Overview A first introduction to Prolog Implementing finite state - PDF document

Overview • A first introduction to Prolog Implementing finite state machines and learning Prolog along the way • Encoding finite state machines in Prolog • Recognition and generation with finite state machines in Prolog • Completing the FSM recognition and generation algorithms to use Detmar Meurers: Intro to Computational Linguistics I • ǫ transitions OSU, LING 684.01 • abbreviations • Encoding finite state transducers in Prolog 2 The Prolog programming language (1) The Prolog programming language (2) PROgrammation LOGique was invented by Alain Colmerauer and colleagues at Marseille and Edinburgh in the early 70s. A Prolog A Prolog program consists of a set of Horn clauses: program is written in a subset of first order predicate logic. There are • unit clauses or facts • constants naming entities – syntax: predicate followed by a dot – syntax : starting with lower-case letter (or number or single quoted) – example: father(tom,mary). – examples: twelve, a, q 1, 14, ’John’ • variables over entities • non-unit clauses or rules – syntax : starting with upper-case letter (or an underscore) – syntax: rel 0 :- rel 1 , ..., rel n . – examples: A, This, – example: grandfather(Old,Young) :- twelve, father(Old,Middle), • predicate symbols naming relations among entities father(Middle,Young). – syntax: predicate name starting with a lower-case letter with parentheses around comma-separated arguments – examples: father(tom,mary) , age(X,15) 3 4 The Prolog programming language (3) A first Prolog program grandfather.pl father(adam,ben). • No global variables: Variables only have scope over a single clause. father(ben,claire). • No explicit typing of variables or of the arguments of predicates. father(ben,chris). • Negation by failure: For \ +(P) Prolog attempts to prove P , and if this grandfather(Old,Young) :- succeeds, it fails. father(Old,Middle), father(Middle,Young). Query: ?- grandfather(adam,X). X = claire ? ; X = chris ? ; no 5 6

Recursive relations in Prolog Recursive relations in Prolog Compound terms as data structures Lists as special compound terms To define recursive relations, one needs a richer data structure than the • empty list: represented by the atom ” [] ” constants (atoms) introduced so far: compound terms . • non-empty list: compound term with ” . ” as binary functor A compound term comprises a functor and a sequence of one or more terms, the argument. 1 Compound terms are standardly written in prefix – first argument: first element of list (“ head ”) notation. 2 – second argument: rest of list (“ tail ”) Example: Example: .(a, .(b, .(c, .(d,[])))) – binary tree: bin tree( mother , l-dtr , r-dtr ) – example: bin tree(s, np, bin tree(vp,v,n)) 1 An atom can be thought of as a functor with arity 0. 2 Infix and postfix operators can also be defined, but need to be declared. 7 8 Abbreviating notations for lists An example for the four notations [a,b,c,d] = .(a, .(b, .(c, .(d,[])))) • bracket notation: [ element1 | restlist ] = [a | [b | [c | [d | []]]]] Example: [a | [b | [c | [d | []]]]] . • element separator: [ element1 , element2 ] = a = [ element1 | [ element2 | []]] . Example: [a, b, c, d] b . c . d [] 9 10 Recursive relations in Prolog Recursive relations in Prolog Example relations I: append Example relations IIa: (naive) reverse • Idea: a relation concatenating two lists • Idea: reverse a list • Example: ?- append([a,b,c],[d,e],X). ⇒ X=[a,b,c,d,e] • Example: ?- reverse([a,b,c],X). ⇒ X=[c,b,a] append([],L,L). naive_reverse([],[]). append([H|T],L,[H|R]) :- naive_reverse([H|T],Result) :- append(T,L,R). naive_reverse(T,Aux), append(Aux,[H],Result). 11 12

Recursive relations in Prolog Some practical matters Example relations IIb: reverse • To start Prolog on the Linguistics Department Unix machines: reverse(A,B) :- • SWI-Prolog: pl (on Mac OSX: swipl ) reverse_aux(A,[],B). • SICStus: prolog or M-x run-prolog in XEmacs reverse_aux([],L,L). • At the Prolog prompt ( ?- ): reverse_aux([H|T],L,Result) :- reverse_aux(T,[H|L],Result). • Trace the next command: trace. • Exit Prolog: halt. • Consult a file in Prolog: [ filename ]. 3 • The manuals are accessible from the course web page. 3 The .pl suffix is added automatically, but use single quotes if name starts with a capital letter or contains special characters such as ”.” or ”–”. For example [’MyGrammar’]. or [’˜/file-1’] . 13 14 Encoding finite state automata in Prolog Prolog representation of a finite state automaton What needs to be represented? The FSA is represented by the following kind of Prolog facts: A finite state automaton is a quintuple ( Q, Σ , E, S, F ) with • initial nodes: initial( nodename ). • Q a finite set of states • final nodes: final( nodename ). • Σ a finite set of symbols, the alphabet • edges: arc( from-node , label , to-node ). • S ⊆ Q the set of start states • F ⊆ Q the set of final states • E a set of edges Q × (Σ ∪ { ǫ } ) × Q 15 16 A simple example An example with two final states FSTN representation of FSM: FSTN representation of FSM: r 1 c 1 d c o l o 0 6 5 4 2 u r 0 a b 3 3 2 Prolog encoding of FSM: Prolog encoding of FSM: initial(0). initial(0). final(1). final(1). final(2). arc(0,c,6). arc(6,o,5). arc(5,l,4). arc(4,o,2). arc(0,c,1). arc(1,d,1). arc(0,a,3). arc(3,b,2). arc(2,r,1). arc(2,u,3). arc(3,r,1). 17 18

