Lecture 7: Definite Clause Grammars

Lecture 7: Definite Clause Grammars Theory Introduce context free grammars and some related concepts Introduce definite clause grammars , the Prolog way of working with context free grammars (and other grammars too) Exercises Exercises of LPN: 7.1, 7.2, 7.3 Practical work

Context free grammars Prolog offers a special notation for defining grammars , namely DCGs or definite clause grammars So what is a grammar? We will answer this question by discussing context free grammars CFGs are a very powerful mechanism, and can handle most syntactic aspects of natural languages (such as English or Italian)

Example of a CFG s  np vp np  det n vp  v np vp  v det  the det  a n  man n  woman v  shoots

Ingredients of a grammar The  symbol is used to define the rules The symbols s , np , vp , det , n , v are called the non-terminal symbols The symbols in italics are the terminal symbols: the, a, man, woman, shoots s  np vp np  det n vp  v np vp  v det  the det  a n  man n  woman v  shoots

A little bit of linguistics The non-terminal symbols in this grammar have a traditional meaning in linguistics: np : noun phrase vp : verb phrase det : determiner n : noun v : verb s : sentence

More linguistics In a linguistic grammar, the non-terminal symbols usually correspond to grammatical categories In a linguistic grammar, the terminal symbols are called the lexical items , or simply words (a computer scientist might call them the alphabet )

Context free rules The grammar contains nine context free rules A context free rule consists of: A single non-terminal symbol followed by  followed by a finite sequence of terminal or non-terminal symbols s  np vp np  det n vp  v np vp  v det  the det  a n  man n  woman v  shoots

Grammar coverage Consider the following string: the woman shoots a man Is this string grammatical according to our grammar? And if it is, what syntactic structure does it have?

Syntactic structure s vp np np det n v det n the woman shoots a man s  np vp np  det n vp  v np vp  v det  the det  a n  man n  woman v  shoots

Parse trees Trees representing the syntactic structure of a string are often called parse trees Parse trees are important: They give us information about the string They gives us information about structure

Grammatical strings If we are given a string of words, and a grammar, and it turns out we can build a parse tree, then we say that the string is grammatical (with respect to the given grammar) E.g., the man shoots is grammatical If we cannot build a parse tree, the given string is ungrammatical (with respect to the given grammar) E.g., a shoots woman is ungrammatical

Generated language The language generated by a grammar consists of all the strings that the grammar classifies as grammatical For instance a woman shoots a man a man shoots belong to the language generated by our little grammar

Recogniser A context free recogniser is a program which correctly tells us whether or not a string belongs to the language generated by a context free grammar To put it another way, a recogniser is a program that correctly classifies strings as grammatical or ungrammatical

Information about structure But both in linguistics and computer science, we are not merely interested in whether a string is grammatical or not We also want to know why it is grammatical: we want to know what its structure is The parse tree gives us this structure

Parser A context free parser correctly decides whether a string belongs to the language generated by a context free grammar And it also tells us what its structure is To sum up: A recogniser just says yes or no A parser also gives us a parse tree

Context free language We know what a context free grammar is, but what is a context free language? Simply: a context free language is a language that can be generated by a context free grammar Some human languages are context free, some others are not English and Italian are probably context free Dutch and Swiss-German are not context free

Theory vs. Practice So far the theory, but how do we work with context free grammars in Prolog? Suppose we are given a context free grammar. How can we write a recogniser for it? How can we write a parser for it? In this lecture we will look at how to define a recogniser

CFG recognition in Prolog We shall use lists to represent strings [a,woman,shoots,a,man] The rule s  np vp can be thought as concatenating an np -list with a vp -list resulting in an s -list We know how to concatenate lists in Prolog: using append/3 So let`s turn this idea into Prolog

CFG recognition using append/3 s (C):- np(A), vp(B), append(A,B,C). np (C):- det(A), n(B), append(A,B,C). vp (C):- v(A), np(B), append(A,B,C). vp (C):- v(C). det([the]). det([a]). n([man]). n([woman]). v([shoots]).

CFG recognition using append/3 ?- s ([the,woman,shoots,a,man]). yes ?- s (C):- np(A), vp(B), append(A,B,C). np (C):- det(A), n(B), append(A,B,C). vp (C):- v(A), np(B), append(A,B,C). vp (C):- v(C). det([the]). det([a]). n([man]). n([woman]). v([shoots]).

CFG recognition using append/3 ?- s (S). S = [the,man,shoots,the,man]; S = [the,man,shoots,the,woman]; S = [the,woman,shoots,a,man] … s (C):- np(A), vp(B), append(A,B,C). np (C):- det(A), n(B), append(A,B,C). vp (C):- v(A), np(B), append(A,B,C). vp (C):- v(C). det([the]). det([a]). n([man]). n([woman]). v([shoots]).

