A Rust library for SAT solving and automated theorem proving. This package is a work in progress, but the following are supported:

1. Datatypes/parsing/printing operations
2. Eval
3. Standard DNF/CNF algorithms,
4. Definitional CNF (preserving equisatisfiability) for propositional logic.
5. The classic DP and DPLL algorithms for testing satisfiability
6. Basic Iterative DPLL as well as Backjumping/Conflict clause learning.
7. Prenex normal form
8. Skolemization
9. A semi-decision procedure for first-order validity checking (Gilmore and Davis-Putnam)

To run the Jupyter notebook README.ipynb, you will need both Jupyter https://jupyter-notebook.readthedocs.io/en/stable/

and a Jupyter Rust kernel, e.g. https://github.com/evcxr/evcxr/blob/main/evcxr_jupyter/README.md.

README.md is generated from README.ipynb by running jupyter nbconvert --execute --to markdown README.ipynb

Acknowlegement: This library was highly informed by John Harrison's text on Automated Theorem Proving (which uses Ocaml).

(Harrison, J. (2009). Handbook of Practical Logic and Automated Reasoning. Cambridge: Cambridge University Press)

:dep harrison_rust = {path = "."}

use std::collections::{HashMap, HashSet};
use std::io::stdout;
use std::path::Path;

use harrison_rust::first_order_logic::{FOValuation, Interpretation, Language, Pred};
use harrison_rust::formula::{DPLISolver, Formula};
use harrison_rust::propositional_logic::Prop;
use harrison_rust::sudoku::{get_board_formulas, parse_sudoku_dataset, Board};
use harrison_rust::utils::run_and_time;

Example 1: Simple formula

let formula = Formula::<Prop>::parse("C \\/ D <=> (~A /\\ B)").unwrap();
formula.pprint();
println!("{}", formula.get_truthtable());
let cnf = Formula::to_cnf(&formula);
cnf.pprint();

println!("Is satisfiable?: {}", formula.dpll_sat());
println!("Is tautology?: {}", formula.dpll_taut());

C \/ D <=> ~A /\ B





A     B     C     D     | formula


---------------------------------


true  true  true  true  | false


true  true  true  false | false


true  true  false true  | false


true  true  false false | true


true  false true  true  | false


true  false true  false | false


true  false false true  | false


true  false false false | true


false true  true  true  | true


false true  true  false | true


false true  false true  | true


false true  false false | false


false false true  true  | false


false false true  false | false


false false false true  | false


false false false false | true


---------------------------------





((((((A \/ C) \/ D) \/ ~B) /\ (B \/ ~C)) /\ (B \/ ~D)) /\ (~A \/ ~C)) /\ (~A \/ ~D)





Is satisfiable?: true


Is tautology?: false

Example 2: A Tautology

let formula = Formula::<Prop>::parse("A \\/ ~A").unwrap();
formula.pprint();
println!("{}", formula.get_truthtable());

println!("Is satisfiable?: {}", formula.dpll_sat());
println!("Is tautology?: {}", formula.dpll_taut());

A \/ ~A





A     | formula


---------------


true  | true


false | true


---------------





Is satisfiable?: true


Is tautology?: true

Example 3: A Contradiction

let formula = Formula::<Prop>::parse("~A /\\ A").unwrap();
formula.pprint();
println!("{}", formula.get_truthtable());

println!("Is satisfiable?: {}", formula.dpll_sat());
println!("Is tautology?: {}", formula.dpll_taut());
println!("Is contradiction?: {}", Formula::not(&formula).dpll_taut());

~A /\ A





A     | formula


---------------


true  | false


false | false


---------------





Is satisfiable?: false


Is tautology?: false


Is contradiction?: true

Example 4: Formula simplification

let formula =
    Formula::<Prop>::parse("((true ==> (x <=> false)) ==> ~(y \\/ (false /\\ z)))").unwrap();
formula.pprint();
println!("...simplifies to...");
let simplified = formula.simplify();

simplified.pprint();

let formula = Formula::<Pred>::parse(
    "forall x. ((true ==> (R(x) <=> false)) ==> exists z. exists y. ~(K(y) \\/ false))",
).unwrap();
formula.pprint();
println!("...simplifies to...");
let simplified = formula.simplify();

simplified.pprint();

(true ==> (x <=> false)) ==> ~(y \/ false /\ z)





...simplifies to...


