s12. s10
ltNegt -gt s13. s11 s -gt s14. s12 ltP0gt -gt
fs15. s13 ltPsgt -gt fs15. s14 e -gt s16. fs15
(no arcs) s16 ltN0gt -gt s12.
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s0 c -gt s1, d -gt s2, f -gt s3, g -gt s4. s1 a
-gt s5. s2 o -gt s6. s3 o -gt s7. s4 ltoegt
-gt s8. s5 t -gt s9. s6 g -gt s9. s7 x -gt
s10. s8 ltoegt -gt s11. s9 ltNsgt -gt s12. s10
ltNegt -gt s13. s11 s -gt s14. s12 ltP0gt -gt
fs15. s13 ltPsgt -gt fs15. s14 e -gt s16. fs15
(no arcs) s16 ltN0gt -gt s12.
fst( s0,c,s1, d,s2, f,s3,
g,s4, s1,a,s5, s2,o,s6, s3,o,s7, s
4,o,e,s8, s5,t,s9, s6,g,s9, s7,x,s1
0, s8,o,e,s11, s9,'N',s,s12, s10,
'N',e,s13, s11,s,s14, s12,'P',0,fs15,
s13,'P',s,fs15, s14,e,s16, fs15,
noarcs, s16,'N',0,s12 ).
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FST 3.9
f o x c a t d o g
PL s
Ne
q4
q1
g o o s e s h e e p m o u s e
SG
Ne
s0
q5
q2
q7
SG
g oe oe s e s h e e p m oi uesc e
Ne
PL
q6
q3
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FST 3.9 (portion)
f o x c a t d o g
s0,f,s1, c,s3, d,s5, s1,o,s2, s2,x,q
1, s3,a,s4, s4,t,q1, s5,o,s6, s6,g
,q1,
q1
s0
o
s1
s2
f
x
a
c
t
s0
q1
s3
s4
d
g
o
s5
s6