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

Example Jurafsky & Martin, Figure 2.10 Loop on q 3 means that it can account for infinite length strings “Deterministic” because in any state, its behaviour is fully predictable q0q0 q1q1 q2q2 q3q3 q4q4 b aa! a

Non-deterministic FSA Jurafsky & Martin, Figure 2.18 At state q 2 with input “a” there is a choice of transitions We can also have “jump” arcs (or empty transitions), which also introduce non- determinism q0q0 q1q1 q2q2 q3q3 q4q4 b aa! a 2.19 ε

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. It’s 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

Augmented Transition Networks Sq1q1 NPq1q1 ex push NP put “num” det put “num” push VP get “num” n put “num” adj q2q2 ε pop NPprep

Exercises q0q0 q1q1 q2q2 q3q3 q4q4 b aa! a fsa([[0,b,1],[1,a,2],[2,a,3],[3,a,3],[3,!,end]]). [0,b,1] [1,a,2] [2,a,3] [3,a,3] [3,!,end]

NDSFA q0q0 q1q1 q2q2 q3q3 q4q4 b aa! ε fsa([[0,b,1],[1,a,2],[2,a,3],[3,empty,2],[3,!,end]]). [0,b,1] [1,a,2] [2,a,3] [3,!,end] [3,empty,2]

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

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 !

| ?- gen(X). FSA and NDFSA programs 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: 2. Choice at state 3. Choose state from (1) [!,end] (2) [empty,2] Select choice number: 1. X = [b,a,a,a,a,!] ? yes

| ?- rec(X,quiet). X = [b,a,a] ? FSA and NDFSA programs ; X = [b,a,a,a] ? ; X = [b,a,a,a,a] ? ; X = [b,a,a,a,a,a] ? yes

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

Regular expressions CharacterMeaningExamples [ ]alternatives/[aeiou]/, /m[ae]n/ ­ range /[a-z]/ [^ ]not/[^pbm]/, /[^ox]s/ ?optionality/Kath?mandu/ *zero or more/baa*!/ +one or more/ba+!/.any character /cat.[aeiou]/ ^, $start, end of line \not special character \.\?\^ |alternate strings/cat|dog/ ( )substring/cit(y|ies)/ etc.

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

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

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

Some conventions Transitions are marked by “:” A non-changing transition “x:x” can be shown simply as “x” Wild-cards are shown as Empty string shown as “ε”

An example J&M Fig. 3.9, p.74 q0q0 q6q6 q5q5 q4q4 q3q3 q2q2 q1q1 q7q7 f o x c a t d o g g o o s e s h e e p m o u s e g o:e o:e s e s h e e p m o:i u:εs:c e N:ε P:^ s # S:# P:# lexical:intermediate

q0q0 q6q6 q5q5 q4q4 q3q3 q2q2 q1q1 q7q7 g o o s e s h e e p m o u s e g o:e o:e s e s h e e p m o:i u:εs:c e N:ε P:^ s # S:# P:# [0] f:f o:o x:x [1] N:ε [4] P:^ s:s #:# [7] [0] f:f o:o x:x [1] N:ε [4] S:# [7] [0] c:c a:a t:t [1] N:ε [4] P:^ s:s #:# [7] [0] s:s h:h e:e p:p [2] N:ε [5] S:# [7] [0] g:g o:o o:o s:s e:e [2] N:ε [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

Lexical:surface mapping J&M Fig. 3.14, p.78 ε  e / {x s z} ^ __ s # f o x N P s # : f o x ^ s # c a t N P s # : c a t ^ s # q5q5 q4q4 q0q0 q2q2 q3q3 q1q1 ^: ε # other z, s, x #, otherz, x ^: ε s ε:e s #

f o x ^ s # f o x e s # c a t ^ s # : c a t ^ s # q5q5 q4q4 q0q0 q2q2 q3q3 q1q1 ^: ε # other z, s, x #, otherz, x ^: ε s ε:e s # [0] f:f [0] o:o [0] x:x [1] ^:ε [2] ε:e [3] s:s [4] #:# [0] [0] c:c [0] a:a [0] t:t [0] ^:ε [0] s:s [0] #:# [0]

FST Can be generated automatically Therefore, slightly different formalism

FST compiler [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 -> s1, d -> s2, f -> s3, g -> s4. s1: a -> s5. s2: o -> s6. s3: o -> s7. s4: -> s8. s5: t -> s9. s6: g -> s9. s7: x -> s10. s8: -> s11. s9: -> s12. s10: -> s13. s11: s -> s14. s12: -> fs15. s13: -> fs15. s14: e -> s16. fs15: (no arcs) s16: -> s12. s0s0 s3s3 s2s2 s1s1 s4s4 c d f g

s0: c -> s1, d -> s2, f -> s3, g -> s4. s1: a -> s5. s2: o -> s6. s3: o -> s7. s4: -> s8. s5: t -> s9. s6: g -> s9. s7: x -> s10. s8: -> s11. s9: -> s12. s10: -> s13. s11: s -> s14. s12: -> fs15. s13: -> fs15. s14: e -> s16. fs15: (no arcs) s16: -> s12. fst([ [s0,[c,s1], [d,s2], [f,s3], [g,s4]], [s1,[a,s5]], [s2,[o,s6]], [s3,[o,s7]], [s4,[[o,e],s8]], [s5,[t,s9]], [s6,[g,s9]], [s7,[x,s10]], [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]] ]).

FST 3.9 s0s0 q6q6 q5q5 q4q4 q3q3 q2q2 q1q1 q7q7 g o o s e s h e e p m o u s e g o:e o:e s e s h e e p m o:i u:εs:c e N:ε PL:^ s # SG:# PL:# f o x c a t d o g

s0s0 q1q1 f o x c a t d o g FST 3.9 (portion) [s0,[f,s1], [c,s3], [d,s5]], [s1,[o,s2]], [s2,[x,q1]], [s3,[a,s4]], [s4,[t,q1]], [s5,[o,s6]], [s6,[g,q1]], s0s0 q1q1 f s1s1 s2s2 s3s3 s4s4 s5s5 s6s6 c d o a o x t g