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//! The **RUP** (Reverse Unit Propagation) linear checker — the fast default trust tier.
//!
//! Modern certified SAT separates an untrusted, fast solver from a tiny, auditable checker
//! that replays the solver's resolution proof by unit propagation alone (Goldberg & Novikov,
//! 2003 — RUP; Wetzler/Heule/Hunt, 2014 — DRAT; Cruz-Filipe et al., 2017 — LRAT; Tan/Heule/
//! Myreen, 2021 — the formally-verified `cake_lpr`). A learned clause `C` is valid iff
//! assuming `¬C` and unit-propagating the current clause database hits a conflict; the
//! refutation is verified iff, after adding every learned clause in order, unit propagation
//! over the whole database derives the empty clause.
//!
//! This module re-derives NOTHING symbolic and builds no proof tree — so it sidesteps the
//! emit-cost wall that sank eager probing. The solve produces the learned-clause trace for
//! free ([`crate::cdcl::Solver::learned`]); here we *check* it, independently of the solver.
//! The checker is deliberately small and naive (linear, fixpoint unit propagation) — its
//! simplicity IS the trust. If it cannot confirm the solver's UNSAT, we fail closed.
use crate::cdcl::{Lit, SolveResult, Solver};
use crate::cnf::Cnf;
use crate::ProofExpr;
/// A certified propositional entailment verdict.
#[derive(Clone, Copy, PartialEq, Eq, Debug)]
pub enum Verdict {
/// `premises ⊨ goal`, and an independent RUP replay confirmed the refutation.
Entailed,
/// `premises ⊭ goal` (the solver found a model of `premises ∧ ¬goal`).
NotEntailed,
}
/// Decide `premises ⊨ goal` with the CDCL solver and **independently certify** an
/// `Entailed` verdict by RUP replay. `None` if the problem is not purely propositional over
/// recognisable atoms, OR if the solver claims UNSAT but the checker cannot confirm it (a
/// solver bug — fail closed, never report a false `Entailed`).
pub fn entails_certified(premises: &[ProofExpr], goal: &ProofExpr) -> Option<Verdict> {
let mut cnf = Cnf::new();
for p in premises {
cnf.assert(p)?;
}
certified_from_cnf(cnf, goal)
}
/// Certify `prepared ⊨ goal` against a premise CNF clausified ONCE by
/// [`Cnf::from_premises`]. Solving a whole puzzle this way pays the of-pair Tseitin cost a
/// single time, then each cell only adds its `¬goal` unit — the incremental win.
pub fn entails_certified_prepared(prepared: &Cnf, goal: &ProofExpr) -> Option<Verdict> {
certified_from_cnf(prepared.clone(), goal)
}
/// Shared core: add `¬goal` to `cnf`, solve, and RUP-certify an UNSAT (entailed) verdict.
fn certified_from_cnf(mut cnf: Cnf, goal: &ProofExpr) -> Option<Verdict> {
cnf.assert_neg(goal)?;
let num_vars = cnf.num_vars();
// Hand the clauses straight to the solver (one copy, inside the solver) — the RUP check
// reads them back via `original_clauses()`, so the clause set is never copied again.
let mut solver = cnf.into_solver();
match solver.solve() {
SolveResult::Sat(_) => Some(Verdict::NotEntailed),
SolveResult::Unsat => {
let learned: Vec<Vec<Lit>> =
solver.learned().iter().map(|c| c.lits.clone()).collect();
if check_refutation(num_vars, solver.original_clauses(), &learned) {
Some(Verdict::Entailed)
} else {
None // solver said UNSAT, the trusted checker can't confirm → fail closed
}
}
}
}
/// Verify a DRAT/LRAT-style refutation: every learned clause is RUP w.r.t. the database
/// built so far, and the full database is then refuted by unit propagation (the empty
/// clause is RUP). Returns `false` if any step does not check — so a corrupted or bogus
/// trace is rejected.
pub fn check_refutation(num_vars: usize, original: &[Vec<Lit>], learned: &[Vec<Lit>]) -> bool {
let mut db: Vec<Vec<Lit>> = original.to_vec();
for c in learned {
if !is_rup(num_vars, &db, c) {
return false;
}
db.push(c.clone());
}
// The empty clause is RUP w.r.t. the final DB iff unit propagation alone conflicts.
is_rup(num_vars, &db, &[])
}
/// Is clause `c` derivable by reverse unit propagation from `db`? Assume `¬c` (set each of
/// `c`'s literals false) and unit-propagate `db`; `c` is RUP iff that reaches a conflict.
pub(crate) fn is_rup(num_vars: usize, db: &[Vec<Lit>], c: &[Lit]) -> bool {
let mut assign: Vec<Option<bool>> = vec![None; num_vars];
for &l in c {
// Assume ¬l. If that already clashes, ¬c is unsatisfiable ⇒ c is (trivially) RUP.
if !set_true(&mut assign, l.negated()) {
return true;
}
}
propagate(db, &mut assign)
}
/// The value of literal `l` under `assign`.
#[inline]
pub(crate) fn lit_val(assign: &[Option<bool>], l: Lit) -> Option<bool> {
assign[l.var() as usize].map(|b| if l.is_positive() { b } else { !b })
}
/// Set `l` true. Returns `false` if `l` is already false (a conflict).
