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#include <u.h> #include <libc.h> #include <sat.h>

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struct SATParam { void (*errfun)(char *msg, void *erraux); void *erraux; long (*randfn)(void *randaux); void *randaux; /* + finetuning parameters, see sat.h */ };

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struct SATSolve { SATParam; /* + internals */ };

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SATSolve* satnew(void); void satfree(SATSolve *s); SATSolve* satadd1(SATSolve *s, int *lit, int nlit); SATSolve* sataddv(SATSolve *s, ...); SATSolve* satrange1(SATSolve *s, int *lit, int nlit, int min, int max); SATSolve* satrangev(SATSolve *s, int min, int max, ...); int satsolve(SATSolve *s); int satmore(SATSolve *s); int satval(SATSolve *s, int lit); int satget(SATSolve *s, int i, int *lit, int nlit); void satreset(SATSolve *s);

DESCRIPTION

Libsat is a solver for the boolean satisfiability problem, i.e. given a boolean formula it will either find an assignment to the variables that makes it true, or report that this is impossible. The input formula must be in conjunctive normal form (CNF), i.e. of the form

(x₁ ∨ x₂ ∨ x₃ ∨ …) ∧ (y₁ ∨ y₂ ∨ y ₃ ∨ …) ∧ …,  

where each x_i or y_i can optionally be negated.

For example, consider

(x₁ ∨ x₂ ∨ x₃) ∧ (¬x₁ ∨ ¬x₂) ∧ (¬x₂ ∨ ¬x₃) ∧ (¬x₁ ∨ ¬x₃).  

This formula encodes the constraint that exactly one of the three variables be true. To represent this as input for libsat we assign positive integers to each variable. Negation is represented by the corresponding negative number, hence our example corresponds to the set of "clauses"

1, 2, 3
-1, -2
-1, -3
-2, -3  

To actually solve this problem we would create a SATSolve structure and add clauses one by one using satadd1 or sataddv (the former takes an int array, the latter a variadic list terminated by 0). The SATSolve is modified inplace but returned for convenience. Passing nil as a first argument will create and return a new structure. Alternatively, satnew will create an empty structure.

Once clauses have been added, satsolve will invoke the actual solver. It returns 1 if it found an assignment and 0 if there is no assignment (the formula is unsatisfiable). If an assignment has been found, further clauses can be added to constrain it further and satsolve rerun. Satmore performs this automatically, excluding the current values of the variables. It is equivalent to satsolve if no variables have assigned values.

Once a solution has been found, satval returns the value of literal lit. It returns 1 for true, 0 for false and -1 for undetermined. If the formula is satisfiable, an undetermined variable is one where either value will satisfy the formula. If the formula is unsatisfiable, all variables are undetermined.

Satrange1 and satrangev function like their satadd brethren but rather than adding a single clause they add multiple clauses corresponding to the constraint that at least min and at most max literals from the provided array be true. For example, the clause from above corresponds to

satrangev(s, 1, 1, 1, 2, 3, 0);  

For debugging purposes, clauses can be retrieved using satget. It stores the literals of the clause with index i (starting from 0) at location lit. If there are more than nlit literals, only the first nlit literals are stored. If it was successful, it returns the total number of literals in the clause (which may exceed nlit). Otherwise (if idx was out of bounds) it returns -1.

Satreset resets all solver state, deleting all learned clauses and variable assignments. It retains all user provided clauses. Satfree deletes a solver structure and frees all associated storage.

There are a number of user-adjustable parameters in the SATParam structure embedded in SATSolve. Randfun is called with argument randaux to generate random numbers between 0 and 2³¹-1; it defaults to lrand (see rand(2)). Errfun is called on fatal errors (see DIAGNOSTICS). Additionally, a number of finetuning parameters are defined in sat.h. By tweaking their values, the run-time for a given problem can be reduced.

EXAMPLE

Find all solutions to the example clause from above:

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SATSolve *s; s = nil; s = sataddv(s, 1, 2, 3, 0); s = sataddv(s, -1, -2, 0); s = sataddv(s, -1, -3, 0); s = sataddv(s, -2, -3, 0); while(satmore(s) > 0) print("x1=%d x2=%d x3=%d\\n", satval(s, 1), satval(s, 2), satval(s, 3)); satfree(s);

SOURCE

/sys/src/libsat

SEE ALSO

Donald Knuth, “The Art of Computer Programming”, Volume 4, Fascicle 6.

DIAGNOSTICS

Satnew returns nil on certain fatal error conditions (such as malloc(2) failure). Other routines will call errfun with an error string and erraux. If no errfun is provided or if it returns, sysfatal (see perror(2)) is called. It is permissible to use setjmp(2) to return from an error condition. Call satfree to clean up the SATSolve structure in this case. Note that calling the satadd or satrange routines with nil first argument will invoke sysfatal on error, since no errfun has been defined yet.

BUGS

Variable numbers should be consecutive numbers starting from 1, since variable data is kept in arrays internally.

Large clauses of several thousand literals are probably inefficient and should be split up using auxiliary variables. Very large clauses exceeding about 16,000 literals will not work at all.

There is no way to remove clauses (since it’s unclear what the semantics should be).

The details about the tuning parameters are subject to change.

Calling satadd or satrange after satsolve or satmore may reset variable values.

Satmore will always return 1 when there are no assigned variables in the solution.

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Some debugging routines called under "shouldn’t happen" conditions are non-reentrant.

HISTORY

Libsat first appeared in 9front in March, 2018.