FS Properties and FSTs

Transcription

1 FS Properties and FSTs Chris Dyer Algorithms for NLP

2 Announcements HW1 has been posted; due in class in 2 weeks

3 Goals of Today s Lecture Understand properties of regular languages Understand Brzozowski derivatives and how to use them to prove languages are regular/not-regular. Understand relations between formal languages and the definition of finite state transducers Understand the operations on finite state transducers (FSTs): union, concatenation, Kleene*, composition, and (sometimes) intersection

4 Properties of FSAs Any NFA-e can be made into a NFA Any NFA can be made into a DFA Thus, all (DFAs NFAs NFA-e s) define the same set. RE s can be converted into NFA-e s (and thus to DFAs). Can we convert any DFA into an RE?

5 DFAs into REs Yes! Algorithm due to Kleene (1956) Given a DFA M = hq,,,q 0,Fi with states numbered 0, 1, 2,, n We will incrementally construct which gives the expression for the language containing all strings for each pair of states (i,j) without going through a state higher than k. [ R = j2f R n 0j R k ij

6 RE construction Base: R 1 ii =( m ") () (q i, 1) =...= (q i, m) = (q i, ") =q i R 1 ij =( m) Recursion: () (q i, 1) =...= (q i, m) =q j Rij k = R k 1 ik (Rk 1 kk )Rk 1 R k 1 kj ij

7 RE construction Base: R 1 ii =( m ") R 1 () (q i, 1) =...= (q i, m) = (q i, ") =q i =( ij m) Thus, REs, DFAs, NFAs, NFA-e s all represent () the (q i, full 1) set =...= of regular (q i, m) languages. =q j Recursion: Rij k = R k 1 ik (Rk 1 kk )Rk 1 R k 1 kj ij

8 Closure Properties Regular languages are closed under Intersection (cf. dual control proof from recitation) Finite union (how would you prove this?) Kleene* Concatenation Complementation (Contruct a DFA. Flip final states for nonfinal states) Difference A B = A \ B

9 Closure Properties Regular languages are closed under Intersection (cf. dual control proof from recitation) Finite union (how would you prove this?) Other definitions of complementation and Kleene* difference are possible. (Exam Q?) Concatenation Complementation (Contruct a DFA. Flip final states for nonfinal states) Difference A B = A \ B

10 Minimization For a regular language L there exists a unique DFA A (up to a renaming of states) that accepts L The proof is constructive and provides an algorithm for DFA minimization See the book for details. You will not be required to understand the details of the proof, but you should be aware that it exists. Intuition: lexicographic sort. DFA minimization is important for practical applications

11 Some other Qs Is a language empty? Determinize, minimize, compare to empty Are two regular languages equivalent? Determinize, minimize, compare Is a language finite? Determinize, minimize, look for loops Is a language regular?

12 Is this language regular? If finite, then yes. If infinite, then we must prove things. The classic intro solution is the Pumping Lemma. It is not a great bit of theory, IMHO: It is necessary, but not sufficient to prove nonregularity (i.e., a non regular language may be pumpable). It is needlessly complex. Many thanks to Adam Lopez (Edinburgh) for pointing this out to me:

13 Brzozowski Derivatives Recall differentiation from calculus: d dx x y z = y z d (xyz + xz) =(yz + z) dx

14 Brzozowski Derivatives Recall differentiation from calculus: d dx x y z = y z d (xyz + xz) =(yz + z) dx Brzozowski s idea is that concatenation is a kind of product, and union is a kind of sum, motivating the following definition: d dw L = {v 2 wv 2 L}

15 Brzozowski Derivatives Brzozowski s idea is that concatenation is a kind of product, and union is a kind of sum, motivating the following definition: d dw L = {v 2 wv 2 L}

16 Brzozowski Derivatives Brzozowski s idea is that concatenation is a kind of product, and union is a kind of sum, motivating the following definition: d dw L = {v 2 wv 2 L} Examples.

17 Brzozowski Derivatives Brzozowski s idea is that concatenation is a kind of product, and union is a kind of sum, motivating the following definition: d dw L = {v 2 wv 2 L} Examples. d {ab, aab, ba, b} = {b, ab} da d da {an : n 0} = {a n : n 0} d {a, ab} = {",b} da

18 Proving Regularity Theorem. A language is regular iff it has a finite number of Brzozowski derivatives.

19 Proving Regularity Theorem. A language is regular iff it has a finite number of Brzozowski derivatives. Lemma. Every derivative of a regular language is regular. Proof. Stripping off a prefix corresponds to changing the start state of a DFA.

20 Proving Regularity Theorem. A language is regular iff it has a finite number of Brzozowski derivatives. Lemma. Every derivative of a regular language is regular. Proof. Stripping off a prefix corresponds to changing the start state of a DFA. Proof of theorem. (=) ) By definition, DFAs have a finite number of, states, and by the lemma, derivatives ~ states, there must be a finite number of derivatives.

