Boolean.h

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/**
 * @file semiring/Boolean.h
 * @brief Boolean semiring ({false, true}, ∨, ∧, false, true).
 *
 * The Boolean semiring is the simplest semiring supported by ProvSQL.
 * Provenance evaluates to @c true if at least one derivation of the
 * tuple exists in the current database instance (i.e., the query answer
 * is "certain"), and @c false otherwise.
 *
 * The semiring is absorptive (∨ is idempotent), so the circuit evaluator
 * can safely deduplicate children of OR gates.
 *
 * Operations:
 * - @c zero()   → @c false
 * - @c one()    → @c true
 * - @c plus()   → logical OR (any_of)
 * - @c times()  → logical AND (all_of)
 * - @c monus()  → @f$x \;\&\; \lnot y@f$
 * - @c delta()  → identity
 *
 * @see https://provsql.org/lean-docs/Provenance/Semirings/Bool.html
 *      Lean 4 verified instance: @c instSemiringWithMonusBool, with
 *      proofs of @c Bool.absorptive, @c Bool.idempotent, and
 *      @c Bool.mul_sub_left_distributive.
 */
#ifndef BOOLEAN_H
#define BOOLEAN_H
 
#include <algorithm>
#include <vector>
 
#include "Semiring.h"
 
namespace semiring {
/**
 * @brief The Boolean semiring over @c bool.
 *
 * Provides the standard Boolean interpretation of provenance circuits.
 */
class Boolean : public semiring::Semiring<bool>
{
public:
virtual value_type zero() const override {
  return false;
}
virtual value_type one() const override {
  return true;
}
virtual value_type plus(const std::vector<value_type> &v) const override {
  return std::any_of(v.begin(), v.end(), [](bool x) {
        return x;
      });
}
virtual value_type times(const std::vector<value_type> &v) const override {
  return std::all_of(v.begin(), v.end(), [](bool x) {
        return x;
      });
}
virtual value_type monus(value_type x, value_type y) const override
{
  return x & !y;
}
virtual value_type delta(value_type x) const override
{
  return x;
}
virtual bool absorptive() const override {
  return true;
}
/**
 * @brief The identity map @c BoolFunc(X) →+* Bool (evaluating a free
 *        Boolean function at a valuation) is an m-semiring
 *        homomorphism, so the safe-query Boolean rewrite preserves
 *        evaluation results in this semiring.
 *
 * Lean: @c Provenance.Semirings.Bool.homomorphism_to_BoolFunc and
 * @c Bool.homomorphism_from_BoolFunc
 * (provenance-lean/Provenance/Semirings/Bool.lean).
 */
virtual bool compatibleWithBooleanRewrite() const override {
  return true;
}
value_type parse_leaf(const char *v) const {
  return *v != 'f' && *v != '0';
}
};
}
 
#endif /* BOOLEAN_H */