Java Antlr4一阶逻辑语法

Question

I want to parse strings in first order logic and turn them into a specific class structure. For instance, I want to parse a formula such as

∀x∃y∃z((R(x,y) ∨ Px)→(Qx→(Px∧Zx))) 

and turn it into a Universal class which has a Variable field and a Formula quantifiedFormula field which stands for the rest of the formula. I have troubles with the grammar, though. When I parse that formula with the antlr generated code I get

line 1:11 extraneous input '(' expecting {'\u2200', '\u2203', '\u00ac'}

'\∀' is ∀, \∃ is ∃ and \¬ is ¬, the negation sign.

This is my grammar file. I put it together following the FOL.g file found on the antlr3 site. I am using antl4, however.

grammar FOL;

options{
    language=Java;
    output=AST;
    ASTLabelType = CommonTree;
    backtrack=true;
}

tokens{
    PREDICATE,
    FUNCTION
}

/*------------------------------------------------------------------
* PARSER RULES
*------------------------------------------------------------------*/

condition: formula EOF ;

formula 
:     (forall | exists)* bidir ;
forall  :   FORALL VARIABLE ;
exists  :   EXISTS VARIABLE ;

bidir   :   implication (BIDIR implication)*;

implication
:   disjunction (IMPL disjunction)*;

disjunction
:   conjunction (OR conjunction)* ;

conjunction
:   negation (AND negation)* ;

negation 
:   NOT (predicate | LPAREN* formula RPAREN*) ;

predicate 
:   PREPOSITION predicateTuple (PREDICATE PREPOSITION predicateTuple)
|   PREPOSITION ;

predicateTuple
:   LPAREN term (',' term)* RPAREN ;

term    :   function | VARIABLE ;

 function:  CONSTANT functionTuple (FUNCTION CONSTANT functionTuple)
|   CONSTANT;

functionTuple
    :   LPAREN (CONSTANT | VARIABLE) (',' (CONSTANT | VARIABLE) )* RPAREN;

/*------------------------------------------------------------------
* LEXER RULES
*------------------------------------------------------------------*/
LPAREN: '(';
RPAREN: ')';
FORALL: '\u2200';
EXISTS: '\u2203';
NOT:'\u00ac';
IMPL: '\u2192';
BIDIR: '\u2194';
OR: '\u2228';
AND: '\u2227';
EQ: '=';


VARIABLE:  (('q'..'z') ) CHARACTER* ;

CONSTANT: (('a'..'p') ) CHARACTER* ;

PREPOSITION: ('A'..'Z') CHARACTER* ;

fragment CHARACTER: ('a'..'z' | 'A'..'Z' | '_') ;

WS : (' ' | '\t' | '\r' | '\n')+ -> skip  ;

Answer 1

That seems unsurprising.

According to your grammar, a formula is some number of exists and forall clauses followed by a bidir . If you trace through the productions for bidir , it becomes clear that it must start with a negation and that, in turn, must start with NOT . So while you scan the formula , you must see clauses headed by one of the three tokens EXISTS , FORALL or NOT .

Your negation needs to include the possibility that it is not a negation. You could, for example, make NOT optional.