definite clause grammars

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Definite Clause Grammars

t.k.prasad@wright.eduhttp://www.knoesis.org/tkprasad/

Review : Difference Lists • Represent list L as a difference of two lists

L1 and L2– E.g., consider L = [a,b,c] and various L1-L2

combinations given below.

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L1 L2[a,b,c|T] T

[a,b,c,d|T] [d|T]

Review: Append using Difference Lists

append(X-Y, Y-Z, X-Z).• Ordinary append complexity = O(length of first list)• Difference list append complexity = O(1)

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X

Y

Z

X-Y

Y-Z

Y

Z

Z

X-Z

Running append goal

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DCGs

• Mechanize attribute grammar formalism (a generalization of CFG)– Executable specification

• Use difference lists for efficiency

• Translation from DCGs to Prolog clauses is automatic

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Sample Applications of DCGs

• Coding recursive descent backtracking parser

• Encoding and checking context-sensitive constraints

• Simple NLP

• In general, enabling syntax directed translation

• E.g., VHDL Parser-Pretty Printer

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DCG Example : Syntaxsentence --> noun_phrase, verb_phrase.noun_phrase --> determiner, noun.verb_phrase --> verb, noun_phrase.determiner --> [a].determiner --> [the].determiner --> [many].noun --> [president].noun --> [cat].noun --> [cats].verb --> [has].verb --> [have].

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DCG to Ordinary Prolog Syntaxsentence(S,R) :-

noun_phrase(S,T), verb_phrase(T,R).noun_phrase(S,T) :- determiner(S,N), noun(N,T).verb_phrase(T,R) :- verb(T,N), noun_phrase(N,R).determiner([a|R],R).determiner([the|R],R).determiner([many|R],R).noun([president|R],R).noun([cat|R],R).noun([cats|R],R).verb([has|R],R).verb([have|R],R).

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Queries?- sentence([the, president, has, a, cat], []).

?- sentence([the, cats, have, a, president], []).

?- sentence([a, cats, has, the, cat, president], [president]).

?- sentence([a, cats, has, the, cat, President], [President]).

• Each non-terminal takes two lists as arguments. • In difference list representation, they together

stand for a single list.

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DCG Example: Number Agreement

sentence --> noun_phrase(N),verb_phrase(N).noun_phrase(N) --> determiner(N), noun(N).verb_phrase(N) --> verb(N), noun_phrase(_).

determiner(sgular) --> [a].determiner(_) --> [the].determiner(plural) --> [many].

noun(sgular) --> [president].noun(sgular) --> [cat].noun(plural) --> [cats].verb(sgular) --> [has].verb(plural) --> [have].

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Extension: AST plus Number agreement

sentence(s(NP,VP)) --> noun_phrase(N, NP),verb_phrase(N, VP).

noun_phrase(N, np(D,NT)) --> determiner(N, D), noun(N, NT).

verb_phrase(N, vp(V,NP)) --> verb(N, V), noun_phrase(_, NP).

determiner(sgular, dt(a)) --> [a].

determiner(_, dt(the)) --> [the].

determiner(plural, dt(many)) --> [many].

noun(sgular, n(president)) --> [president].

noun(sgular, n(cat)) --> [cat].

noun(plural, n(cats)) --> [cats].

verb(sgular, v(has)) --> [has].

verb(plural, v(have)) --> [have].

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Queries?- sentence(T,[the, president, has, a, cat], []).

T = s(np(dt(the), n(president)), vp(v(has), np(dt(a), n(cat)))) ;

?- sentence(T,[the, cats, have, a, president|X], X).

?- sentence(T,[a, cats, has, the, cat, preside], [preside]).

• Each non-terminal takes two lists as arguments for input sentences, and additional arguments for the static semantics (e.g., number and AST).

• Number disagreement causes the last query to fail.

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Prefix Expression DCG

expr --> [if], expr, [then], expr,

[else], expr.

expr --> [’+’], expr, expr.

expr --> [’*’], expr, expr.

expr --> [m].

expr --> [n].

expr --> [a].

expr --> [b].

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Queries

?-expr([’*’, m, n], []).

?-expr([m, ’*’, n], []).

?-expr([’*’, m, ’+’, ’a’, n, n], [n]).

?-expr([if, a, then, m, else, n], []).

?-expr([if, a, then, a, else, ’*’, m, n], []).

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Prefix Expression DCG : Type Checking Version

tExpr(T) --> [if], tExpr(bool), [then], tExpr(T),

[else], tExpr(T).

tExpr(T) --> [’+’], tExpr(T), tExpr(T).

tExpr(T) --> [’*’], tExpr(T), tExpr(T).

tExpr(int) --> [m].

tExpr(int) --> [n].

tExpr(bool) --> [a].

tExpr(bool) --> [b].

• Assume that + and * are overloaded for int and bool.

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Queries

?-tExpr(T,[’*’, m, n], []).

?-tExpr(T,[m, ’*’, n], []).

?-tExpr(T,[’*’, m, ’+’, ’a’, n, n], [n]).

?-tExpr(T,[if, a, then, m, else, n], []).

T = int ;

?-tExpr(T,[if, a, then, b, else, ’*’, m, n], []).