~x ==> ~y





forall x. ((true ==> (R(x) <=> false)) ==> (exists z y. ~(K(y) \/ false)))


...simplifies to...


forall x. (~R(x) ==> (exists y. ~K(y)))

Example 5: Solve a hard sudoku board

let path_str: &str = "./data/sudoku.txt";
let path: &Path = Path::new(path_str);
let boards: Vec<Board> = parse_sudoku_dataset(path, Some(1));
let clauses = get_board_formulas(&boards, 9, 3)[0].clone();
let mut solver = DPLISolver::new(&clauses);
let num_props = solver.num_props();
println!("(Sukoku sentence has {num_props} propositional variables)");
let is_sat = run_and_time(|| solver.dplb_solve());
println!("Is satisfiable?: {is_sat}");

let formula = Formula::formulaset_to_cnf_formula(clauses);
let check = formula.eval(&solver.get_valuation().unwrap());
println!("Check: Solution satisfies original constraints?: {check}");

(Sukoku sentence has 729 propositional variables)


Run time is 5.664883334s.





Is satisfiable?: true


Check: Solution satisfies original constraints?: true

Example 6: Arithmetic mod n (n >= 2)

fn integers_mod_n(n: u32) -> Interpretation<u32> {
    assert!(n > 1);

    type FuncType = dyn Fn(&[u32]) -> u32;
    type RelType = dyn Fn(&[u32]) -> bool;

    let lang = Language {
        func: HashMap::from([
            ("+".to_string(), 2),
            ("*".to_string(), 2),
            ("0".to_string(), 0),
            ("1".to_string(), 0),
        ]),
        rel: HashMap::from([("=".to_string(), 2)]),
    };

    let domain: HashSet<u32> = HashSet::from_iter(0..n);

    let addition = move |inputs: &[u32]| -> u32 { (inputs[0] + inputs[1]) % n };
    let multiplication = move |inputs: &[u32]| -> u32 { (inputs[0] * inputs[1]) % n };
    let zero = |_inputs: &[u32]| -> u32 { 0 };
    let one = |_inputs: &[u32]| -> u32 { 1 };

    fn equality(inputs: &[u32]) -> bool {
        inputs[0] == inputs[1]
    }

    let funcs: HashMap<String, Box<FuncType>> = HashMap::from([
        ("+".to_string(), Box::new(addition) as Box<FuncType>),
        ("*".to_string(), Box::new(multiplication) as Box<FuncType>),
        ("0".to_string(), Box::new(zero) as Box<FuncType>),
        ("1".to_string(), Box::new(one) as Box<FuncType>),
    ]);
    let rels: HashMap<String, Box<RelType>> =
        HashMap::from([("=".to_string(), Box::new(equality) as Box<RelType>)]);

    Interpretation::new(&lang, domain, funcs, rels)
}

// Let's verify (for n < 20) that the integers mod n form a field
// (have multiplicative inverses) if and only if n is prime.
let mult_inverse = "forall x. (~(x = 0) ==> exists y. x * y = 1)";
let mult_inverse_formula = Formula::<Pred>::parse(mult_inverse).unwrap();
println!("Definition of multiplicative inverses:");
mult_inverse_formula.pprint();

let empty_valuation = FOValuation::new();
println!("Model:         |  Is a field?");
for n in 2..20 {
    let interpretation = integers_mod_n(n);
    let sat = mult_inverse_formula.eval(&interpretation, &empty_valuation);
    println!("Integers mod {n}:  {sat}");
}

Definition of multiplicative inverses:


forall x. (~x = 0 ==> (exists y. x * y = 1))


Model:         |  Is a field?