Recognition with FSMs in Prolog Generation with FSMs in Prolog fstn traversal basic.pl generate :- test(Words) :- test(X), initial(Node), write(X), recognize(Node,Words). nl, fail. recognize(Node,[]) :- final(Node). recognize(FromNode,String) :- arc(FromNode,Label,ToNode), traverse(Label,String,NewString), recognize(ToNode,NewString). traverse(First,[First|Rest],Rest). 19 20 Encoding finite state transducers in Prolog Prolog representation of a transducer What needs to be represented? The only change compared to automata, is an additional argument in the representation of the arcs: arc( from-node , label-in , to-node , label-out ). A finite state transducer is a 6-tuple ( Q, Σ 1 , Σ 2 , E, S, F ) with Example: • Q a finite set of states initial(1). • Σ 1 a finite set of symbols, the input alphabet final(5). • Σ 2 a finite set of symbols, the output alphabet arc(1,2,where,ou). arc(2,3,is,est). • S ⊆ Q the set of start states arc(3,4,the,la). arc(4,5,exit,sortie). • F ⊆ Q the set of final states arc(4,5,shop,boutique). arc(4,5,toilet,toilette). • E a set of edges Q × (Σ 1 ∪ { ǫ } ) × Q × (Σ 2 ∪ { ǫ } ) arc(3,6,the,le). arc(6,5,policeman,gendarme). 21 22 Processing with a finite state transducer FSMs with ǫ transitions and abbreviations Defining Prolog representations test(Input,Output) :- initial(Node), transduce(Node,Input,Output), 1. Decide on a symbol to use to mark ǫ transitions: ’#’ write(Output),nl. 2. Define abbreviations for labels: transduce(Node,[],[]) :- macro(Label,Word). final(Node). 3. Define a relation special/1 to recognize abbreviations and epsilon transduce(Node1,String1,String2) :- transitions: arc(Node1,Node2,Label1,Label2), traverse2(Label1,Label2,String1,NewString1, String2,NewString2), special(’#’). transduce(Node2,NewString1,NewString2). special(X) :- macro(X,_). traverse2(Word1,Word2,[Word1|RestString1],RestString1, [Word2|RestString2],RestString2). 23 24

traverse(Label,[Label|RestString],RestString) :- FSMs with ǫ transitions and abbreviations \+ special(Label). Extending the recognition algorithm traverse(Abbrev,[Label|RestString],RestString) :- macro(Abbrev,Label). test(Words) :- traverse(’#’,String,String). initial(Node), recognize(Node,Words). special(’#’). special(X) :- recognize(Node,[]) :- macro(X,_). final(Node). recognize(FromNode,String) :- arc(FromNode,Label,ToNode), traverse(Label,String,NewString), recognize(ToNode,NewString). 25 26 A tiny English fragment as an example Reading assignment (fsa/ex simple engl.pl) arc(7,n,9). macro(n,man). initial(1). • Pages 1–26 of Fernando Pereira and Stuart Shieber (1987): Prolog final(9). arc(8,adj,9). macro(n,woman). and Natural-Language Analysis . Stanford: CSLI. arc(1,np,3). arc(8,mod,8). macro(pv,is). arc(9,cnj,4). macro(pv,was). arc(1,det,2). arc(2,n,3). arc(9,cnj,1). macro(cnj,and). macro(cnj,or). arc(3,pv,4). arc(4,adv,5). macro(np,kim). macro(adj,happy). macro(np,sandy). macro(adj,stupid). arc(4,’#’,5). arc(5,det,6). macro(np,lee). macro(mod,very). arc(5,det,7). macro(det,a). macro(adv,often). arc(5,’#’,8). macro(det,the). macro(adv,always). arc(6,adj,7). macro(det,her). macro(adv,sometimes). macro(n,consumer). arc(6,mod,6). 27 28