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Stabler - Lx 185/209 2003 (9) Example: Let Γ be the following sequence of definite clauses: socrates_is_a_man. socrates_is_dangerous. socrates_is_mortal :- socrates_is_a_man. socrates_is_a_dangerous_man :socrates_is_a_man, socrates_is_dangerous. Clearly, we can prove socrates_is_mortal and socrates_is_a_dangerous_man. The proof can be depicted with a tree like this: (10) Example: Acontextfreegrammar where socrates_is_a_dangerous_man socrates_is_a_man socrates_is_dangerous G =〈Σ,N,→, S〉, 1. Σ,N are finite, nonempty sets, 2. S is some symbol in N, 3. the binary relation (→) ⊆ N × (Σ ∪ N) ∗ is also finite (i.e. it has finitely many pairs), For example, ip → dp i1 i1 → i0 vp i0 → will dp → d1 d1 → d0 np d0 → the np → n1 n1 → n0 n1 → n0 cp n0 → idea vp → v1 v1 → v0 v0 → suffice cp → c1 c1 → c0 ip c0 → that Intuitively, if ip is to be read as “there is an ip,” and similarly for the other categories, then the rewrite arrow cannot be interpreted as implies, since there are alternative derivations. That is, the rules (n1 → n0) and (n1 → n0 cp) signify that a given constituent can be expanded either one way or the other. In fact, we get an appropriate logical reading of the grammar if we treat the rewrite arrow as meaning “if.” With that reading, we can also express the grammar as a prolog theory. /* * file: th2.pl */ ip :- dp, i1. i1 :- i0, vp. i0 :- will. will. dp :- d1. d1 :- d0, np. d0 :- the. the. np :- n1. n1 :- n0. n0 :- idea. idea. n1 :- n0, cp. vp :- v1. v1 :- v0. v0 :- suffice. suffice. cp :- c1. c1 :- c0, ip. c0 :- that. that. In this theory, the proposition idea can be read as saying that this word is in the language, and ip :dp, i1 says that ip is in the language if dp and i1 are. The proposition ip follows from this theory. After loading this set of axioms, we can prove ?- ip. Finding a proof corresponds exactly to finding a derivation from the grammar. 9
Stabler - Lx 185/209 2003 In fact, there are infinitely many proofs of ?- ip. The first proof that prolog finds can be depicted with atreelikethis: 1.3 Using prolog d0 the dp d1 np n1 n0 idea Here is a session log, where I put the things I typed in bold: ip i0 will 1%pl Welcome to SWI-Prolog (Version 4.0.11) Copyright (c) 1990-2000 University of Amsterdam. Copy policy: GPL-2 (see www.gnu.org) For help, use ?- help(Topic). or ?- apropos(Word). ?- write(’hello world’). hello world Yes ?- halt. 2%emacs test.pl 3%cat test.pl % OK, a test p :- q,r. r :- s. q :- t. s. t. 4%pl Welcome to SWI-Prolog (Version 4.0.11) Copyright (c) 1990-2000 University of Amsterdam. Copy policy: GPL-2 (see www.gnu.org) For help, use ?- help(Topic). or ?- apropos(Word). ?- [test]. % test compiled 0.00 sec, 1,180 bytes Yes ?- listing. 10 i1 vp v1 v0 suffice
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