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Prefix Expression DCG : Type Checking and Evaluation Version

evalExpr(V) --> etExpr(V,_).

etExpr(V,T) --> [if], etExpr(B,bool), [then],

etExpr(V1,T), [else], etExpr(V2,T),

{B==true -> V = V1 ; V = V2}.

etExpr(V,bool) --> [’+’],

etExpr(V1,bool), etExpr(V2,bool),

{or(V1,V2,V)}.

etExpr(V,int) --> [’+’],

etExpr(V1,int), etExpr(V2,int),

{V is V1 + V2}.

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(cont’d)

etExpr(V,bool) --> [’*’],

etExpr(V1,bool), etExpr(V2,bool),

{and(V1,V2,V)}.

etExpr(V,bool) --> [’*’],

etExpr(V1,int), etExpr(V2,int),

{V is V1 * V2}.

etExpr(V,int) --> [m], {value(m,V)}.

etExpr(V,int) --> [n], {value(n,V)}.

etExpr(V,bool) --> [a], {value(a,V)}.

etExpr(V,bool) --> [b], {value(b,V)}.

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(cont’d)

value(m,10). value(n,5).value(a,true).value(b,false).

and(true,true,true). and(true,false,false).and(false,true,false).

and(false,false,false).

or(true,true,true). or(true,false,true).or(false,true,true). or(false,false,false).

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Prefix Expression DCG : AST Version

treeExpr(V) --> trExpr(V,_).

trExpr(cond(B,V1,V2),T) -->

[if], trExpr(B,bool), [then], trExpr(V1,T),

[else], trExpr(V2,T).

trExpr(or(V1,V2),bool) --> [’+’],

trExpr(V1,bool), trExpr(V2,bool).

trExpr(plus(V1,V2),int) --> [’+’],

trExpr(V1,int), trExpr(V2,int).

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(cont’d)

trExpr(and(V1,V2),bool) --> [’*’],

trExpr(V1,bool), trExpr(V2,bool).

trExpr(mul(V1,V2),int) --> [’*’],

trExpr(V1,int), trExpr(V2,int).

trExpr(m,int) --> [m].

trExpr(n,int) --> [n].

trExpr(a,bool) --> [a].

trExpr(b,bool) --> [b].

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Other Compiler Operations• From parse tree and type information, one

can:– compute (stack) storage requirements for

variables and for expression evaluation– generate assembly code (with coercion

instructions if necessary)– transform/simplify expression

• Ref: http://www.cs.wright.edu/~tkprasad/papers/Attribute-Grammars.pdf

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Variation on Expression GrammarsInefficient

Backtracking Parser Exists

E -> T + E | T T -> F * T | F F -> (E) | x | y

Unsuitable Grammar

E -> E + E | E * E | x | y

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RELATIONSHIP TO ATTRIBUTE GRAMMARS

4

5

Attribute Grammars

• Formalism for specifying semantics based on context-free grammars (BNF) Static semantics (context-sensitive aspects)

Type checking and type inference Compatibility between procedure definition and call

Dynamic semantics

• Associate attributes with terminals and non-terminals

• Associate attribute computation rules with productions

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• AttributesAttributes A(X)– Synthesized S(X)– Inherited I(X)

• Attribute computation rulesAttribute computation rules (Semantic functions)

X0 -> X1 X2 … Xn

S(X0) = f( I(X0), A(X1), A(X2), …, A(Xn) )I(Xj) = Gj( I(X0), A(X1), A(X2), …, A(Xj-1)) for all j in 1..n P( A(X0), A(X1), A(X2), …, A(Xn) )

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Information Flowinherited

synthesized

computedcomputed

availableavailable

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• Synthesized AttributesSynthesized Attributes Pass information up the parse tree

• Inherited AttributesInherited Attributes Pass information down the parse tree or

from left siblings to the right siblings• Attribute values assumed to be available from the

context.• Attribute values computed using the semantic rules

provided. The constraints on the attribute evaluation rules

permit top-down left-to-right (one-pass) traversal of the parse tree to compute the meaning.

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An Extended Example• Distinct identifiers in a straight-line program.BNF<exp> ::= <var> | <exp> + <exp><stm> ::= <var> := <exp> | <stm> ; <stm>

Attributes <var> id <exp> ids <stm> ids num

• Semantics specified in terms of sets (of identifiers).

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<exp> ::= <var><exp>.ids = { <var>.id } <exp> ::= <exp1> + <exp2><exp>.ids = <exp>.ids U <exp>.ids <stm> ::= <var> := <exp><stm>.ids ={ <var>.id } U <exp>.ids<stm>.num = | <stm>.ids | <stm> ::= <stm1> ; <stm2><stm>.ids = <stm1>.ids U <stm2>.ids<stm>.num = | <stm>.ids |

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Alternative Semantics using lists• Attributes

envi : list of vars in preceding context envo : list of vars for following context dnum : number of new variables

<exp> ::= <var><exp>.envo = if member(<var>.id,<exp>.envi) then <exp>.envi else cons(<var>.id,<exp>.envi)

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<exp> ::= <exp1> + <exp2>

envi envi envi envo envo envo dnum dnum dnum

<exp1>.envi = <exp>.envi <exp2>.envi = <exp1>.envo <exp>.envo = <exp2>.envo <exp>.dnum = length(<exp>.envo)

Attribute Computation Rules