I am working on modal logic tableau solver which is implemented in python (2.7.5 version). So I already have a function that translates an input string to tableau format that is:
Input：
~p ^ q
Parsed:
['and',('not', 'p'), 'q']
Parsed and alpha rule applied:
[('not', 'p'), 'q']
Now, I dealt with alpha formulas that is intersection, double negations etc. The problem that I am encountering now are the beta formulas , for example Union.
For the Union formula I need to write a function that splits one list in two lists, so for example:
Input:
('and', 's', ('or', (not,'r'), 'q'))
Outputs:
1st list ('s',('not','r'))
2nd list ('s','q')
I can easily do it once, but how can I recursively scan the formula and generate these list so that later I can scan them and verify whether they are closed or not?
The final goal of this is to create a tableau solver which generates a graph that is a Model or return an answer that formula is unsatisfiable.
It' a very interesting project :). I'm myself in thesis about modal logic right now.
I'm first of all, advising you to use the InToHyLo input format, which is quite a standard in the existing solvers.
The InToHyLo format is looking as follow:
file ::= ['begin'] dml ['end']
fml ::= '(' fml ')' (* parentheses *)
| '1' | 'true' | 'True' | 'TRUE' (* truth *)
| '0' | 'false' | 'False' | 'FALSE' (* falsehood *)
| '~' fml | '-' fml (* negation *)
| '<>' fml | '<' id '>' fml (* diamonds *)
| '[]' fml | '[' id ']' fml (* boxes *)
| fml '&' fml (* conjunction *)
| fml '|' fml (* disjunction *)
| fml '->' fml (* implication *)
| fml '<->' fml (* equivalence *)
| id (* prop. var. *)
where identifiers (id) are arbitrary nonempty alphanumeric sequences: (['A'-'Z' 'a'-'z' '0'-'9']+)
To simplify the parsing of your formula and to focus on the real problem : Solving the instance. I advise you to use an existing parser like flex/bison.
By looking on Internet for your problem, (I'm far from being an expert in Python) it looks like the library "http://pyparsing.wikispaces.com" is the reference for parsing.
And after, just by using Bison as follow, your file will be fully parsed.
Here is my Bison file (for using Flex/Bison in a solver C++):
/*
*
* Compile with bison.
*/
/*** Code inserted at the begin of the file. ***/
%{
#include <stdlib.h>
#include <list>
#include "Formula.h"
// yylex exists
extern int yylex();
extern char yytext[];
void yyerror(char *msg);
%}
/*** Bison declarations ***/
%union
{
bool bval;
operator_t opval;
char *sval;
TermPtr *term;
}
%token LROUND RROUND
%left IFF
%left IMP
%left OR
%left AND
%right DIAMOND
%right BOX
%right NOT
%token VALUE
%token IDENTIFIER
%type<bval> VALUE
%type<sval> IDENTIFIER
%type<term> Formula BooleanValue BooleanFormula ModalFormula PropositionalVariable UnaryFormula
%type<opval> BinaryBoolOperator UnaryBoolOperator ModalOperator
%start Start
%%
Start:
| Formula { (Formula::getFormula()).setRoot(*$1); }
;
Formula: BooleanFormula { $$ = $1; }
| ModalFormula { $$ = $1; }
| UnaryFormula { $$ = $1; }
| LROUND Formula RROUND { $$ = $2; }
;
BooleanValue: VALUE { $$ = new TermPtr( (Term*) new BooleanValue($1) ); }
;
PropositionalVariable: IDENTIFIER { $$ = new TermPtr( (Term*) new PropositionalVar($1) ); }
;
BooleanFormula: Formula BinaryBoolOperator Formula {
$$ = new TermPtr( (Term*) new BooleanOp(*$1, *$3, $2) ); /* can be (A OR B) or (A AND B) */
delete($3);
delete($1);
}
| Formula IMP Formula {
($1)->Negate();
$$ = new TermPtr( (Term*) new BooleanOp(*$1, *$3, O_OR) ); /* A -> B can be written : (¬A v B) */
delete($3);
delete($1);
}
| PropositionalVariable IFF PropositionalVariable {
PropositionalVar *Copy1 = new PropositionalVar( *((PropositionalVar*)$1->getPtr()) );
PropositionalVar *Copy3 = new PropositionalVar( *((PropositionalVar*)$3->getPtr()) );
TermPtr Negated1( (Term*)Copy1, $1->isNegated() );
TermPtr Negated3( (Term*)Copy3, $3->isNegated() );
Negated1.Negate();
Negated3.Negate();
TermPtr Or1( (Term*) new BooleanOp(Negated1, *$3, O_OR) ); /* Or1 = (¬A v B) */
TermPtr Or2( (Term*) new BooleanOp(Negated3, *$1, O_OR) ); /* Or2 = (¬B v A) */
$$ = new TermPtr( (Term*) new BooleanOp(Or1, Or2, O_AND) ); /* We add : (Or1 AND OrB) */
delete($3);
delete($1);
}
;
ModalFormula: ModalOperator LROUND Formula RROUND {
$$ = new TermPtr( (Term*) new ModalOp(*$3, $1) );
delete($3);
}
|
ModalOperator ModalFormula {
$$ = new TermPtr( (Term*) new ModalOp(*$2, $1) );
delete($2);
}
|
ModalOperator UnaryFormula {
$$ = new TermPtr( (Term*) new ModalOp(*$2, $1) );
delete($2);
}
;
UnaryFormula: BooleanValue { $$ = $1; }
| PropositionalVariable { $$ = $1; }
|
UnaryBoolOperator UnaryFormula {
if ($1 == O_NOT) {
($2)->Negate();
}
$$ = $2;
}
|
UnaryBoolOperator ModalFormula {
if ($1 == O_NOT) {
($2)->Negate();
}
$$ = $2;
}
|
UnaryBoolOperator LROUND Formula RROUND {
if ($1 == O_NOT) {
($3)->Negate();
}
$$ = $3;
}
;
ModalOperator: BOX { $$ = O_BOX; }
| DIAMOND { $$ = O_DIAMOND; }
;
BinaryBoolOperator: AND { $$ = O_AND; }
| OR { $$ = O_OR; }
;
UnaryBoolOperator: NOT { $$ = O_NOT; }
;
/*** Code inserted at the and of the file ***/
%%
void yyerror(char *msg)
{
printf("PARSER: %s", msg);
if (yytext[0] != 0)
printf(" near token '%s'\n", yytext);
else
printf("\n");
exit(-1);
}
By adapting it, you will be able to parse fully and recursively a modal logic formula :).
And if later, you want to challenge your solver to existing tableau solver (like Spartacus for example). Dont forget that theses solvers are almost all the time, answering a maximal open Tableau, so they will be for sure faster that you if you want to find a Kripke model of the solution ;)
I hope I manage to help you with your question, I would like to be less theoretical, but I unfortunately don't master python for this :/.
Wish you the best with your solver;
Best Regards.
If you accept my proposition of using the InToHyLo, I worked recently on a Checker of Kripke models for the modal logic K. That you can find here: http://www.cril.univ-artois.fr/~montmirail/mdk-verifier/
It has been published recently in PAAR'2016:
On Checking Kripke Models for Modal Logic K, Jean-Marie Lagniez, Daniel Le Berre, Tiago de Lima, and Valentin Montmirail, Proceedings of the Fifth Workshop on Pratical Aspect of Automated Reasoning (PAAR 2016)