1// weight.h 2// 3// Licensed under the Apache License, Version 2.0 (the "License"); 4// you may not use this file except in compliance with the License. 5// You may obtain a copy of the License at 6// 7// http://www.apache.org/licenses/LICENSE-2.0 8// 9// Unless required by applicable law or agreed to in writing, software 10// distributed under the License is distributed on an "AS IS" BASIS, 11// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 12// See the License for the specific language governing permissions and 13// limitations under the License. 14// 15// 16// \file 17// General weight set and associated semiring operation definitions. 18// 19// A semiring is specified by two binary operations Plus and Times and 20// two designated elements Zero and One with the following properties: 21// Plus: associative, commutative, and has Zero as its identity. 22// Times: associative and has identity One, distributes w.r.t. Plus, and 23// has Zero as an annihilator: 24// Times(Zero(), a) == Times(a, Zero()) = Zero(). 25// 26// A left semiring distributes on the left; a right semiring is 27// similarly defined. 28// 29// A Weight class is required to be (at least) a left or right semiring. 30// 31// In addition, the following should be defined for a Weight: 32// Member: predicate on set membership. 33// >>: reads textual representation of a weight. 34// <<: prints textual representation of a weight. 35// Read(istream &): reads binary representation of a weight. 36// Write(ostrem &): writes binary representation of a weight. 37// Hash: maps weight to ssize_t. 38// ApproxEqual: approximate equality (for inexact weights) 39// Quantize: quantizes wrt delta (for inexact weights) 40// Divide: for all a,b,c s.t. Times(a, b) == c 41// --> b = Divide(c, a, DIVIDE_LEFT) if a left semiring and b.Member() 42// --> a = Divide(c, b, DIVIDE_RIGHT) if a right semiring and a.Member() 43// --> b = Divide(c, a) 44// = Divide(c, a, DIVIDE_ANY) 45// = Divide(c, a, DIVIDE_LEFT) 46// = Divide(c, a, DIVIDE_RIGHT) if a commutative semiring 47// ReverseWeight: the type of the corresponding reverse weight. 48// Typically the same type as Weight for a (both left and right) semiring. 49// For the left string semiring, it is the right string semiring. 50// Reverse: a mapping from Weight to ReverseWeight s.t. 51// --> Reverse(Reverse(a)) = a 52// --> Reverse(Plus(a, b)) = Plus(Reverse(a), Reverse(b)) 53// --> Reverse(Times(a, b)) = Times(Reverse(b), Reverse(a)) 54// Typically the identity mapping in a (both left and right) semiring. 55// In the left string semiring, it maps to the reverse string 56// in the right string semiring. 57// Properties: specifies additional properties that hold: 58// LeftSemiring: indicates weights form a left semiring. 59// RightSemiring: indicates weights form a right semiring. 60// TimesCommutative: for all a,b: Times(a,b) == Times(b,a) 61// Idempotent: for all a: Plus(a, a) == a. 62// Path Property: for all a, b: Plus(a, b) == a or Plus(a, b) == b. 63 64 65#ifndef FST_LIB_WEIGHT_H__ 66#define FST_LIB_WEIGHT_H__ 67 68#include <cctype> 69#include <cmath> 70#include <iostream> 71#include <sstream> 72 73#include "fst/lib/compat.h" 74 75#include "fst/lib/util.h" 76 77namespace fst { 78 79// 80// CONSTANT DEFINITIONS 81// 82 83// A representable float near .001 84const float kDelta = 1.0F/1024.0F; 85 86// For all a,b,c: Times(c, Plus(a,b)) = Plus(Times(c,a), Times(c, b)) 87const uint64 kLeftSemiring = 0x0000000000000001ULL; 88 89// For all a,b,c: Times(Plus(a,b), c) = Plus(Times(a,c), Times(b, c)) 90const uint64 kRightSemiring = 0x0000000000000002ULL; 91 92const uint64 kSemiring = kLeftSemiring | kRightSemiring; 93 94// For all a,b: Times(a,b) = Times(b,a) 95const uint64 kCommutative = 0x0000000000000004ULL; 96 97// For all a: Plus(a, a) = a 98const uint64 kIdempotent = 0x0000000000000008ULL; 99 100// For all a,b: Plus(a,b) = a or Plus(a,b) = b 101const uint64 kPath = 0x0000000000000010ULL; 102 103 104// Determines direction of division. 105enum DivideType { DIVIDE_LEFT, // left division 106 DIVIDE_RIGHT, // right division 107 DIVIDE_ANY }; // division in a commutative semiring 108 109// NATURAL ORDER 110// 111// By definition: 112// a <= b iff a + b = a 113// The natural order is a monotonic and negative partial order iff the 114// semiring is idempotent and (left and right) distributive. It is a 115// total order iff the semiring has the path property. See Mohri, 116// "Semiring Framework and Algorithms for Shortest-Distance Problems", 117// Journal of Automata, Languages and Combinatorics 7(3):321-350, 118// 2002. We define the strict version of this order below. 119 120template <class W> 121class NaturalLess { 122 public: 123 typedef W Weight; 124 125 NaturalLess() { 126 uint64 props = kIdempotent | kLeftSemiring | kRightSemiring; 127 if (W::Properties() & props != props) 128 LOG(ERROR) << "NaturalLess: Weight type is not idempotent and " 129 << "(left and right) distributive: " << W::Type(); 130 } 131 132 bool operator()(const W &w1, const W &w2) const { 133 return (Plus(w1, w2) == w1) && w1 != w2; 134 } 135}; 136 137} // namespace fst; 138 139#endif // FST_LIB_WEIGHT_H__ 140