CFG recognition using append/3 ?- np ([the,woman]). yes ?- np (X). X = [the,man]; X = [the,woman] s (C):- np(A), vp(B), append(A,B,C). np (C):- det(A), n(B), append(A,B,C). vp (C):- v(A), np(B), append(A,B,C). vp (C):- v(C). det([the]). det([a]). n([man]). n([woman]). v([shoots]).

Problems with this recogniser It doesn`t use the input string to guide the search Goals such as np (A) and vp(B) are called with uninstantiated variables Moving the append/3 goals to the front is still not very appealing --- this will only shift the problem --- there will be a lot of calls to append/3 with uninstantiated variables

Difference lists A more efficient implementation can be obtained by using difference lists This is a sophisticated Prolog technique for representing and working with lists Examples: [a,b,c]-[ ] is the list [a,b,c] [a,b,c,d]-[d] is the list [a,b,c] [a,b,c|T]-T is the list [a,b,c] X-X is the empty list [ ]

CFG recognition using difference lists ?- s ([the,man,shoots,a,man]-[ ]). yes ?- s (A-C):- np(A-B), vp(B-C). np (A-C):- det(A-B), n(B-C). vp (A-C):- v(A-B), np(B-C). vp (A-C):- v(A-C). det([the|W]-W). det([a|W]-W). n([man|W]-W). n([woman|W]-W). v([shoots|W]-W).

CFG recognition using difference lists ?- s (X-[ ]). S = [the,man,shoots,the,man]; S = [the,man,shoots,a,man]; … . s (A-C):- np(A-B), vp(B-C). np (A-C):- det(A-B), n(B-C). vp (A-C):- v(A-B), np(B-C). vp (A-C):- v(A-C). det([the|W]-W). det([a|W]-W). n([man|W]-W). n([woman|W]-W). v([shoots|W]-W).

Summary so far The recogniser using difference lists is a lot more efficient than the one using append/3 However, it is not that easy to understand and it is a pain having to keep track of all those difference list variables It would be nice to have a recogniser as simple as the first and as efficient as the second This is possible: using DCGs

Definite Clause Grammars What are DCGs? Quite simply, a nice notation for writing grammars that hides the underlying difference list variables Let us look at three examples

DCGs: first example s --> np, vp. np --> det, n. vp --> v, np. vp --> v. det --> [the]. det --> [a]. n --> [man]. n --> [woman]. v --> [shoots].

DCGs: first example ?- s ([a,man,shoots,a,woman],[ ]). yes ?- s --> np, vp. np --> det, n. vp --> v, np. vp --> v. det --> [the]. det --> [a]. n --> [man]. n --> [woman]. v --> [shoots].

DCGs: first example ?- s (X,[ ]). S = [the,man,shoots,the,man]; S = [the,man,shoots,a,man]; … . s --> np, vp. np --> det, n. vp --> v, np. vp --> v. det --> [the]. det --> [a]. n --> [man]. n --> [woman]. v --> [shoots].

DCGs: second example We added some recursive rules to the grammar… What and how many sentences does this grammar generate? What does Prolog do with this DCG? s --> s, conj, s. s --> np, vp. np --> det, n. vp --> v, np. vp --> v. det --> [the]. det --> [a]. n --> [man]. n --> [woman]. v --> [shoots]. conj --> [and]. conj --> [or]. conj --> [but].

DCG without left-recursive rules s --> simple _ s, conj, s. s --> simple _ s. simple _ s --> np, vp. np --> det, n. vp --> v, np. vp --> v. det --> [the]. det --> [a]. n --> [man]. n --> [woman]. v --> [shoots]. conj --> [and]. conj --> [or]. conj --> [but].

DCGs are not magic! The moral: DCGs are a nice notation, but you cannot write arbitrary context-free grammars as a DCG and have it run without problems DCGs are ordinary Prolog rules in disguise So keep an eye out for left-recursion!

DCGs: third example We will define a DCG for a formal language A formal language is simple a set of strings Formal languages are objects that computer scientist and mathematicians define and study Natural languages are languages that human beings normally use to communicate We will define the language a n b n

DCGs: third example s --> []. s --> l,s,r. l --> [a]. r --> [b]. ?- s ([a,a,a,b,b,b],[ ]). yes ?- s ([a,a,a,a,b,b,b],[ ]). no We will define the formal language a n b n

DCGs: third example s --> []. s --> l,s,r. l --> [a]. r --> [b]. ?- s(X,[ ]). X = [ ]; X = [a,b]; X = [a,a,b,b]; X = [a,a,a,b,b,b] … . We will define the formal language a n b n

Exercises LPN 7.1 LPN 7.2 LPN 7.3

Summary of this lecture We explained the idea of grammars and context free grammars are We introduced the Prolog technique of using difference lists We showed that difference lists can be used to describe grammars Definite Clause Grammars is just a nice Prolog notation for programming with difference lists

Next lecture More Definite Clause Grammars Examine two important capabilities offered by DCG notation Extra arguments Extra tests Discuss the status and limitations of definite clause grammars

Lecture 7: Definite Clause Grammars

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Lecture 7: Definite Clause Grammars