Definite-clause grammars: PSGs in Prolog (in principle, string rewrite described as a logical formula).

Transcription

1 Computational Encoding of PSGs Logical (declarative) Network-based (procedural) Definite-clause grammars: PSGs in Prolog (in principle, string rewrite described as a logical formula). For a grammar G = (V, Σ, P, S), a rule in P as (a) string rewrite: X 0 X 1 X 2 X n, X i (V Σ) (b) logical implication: (X 1 X 2 X n ) X 0, where X i are logical predicates.

2 clause: disjunction of literals literal: atomic formula or a negation of it. definite clause: a clause with at most one positive literal. A B C D not a DC B C D is a DC B C D is a DC p q p q, thus (X 1 X 2 X n ) X 0 X 1 X 2 X 0 is a definite clause if X i, i > 0 are positive literals.

3 for a headless definite clause X 1 X n (X 1 X 2 X n ) false A rule of the form (X 1 X 2 X n ) X 0 translates to x0:- x1, x2,.., xn. in Prolog String rewrite systems are order-sensitive, whereas in logic, it does not matter in which order you satisfy the clauses. Prolog s depth-first top-down strategy of rule evaluation suggests a natural way to impose order.

4 friend(x,y):- likes(x,y), likes(y,x). likes(john,mary). likes(john,sue). likes(sue,john). likes(sue,mary). In First-order logic: x y(likes(x, y) likes(y, x)) friend(x, y) Facts are in fact truth assertions of the kind true F i ( bodyless rules).

5 queries:?- friend(john,sue). match head of a clause (fact or rule head): friend(x,y)=friend(john,sue) for X/john Y/sue: satisfy subgoals likes(john,sue) matches a fact (satisfied) likes(sue,john) matches a fact yes.?- friend(john,mary). no. In parsing, we are concerned with not only whether we ve satisfied, say X i, and in what order, but also in which positions X i is found

6 (after all, the input to our logical formulae is a sequence of facts; which sub-sequence of facts satisfy which sub-formula?). X0 X0 X1 X2... Xn p0 p1 p2 pn... X1 X2 Xn p0 p1 p2 pn in Prolog xo(p0,p1):- x1(p0,p1), x2(p1,p2),..,xn(pn-1,pn). where P i is a list-valued prolog variable (denoting the subsequence covered by the phrase x). DCG is a special notation for this:

7 xo --> x1,x2,..,xn. In DCG notation (rules written in -->), every clause gets two hidden arguments showing the chaining of sequences. There is in fact a more perspicuous (and computationally more efficient) notation for showing the subsequence as a list difference of subsequences. Every x has one argument defined as a difference of two lists. Lists in Prolog: [H T] is a list whose head is H and whose tail is the list T.

8 ex: x([the,man,on,the,moon]). x([a,girl]). x([]).?- x([a]).?- x([a,b]).?- x([a T]).?- x([my T]).?- x([h T]).?- x([a,b,c T]).?- x(a).

9 L1-L2 is the list difference of L1 and L2 (all the elements in L1 up to but not including those in L2). we are interested in the most general difference list (the last one below. All others are instances of it). ex: [a,b,c,d]-[d] [a,b,c]-[] [a,b,c T]-T [a,b,c,d,e]-[d,e] [a,b,c,d,e,f]-[d,e,f] Concatenation of lists is much easier with difference lists: conc_list([],l,l). conc_list([h1 T1],L2,[H1 T]):- conc_list(t1,l2,t). conc_dl(l1-l2,l2-l3,l1-l3).

10 Compare: x0(p0,pn):- x1(p0,p1), x2(p1,p2),.. xn(pn_1,pn). x0(p0-pn):- x1(p0-p1), x2(p1-p2), xn(pn_1-pn). CFG DCG S -> NP VP s --> np,vp. VP -> V NP vp--> v,np. V -> likes v --> [likes]. NP -> john mary np--> [john]. np--> [mary]. native Prolog: s (P0,P2):- np(p0,p1), vp(p1,p2). vp(p0,p2):- v(p0,p1), np(p1,p2). v([likes T],T).

11 np([john T],T). np([mary T],T). Rules with arguments. They carry the information from one symbol to another. DCGs are Turing-equivalent in power when they have arguments. They have CF power without arguments. s --> np(agr), vp(agr). s(p0,p2):- np(agr,p0,p1), vp(agr,p1,p2). When instantiated, an argument can act as a constraint on the derivations. As variables, they represent information requirements.

12 They can model local constraints or non-local constraints. s --> np1s, vp1s. s --> np(agr), vp(agr). s --> np1p, vp1p.... s --> np3s, vp3s. np((person,number))--> pronoun(person,number).... np1p --> [we]. np3s --> det, noun(sing). Arguments can be used to synthesize information too, e.g., to return a parse tree. Prolog structures: name(arguments) s(s(nppt,vppt))--> np(nppt), vp(vppt). np(np(detpt,npt)) --> det(detpt), n(npt). vp(vp(vpt,nppt)) --> v(vpt), np(nppt).

13 v(v(liked)) --> [liked]. n(n(man))--> [man]. det(det(the)) --> [the].?- s(spt,[the,man,liked,the,woman],[]). SPT=s(NPPT,VPPT) NPPT=np(DetPT,NPT) DETPT=det(the) NPT=n(man)... SPT=s(np(det(the),n(man)), vp(v(liked),np(det(the),n(woman))))

14 S NP Det the N man VP V liked Det NP N the woman Lexical organization in DCGs: Try to achieve abstraction in the lexicon to capture generalizations. Also simplifies syntax. v--> [chased]. v--> [Word], {verb(word)}.

15 v--> [ran].... verb(chased). verb(ran). ex: generalization. All forms of the same verb carry the same valence. Why carry this information in five different places? verb(tv,write,writes,wrote,written,writing). % verb database verb(iv,sleep,sleeps,slept,slept,sleeping). verb(dv,give,gives,gave,given,giving). vp --> v(iv). vp --> v(tv), np. vp --> v(dv), np, np. % e.g., consuming the s-form of a verb and carrying valence v(valence)--> [Word], {verb(valence,_,word,_,_,_)}.

16 alternative: separate form and valence information in the lexicon. verb(write,writes,wrote,written,writing). % verb database verb(sleep,sleeps,slept,slept,sleeping). verb(give,gives,gave,given,giving). trans(write). intrans(sleep). ditrans(give). vp --> v(base), {intrans(base)}. vp --> v(base), {trans(base)},np. vp --> v(base), {ditrans(base)},np, np. % e.g., consuming the s-form of a verb and carrying base v(base)--> [Word], {verb(base,word,_,_,_)}. Yet another alternative: flip the verb table. carry the form information and base in individual entries.

17 verb(write,writes,(3,sing)). verb(write,write,(_,plu)). verb(write,wrote,(_,_)). trans(write). intrans(sleep). ditrans(give). vp(agr) --> v(base,agr), {intrans(base)}. vp(agr) --> v(base,agr), {trans(base)},np. vp(agr) --> v(base,agr), {ditrans(base)},np, np. v(base,agr)--> [Word], {verb(base,word,agr)}. s --> np(agr), vp(agr). pronoun(he,3,sing,nom). pronoun(him,3,sing,acc).