21 Proving Regularity Proof of theorem. ( (= ) By construction. Given a language L on alphabet. States of the DFA are languages. q 0 = L (q, )= d d (q) F = {q 2 Q " 2 q}

22 Proving Regularity Proof of theorem. ( (= ) By construction. Given a language L on alphabet. States of the DFA are languages. q 0 = L (q, )= d d (q) F = {q 2 Q " 2 q} Proof that construction is correct. By induction on length of w. (try at home).

23 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No.

24 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No.

25 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No.

26 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No. Is the following language regular? L = a i b j i 0 ^ j 0 d d(a n ) L = ai b j i 0 ^ j 0 8n 1 Yes.

27 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No. Is the following language regular? L = a i b j i 0 ^ j 0 d d(a n ) L = ai b j i 0 ^ j 0 8n 1 Yes.

28 Examples Is the following language regular? L = a i b i i 0 d d(a n ) L = ai b i+n i 0 8n 1 No. Is the following language regular? L = a i b j i 0 ^ j 0 d d(a n ) L = ai b j i 0 ^ j 0 8n 1 Yes.

29 Transducers In NLP we often want to transduce between different representations Tokenization, POS tagging, grapheme to phoneme conversion, morphological analysis, spelling correction, translation, The formal concept we will rely on is that of a relation (generalization of a function) We will refer to automata that define relations/perform transduction as transducers

30 a b ab aba aaa bbb...

31 a b ab aba aaa bbb

32 Relation a b ab aba aaa bbb In general, a relation is a many-to-many mapping.

33 Finite State Transducers a :0 b : " q 0 b :1 a :1 b :0 a :2

34 Finite State Transducers a :0 b : " q 0 b :1 a :1 b :0 a :2

35 Notational Equivalence a : a a

36 Finite State Transducers A finite state transducer is a 6-tuple T = hq,,,,q 0,Fi where Q is a finite set of states is the finite input alphabet : is Qthe finite! 2output Q is the a transition alphabet transition function relation : Q ( [ {"}) ( [ {"})! 2 Q is the transition relation q 0 2 Q F Q is the start (initial) state is the set of final (accept) states 24

37 Generalized Transitions As with FSAs, we can provide a generalized definition of the transition function. ˆ(q, ", ") =q ˆ(q, a, b) = (q, a, b) ˆ(q, xa, yb) =s () ˆ(q, x, y) =r ^ (r, a, b) =s Given x 2 and y 2 we say that T transduces x to y and we write x[t ]y iff there is a path from q 0 to some final state producing the string pair x, y, i.e. ˆ(q0, x, y) \ F 6= ; 25

38 Regular Relations R(T ) R(T )={x 2 : y 2 x[t ]y} where T is a finite state transducer. 26

39 Operations on FSTs Given FSTs T and S and w, x 2 and y, z 2, the following FSTs exist: 27

40 Operations on FSTs Given FSTs T and S and w, x 2 and y, z 2, the following FSTs exist: (union) x[t [ S]y () x[t ]y _ x[s]y 27

41 Operations on FSTs Given FSTs T and S and w, x 2 and y, z 2, the following FSTs exist: (union) (concatenation) x[t [ S]y () x[t ]y _ x[s]y wx[t.s]yz () w[t ]y ^ x[s]z 27

42 Operations on FSTs Given FSTs T and S and w, x 2 and y, z 2, the following FSTs exist: (union) (concatenation) x[t [ S]y () x[t ]y _ x[s]y wx[t.s]yz () w[t ]y ^ x[s]z (Kleene*) "[T ]" w[t ]y ^ x[t ]z =) wx[t ]yz 27

43 Operations on FSTs Given FSTs T and S and w, x 2 and y, z 2, the following FSTs exist: (union) (concatenation) x[t [ S]y () x[t ]y _ x[s]y wx[t.s]yz () w[t ]y ^ x[s]z (Kleene*) "[T ]" w[t ]y ^ x[t ]z =) wx[t ]yz Given FSTs T and S, with alphabets (, ) and (, ) the following FSTs exist: (composition) x[t S]y () 9z 2 s.t. x[t ]z ^ z[s]y 27

44 Other Operations In contrast to regular languages, regular relations are not closed under Intersection Complementation Difference 28

45 Intersection Theorem. Regular relations aren t closed under intersection. Proof. By counterexample. a : b " : c " : b a : c " : c a : c q 0 q 0 T S T [ S = {(a n,b n c n ):n 0} 29

46 Intersection However, certain restricted classes of FSTs are closed under intersection And FST intersection is extremely useful, as you will see in later lectures 30

47 Operations on FSTs FSTs can be inverted by swapping the input and output labels FSTs can be determinized such that each state has a single outgoing transition with a single label in the input language (outputs may not be deterministic); FST must be a functional relation. [construction similar to powerset construction] Deterministic FSTs can be minimized. FSTs can be projected to FSAs to yield the input or output languages 31

48 Building FSTs FSTs are often constructed modularly to deal with certain phenomena and then composed. Enables a divide and conquer approach to design. 32