Integers mod 2:  true


Integers mod 3:  true


Integers mod 4:  false


Integers mod 5:  true


Integers mod 6:  false


Integers mod 7:  true


Integers mod 8:  false


Integers mod 9:  false


Integers mod 10:  false


Integers mod 11:  true


Integers mod 12:  false


Integers mod 13:  true


Integers mod 14:  false


Integers mod 15:  false


Integers mod 16:  false


Integers mod 17:  true


Integers mod 18:  false


Integers mod 19:  true

Example 7: Prenex Normal Form

let formula =
    Formula::<Pred>::parse("(exists x. F(x, z)) ==> (exists w. forall z. ~G(z, x))").unwrap();
formula.pprint();
println!("In prenex normal form:");
let result = formula.pnf();
result.pprint();

(exists x. F(x, z)) ==> (exists w. (forall z. ~G(z, x)))


In prenex normal form:


forall x' z'. (~F(x', z) \/ ~G(z', x))

Example 8: Skolemization

let formula = Formula::<Pred>::parse(
    "R(F(y)) \\/ (exists x. P(f_w(x))) /\\ exists n. forall r. forall y. exists w. M(G(y, w)) 
    \\/ exists z. ~M(F(z, w))",
)
.unwrap();
formula.pprint();
println!("Skolemized:");
let result = formula.skolemize();
result.pprint();

R(F(y)) \/ (exists x. P(f_w(x))) /\ (exists n. (forall r y. (exists w. M(G(y, w))))) \/ (exists z. ~M(F(z, w)))


Skolemized:


R(F(y)) \/ P(f_w(c_x)) /\ M(G(y', f_w'(y'))) \/ ~M(F(f_z(w), w))

Example 9: Test a first order formula for validity (valid formula)

let string = "(forall x y. exists z. forall w. (P(x) /\\ Q(y) ==> R(z) /\\ U(w))) 
    ==> (exists x y. (P(x) /\\ Q(y))) ==> (exists z. R(z))";
let formula = Formula::<Pred>::parse(string).unwrap();
let compute_unsat_core = true;
let max_depth = 10;
Formula::davis_putnam(&formula, compute_unsat_core, max_depth);

Skolemized negation is ((~P(x) \/ ~Q(y)) \/ R(f_z(x, y)) /\ U(w)) /\ (P(c_x) /\ Q(c_y)) /\ ~R(x)


Generating tuples for next level 0


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, c_x)) \\/ ~P(c_x)) \\/ ~Q(c_x))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(c_x))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, c_y)) \\/ ~P(c_x)) \\/ ~Q(c_y))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(c_y))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, c_x)) \\/ ~P(c_y)) \\/ ~Q(c_x))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(c_x))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, c_y)) \\/ ~P(c_y)) \\/ ~Q(c_y))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(c_y))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, c_x)) \\/ ~P(c_x)) \\/ ~Q(c_x))) /\\ ((U(c_y) \\/ ~P(c_x)) \\/ ~Q(c_x))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, c_y)) \\/ ~P(c_x)) \\/ ~Q(c_y))) /\\ ((U(c_y) \\/ ~P(c_x)) \\/ ~Q(c_y))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, c_x)) \\/ ~P(c_y)) \\/ ~Q(c_x))) /\\ ((U(c_y) \\/ ~P(c_y)) \\/ ~Q(c_x))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, c_y)) \\/ ~P(c_y)) \\/ ~Q(c_y))) /\\ ((U(c_y) \\/ ~P(c_y)) \\/ ~Q(c_y))) /\\ ~R(c_y)"


Generating tuples for next level 1


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, f_z(c_x, c_x))) \\/ ~P(c_x)) \\/ ~Q(f_z(c_x, c_x)))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(f_z(c_x, c_x)))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, f_z(c_x, c_y))) \\/ ~P(c_x)) \\/ ~Q(f_z(c_x, c_y)))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(f_z(c_x, c_y)))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, f_z(c_y, c_x))) \\/ ~P(c_x)) \\/ ~Q(f_z(c_y, c_x)))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(f_z(c_y, c_x)))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_x, f_z(c_y, c_y))) \\/ ~P(c_x)) \\/ ~Q(f_z(c_y, c_y)))) /\\ ((U(c_x) \\/ ~P(c_x)) \\/ ~Q(f_z(c_y, c_y)))) /\\ ~R(c_x)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, f_z(c_x, c_x))) \\/ ~P(c_y)) \\/ ~Q(f_z(c_x, c_x)))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(f_z(c_x, c_x)))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, f_z(c_x, c_y))) \\/ ~P(c_y)) \\/ ~Q(f_z(c_x, c_y)))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(f_z(c_x, c_y)))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, f_z(c_y, c_x))) \\/ ~P(c_y)) \\/ ~Q(f_z(c_y, c_x)))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(f_z(c_y, c_x)))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(c_y, f_z(c_y, c_y))) \\/ ~P(c_y)) \\/ ~Q(f_z(c_y, c_y)))) /\\ ((U(c_x) \\/ ~P(c_y)) \\/ ~Q(f_z(c_y, c_y)))) /\\ ~R(c_y)"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(f_z(c_x, c_x), c_x)) \\/ ~P(f_z(c_x, c_x))) \\/ ~Q(c_x))) /\\ ((U(c_x) \\/ ~P(f_z(c_x, c_x))) \\/ ~Q(c_x))) /\\ ~R(f_z(c_x, c_x))"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(f_z(c_x, c_x), c_y)) \\/ ~P(f_z(c_x, c_x))) \\/ ~Q(c_y))) /\\ ((U(c_x) \\/ ~P(f_z(c_x, c_x))) \\/ ~Q(c_y))) /\\ ~R(f_z(c_x, c_x))"