#[inline]
pub(crate) fn set_true(assign: &mut [Option<bool>], l: Lit) -> bool {
match lit_val(assign, l) {
Some(true) => true,
Some(false) => false,
None => {
assign[l.var() as usize] = Some(l.is_positive());
true
}
}
}
/// Fixpoint unit propagation over `db`. Returns `true` on conflict (some clause all-false).
/// Naive on purpose: a checker you can read in one sitting is a checker you can trust.
pub(crate) fn propagate(db: &[Vec<Lit>], assign: &mut [Option<bool>]) -> bool {
loop {
let mut changed = false;
for clause in db {
// Evaluate the clause robustly to DUPLICATE literals (`x ∨ x` is the unit `x`) and
// TAUTOLOGIES (`x ∨ ¬x` is always satisfied) — both are legal CNF, and a trusted checker
// must count *distinct* unset literals or it will miss real unit propagations.
let mut satisfied = false;
let mut unset: Vec<Lit> = Vec::new();
for &l in clause {
match lit_val(assign, l) {
Some(true) => {
satisfied = true;
break;
}
Some(false) => {}
None => {
if unset.contains(&l.negated()) {
satisfied = true; // tautology — never propagates, never conflicts
break;
}
if !unset.contains(&l) {
unset.push(l);
}
}
}
}
if satisfied {
continue;
}
if unset.is_empty() {
return true; // every literal false ⇒ conflict
}
if unset.len() == 1 {
set_true(assign, unset[0]);
changed = true;
}
}
if !changed {
return false;
}
}
}
#[cfg(test)]
mod tests {
use super::*;
use crate::cdcl::Lit;
use crate::{ProofExpr, ProofTerm};
fn c(s: &str) -> ProofTerm {
ProofTerm::Constant(s.to_string())
}
fn pred(name: &str, args: Vec<ProofTerm>) -> ProofExpr {
ProofExpr::Predicate { name: name.to_string(), args, world: None }
}
fn not(e: ProofExpr) -> ProofExpr {
ProofExpr::Not(Box::new(e))
}
fn or(a: ProofExpr, b: ProofExpr) -> ProofExpr {
ProofExpr::Or(Box::new(a), Box::new(b))
}
fn imp(a: ProofExpr, b: ProofExpr) -> ProofExpr {
ProofExpr::Implies(Box::new(a), Box::new(b))
}
#[test]
fn certified_disjunctive_syllogism() {
let p = pred("P", vec![c("a")]);
let q = pred("Q", vec![c("a")]);
let prem = vec![or(p.clone(), q.clone()), not(p.clone())];
assert_eq!(entails_certified(&prem, &q), Some(Verdict::Entailed));
assert_eq!(entails_certified(&[or(p, q.clone())], &q), Some(Verdict::NotEntailed));
}
#[test]
fn certified_two_value_grid() {
let in_ = |t: &str, s: &str| pred("In", vec![c(t), c(s)]);
let (fl, me) = ("Florida", "Maine");
let prem = vec![
or(in_("Alpha", fl), in_("Alpha", me)),
or(in_("Beta", fl), in_("Beta", me)),
imp(in_("Alpha", fl), not(in_("Beta", fl))),
imp(in_("Alpha", me), not(in_("Beta", me))),
or(in_("Alpha", fl), in_("Beta", fl)),
or(in_("Alpha", me), in_("Beta", me)),
in_("Alpha", fl),
];
assert_eq!(entails_certified(&prem, &in_("Beta", me)), Some(Verdict::Entailed));
assert_eq!(entails_certified(&prem, &in_("Beta", fl)), Some(Verdict::NotEntailed));
}
#[test]
fn rup_rejects_a_bogus_proof() {
// Original `(p ∨ q)` is SATISFIABLE. A "proof" that bolts on the un-entailed units
// `¬p`, `¬q` must NOT verify — the checker rejects clauses that are not RUP.
let p = Lit::pos(0);
let q = Lit::pos(1);
let original = vec![vec![p, q]];
let bogus_learned = vec![vec![p.negated()], vec![q.negated()]];
assert!(
!check_refutation(2, &original, &bogus_learned),
"a non-RUP learned clause must be rejected"
);
}
#[test]
fn rup_needs_the_proof_not_just_the_clauses() {
// A genuinely UNSAT formula whose unsatisfiability is NOT visible to unit
// propagation alone (all four 2-literal clauses over p,q) must fail to verify
// WITHOUT the solver's learned clauses — proving the checker really replays the
// proof rather than re-searching.
let p = Lit::pos(0);
let q = Lit::pos(1);
let original = vec![
vec![p, q],
vec![p, q.negated()],
vec![p.negated(), q],
vec![p.negated(), q.negated()],
];
assert!(!check_refutation(2, &original, &[]), "UP alone cannot refute this; needs the proof");
// With the resolvents the solver would learn (`p`, then `¬p` ⇒ empty), it checks.
let learned = vec![vec![p], vec![p.negated()]];
assert!(check_refutation(2, &original, &learned), "the learned resolvents make it RUP");
}
}