// weight.h
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//
//
// \file
// General weight set and associated semiring operation definitions.
//
// A semiring is specified by two binary operations Plus and Times and
// two designated elements Zero and One with the following properties:
//   Plus: associative, commutative, and has Zero as its identity.
//   Times: associative and has identity One, distributes w.r.t. Plus, and
//     has Zero as an annihilator:
//          Times(Zero(), a) == Times(a, Zero()) = Zero().
//
//  A left semiring distributes on the left; a right semiring is
//  similarly defined.
//
// A Weight class is required to be (at least) a left or right semiring.
//
// In addition, the following should be defined for a Weight:
//   Member: predicate on set membership.
//   >>: reads textual representation of a weight.
//   <<: prints textual representation of a weight.
//   Read(istream &): reads binary representation of a weight.
//   Write(ostrem &): writes binary representation of a weight.
//   Hash: maps weight to ssize_t.
//   ApproxEqual: approximate equality (for inexact weights)
//   Quantize: quantizes wrt delta (for inexact weights)
//   Divide: for all a,b,c s.t. Times(a, b) == c
//     --> b = Divide(c, a, DIVIDE_LEFT) if a left semiring and b.Member()
//     --> a = Divide(c, b, DIVIDE_RIGHT) if a right semiring and a.Member()
//     --> b = Divide(c, a)
//           = Divide(c, a, DIVIDE_ANY)
//           = Divide(c, a, DIVIDE_LEFT)
//           = Divide(c, a, DIVIDE_RIGHT) if a commutative semiring
//   ReverseWeight: the type of the corresponding reverse weight.
//     Typically the same type as Weight for a (both left and right) semiring.
//     For the left string semiring, it is the right string semiring.
//   Reverse: a mapping from Weight to ReverseWeight s.t.
//     --> Reverse(Reverse(a)) = a
//     --> Reverse(Plus(a, b)) = Plus(Reverse(a), Reverse(b))
//     --> Reverse(Times(a, b)) = Times(Reverse(b), Reverse(a))
//     Typically the identity mapping in a (both left and right) semiring.
//     In the left string semiring, it maps to the reverse string
//     in the right string semiring.
//   Properties: specifies additional properties that hold:
//      LeftSemiring: indicates weights form a left semiring.
//      RightSemiring: indicates weights form a right semiring.
//      TimesCommutative: for all a,b: Times(a,b) == Times(b,a)
//      Idempotent: for all a: Plus(a, a) == a.
//      Path Property: for all a, b: Plus(a, b) == a or Plus(a, b) == b.


#ifndef FST_LIB_WEIGHT_H__
#define FST_LIB_WEIGHT_H__

#include <cctype>
#include <cmath>
#include <iostream>
#include <sstream>

#include "fst/lib/compat.h"

#include "fst/lib/util.h"

namespace fst {

//
// CONSTANT DEFINITIONS
//

// A representable float near .001
const float kDelta =                   1.0F/1024.0F;

// For all a,b,c: Times(c, Plus(a,b)) = Plus(Times(c,a), Times(c, b))
const uint64 kLeftSemiring =           0x0000000000000001ULL;

// For all a,b,c: Times(Plus(a,b), c) = Plus(Times(a,c), Times(b, c))
const uint64 kRightSemiring =          0x0000000000000002ULL;

const uint64 kSemiring = kLeftSemiring | kRightSemiring;

// For all a,b: Times(a,b) = Times(b,a)
const uint64 kCommutative =       0x0000000000000004ULL;

// For all a: Plus(a, a) = a
const uint64 kIdempotent =             0x0000000000000008ULL;

// For all a,b: Plus(a,b) = a or Plus(a,b) = b
const uint64 kPath =                   0x0000000000000010ULL;


// Determines direction of division.
enum DivideType { DIVIDE_LEFT,   // left division
                  DIVIDE_RIGHT,  // right division
                  DIVIDE_ANY };  // division in a commutative semiring

// NATURAL ORDER
//
// By definition:
//                 a <= b iff a + b = a
// The natural order is a monotonic and negative partial order iff the
// semiring is idempotent and (left and right) distributive. It is a
// total order iff the semiring has the path property. See Mohri,
// "Semiring Framework and Algorithms for Shortest-Distance Problems",
// Journal of Automata, Languages and Combinatorics 7(3):321-350,
// 2002. We define the strict version of this order below.

template <class W>
class NaturalLess {
 public:
  typedef W Weight;

  NaturalLess() {
    uint64 props = kIdempotent | kLeftSemiring | kRightSemiring;
    if (W::Properties() & props != props)
      LOG(ERROR) << "NaturalLess: Weight type is not idempotent and "
                 << "(left and right) distributive: " << W::Type();
  }

  bool operator()(const W &w1, const W &w2) const {
    return (Plus(w1, w2) == w1) && w1 != w2;
  }
};

}  // namespace fst;

#endif  // FST_LIB_WEIGHT_H__