Adding new formula to set: "(((P(c_x) /\\ Q(c_y)) /\\ ((R(f_z(f_z(c_x, c_y), c_x)) \\/ ~P(f_z(c_x, c_y))) \\/ ~Q(c_x))) /\\ ((U(c_x) \\/ ~P(f_z(c_x, c_y))) \\/ ~Q(c_x))) /\\ ~R(f_z(c_x, c_y))"


Found 2 inconsistent ground instances of skolemized negation:


(((P(c_x) /\ Q(c_y)) /\ ((R(f_z(f_z(c_x, c_y), c_x)) \/ ~P(f_z(c_x, c_y))) \/ ~Q(c_x))) /\ ((U(c_x) \/ ~P(f_z(c_x, c_y))) \/ ~Q(c_x))) /\ ~R(f_z(c_x, c_y))


(((P(c_x) /\ Q(c_y)) /\ ((R(f_z(c_x, c_y)) \/ ~P(c_x)) \/ ~Q(c_y))) /\ ((U(c_y) \/ ~P(c_x)) \/ ~Q(c_y))) /\ ~R(c_x)


Formula is valid.

Example 10: Test a first order formula for validity (invalid formula)

let string = "forall boy. exists girl. (Loves(girl, friend(boy)))";
let formula = Formula::<Pred>::parse(string).unwrap();
let compute_unsat_core = true;
let max_depth = 20;
let result = Formula::davis_putnam(&formula, compute_unsat_core, max_depth);
println!("{:?}", result);

Skolemized negation is ~Loves(girl, friend(c_boy))


Generating tuples for next level 0


Adding new formula to set: "~Loves(c_boy, friend(c_boy))"


Generating tuples for next level 1


Adding new formula to set: "~Loves(friend(c_boy), friend(c_boy))"


Generating tuples for next level 2


Adding new formula to set: "~Loves(friend(friend(c_boy)), friend(c_boy))"


Generating tuples for next level 3


Adding new formula to set: "~Loves(friend(friend(friend(c_boy))), friend(c_boy))"


Generating tuples for next level 4


Adding new formula to set: "~Loves(friend(friend(friend(friend(c_boy)))), friend(c_boy))"


Generating tuples for next level 5


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(c_boy))))), friend(c_boy))"


Generating tuples for next level 6


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(c_boy)))))), friend(c_boy))"


Generating tuples for next level 7


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(c_boy))))))), friend(c_boy))"


Generating tuples for next level 8


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))), friend(c_boy))"


Generating tuples for next level 9


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))), friend(c_boy))"


Generating tuples for next level 10


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))), friend(c_boy))"


Generating tuples for next level 11


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))))), friend(c_boy))"


Generating tuples for next level 12


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))))), friend(c_boy))"


Generating tuples for next level 13


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))))))), friend(c_boy))"


Generating tuples for next level 14


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))))))), friend(c_boy))"


Generating tuples for next level 15


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))))))))), friend(c_boy))"


Generating tuples for next level 16


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))))))))), friend(c_boy))"


Generating tuples for next level 17


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))))))))))), friend(c_boy))"


Generating tuples for next level 18


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))))))))))), friend(c_boy))"


Generating tuples for next level 19


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy))))))))))))))))))), friend(c_boy))"


Generating tuples for next level 20


Adding new formula to set: "~Loves(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(friend(c_boy)))))))))))))))))))), friend(c_boy))"


After searching to bound depth 20, set of ground instances (of negation) is still satisfiable. Giving up.


BoundReached(20)