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Push

Description

This operation produces an equivalent transducer by pushing the weights and/or the labels towards the initial state or toward the final states.

Weight pushing

When pushing weights towards the initial state, the sum of the weight of the outgoing transitions and final weight at any non-initial state is equal to $1$ in the resulting machine. When pushing weights towards the final states, the sum of the weight of the incoming transitions at any state is equal to $1$.

Weight needs to be left distributive when pushing towards the initial state and right distributive when pushing towards the final states.

Label pushing

Pushing labels towards the initial state consists in minimizing at every state the length of the longest common prefix of the output labels of the outgoing paths. Pushing labels towards the final states consists in minimizing at every state the length of the longest common suffix of the output labels of the incoming paths.

Usage

const uint32 kPushWeights = 0x0001;
const uint32 kPushLabels =  0x0002;

enum ReweightType { REWEIGHT_TO_INITIAL, REWEIGHT_TO_FINAL };

template <class Arc, ReweightType rtype>
void Push(const Fst<Arc> &ifst, MutableFst<Arc> *ofst, uint32 ptype);
fstpush [--opts] a.fst out.fst
    --to_final: type = bool, default = false
      Push/reweight to final (vs. to initial) states
    --push_labels: type = bool, default = false
      Push output labels
    --push_weights: type = bool, default = false
      Push weights

Examples

A:

Input FST A.

Push weights of A to initial state:

Result of pushing weights to the initial state.

Push<Arc, REWEIGHT_TO_INITIAL>(A, &B, kPushWeights);
fstpush --push_weights a.fst out.fst

Push labels of A to initial state:

Result of pushing labels to the initial state.

Push<Arc, REWEIGHT_TO_INITIAL>(A, &B, kPushLabels);
fstpush --push_labels a.fst out.fst

Push weights and labels of A to final states

Result of pushing weights and labels to the final state.

Push<Arc, REWEIGHT_TO_FINAL>(A, &B, kPushWeights|kPushLabels);
fstpush --push_weights --push_labels --to_final a.fst out.fst

Complexity

Push:

  • Weight pushing:

  • Time:

  • Acyclic: $O(V + E)$

  • Tropical semiring: $O(V \log V + E)$

  • General: exponential

  • Space: $O(V + E)$

  • Label pushing:

  • Time: polynomial

  • Space: $O(V^2 + E)$

where $V$ = # of states and $E$ = # of arcs.