# FST optimization

There are several ways to "optimize" a weighted finite-state transducer (WFST). The C++ template function `fst::Optimize` underlies Pynini's `optimize` instance method and Thrax's `Optimize` function. But what exactly does this make more optimal [sic]? How does it accomplish that? And what, if anything, can we assert about the resulting transducer?

## What makes an WFST suboptimal...

Tolstoy wrote that "All happy families are alike; each unhappy family is unhappy in its own way", and that's true of WFSTs too.

• WFSTs may have identically labeled multi-arcs, arcs with the same source state, destination state, input label, and output-label. Arc-sum mapping merges such arcs using ⊕ to combine their weights, making the WFST multi-arc-free.
• WFST operations (including composition, concatenation, and union) introduce epsilon-transitions, arcs with epsilon input and output labels (e.g., all but two arcs here). Epsilon-removal removes these transitions.
• A WFST is deterministic if for each state, there is never more than one arc with a given label leaving that state (e.g., state 1 here). Determinization makes an WFST deterministic. While this can actually increase the size of the WFST, it is a prerequisite for the next algorithm.
• A WFST is minimal if it has the minimum number of states possible for expressing its relation. Minimization modifies a deterministic WFST's topology so that it becomes minimal (e.g., the examples here).

## ...what we can do about it...

`fst::Optimize` checks whether the WFST is known to be epsilon-free; if not, epsilon-removal is performed. Then, arc-sum mapping is performed. Then, it attempts to perform minimization, determinizing along the way, if necessary, as follows. If the WFST is (known to be) an acyclic acceptor but not (known to be) deterministic, we determinize it, then minimize it. This works because any acyclic transducer (over a zero-sum-free semiring) is determinizable (Mohri 2009).

However, if the WFST is a transducer, it may be non-functional (i.e., it describes a one-to-many relation); non-functional transducers can be determinized, but this is quite inefficient. And, if the WFST has weighted cycles (whether or not it's an acceptor or a transducer), it may lack the twins property, in which case the determinization algorithm will never terminate. However, in such a case, it is still usually possible to reduce the number of arcs and states in the WFST (Allauzen et al. 2004): the WFST is determinized and minimized as if it were an acceptor (and if the WFST has weighted cycles, as if it were an unweighted acceptor). (Alternatively, we could have checked for the twins property and/or for non-functionalism, but both checks are themselves quite expensive, so we have chosen not to.) In that case, it is also necessary to perform another round of arc-sum mapping as the aforementioned trick may may introduce new multi-arcs.

## ...and what we can assert about an optimized transducer

Once we have performed optimization on a WFST, it is free of multi-arcs and unnecessary epsilon-epsilon arcs. And, if the WFST was an acyclic acceptor, it is also deterministic and minimal.

We cannot be sure that it is smaller (in terms of the number of states and arcs) than the input.

## Other behind-the-scenes optimizations

Pynini and Thrax silently perform two other types of WFST optimization. A cross-product transducer—what you get when you do `t("foo", "bar")` in Pynini or `"foo" : "bar"` in Thrax—is constructed as follows:

• Replace the output labels of the left-hand side argument with epsilons,
• replace the input labels of the right-hand side argument with epsilons, and
• concatenate the result.

When both arguments are in fact strings represented as WFSTs, this will produce as much as twice as many states and arcs as strictly necessary. Pynini and Thrax therefore perform label-pushing and epsilon-removal to remove these unnecessary arcs and states. (If either of the components in the cross-product isn't a string, this isn't guaranteed to help, so we don't bother.)

When computing the difference of two WFSTs, the right-hand side must be epsilon-free and deterministic. If it is not known to have these properties, Thrax and Pynini perform epsilon-removal and attempt determinization.

## References

Allauzen, C., Mohri, M., Riley, M., and Roark, B. 2004. A generalized construction of integrated speech recognition transducers. In ICASSP, 761-764.

Mohri, M. 2009. Weighted automata algorithms. In M. Droste, W. Kuich, and H. Vogler, Handbook of weighted automata, pages 213-254. New York: Springer.

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Topic revision: r2 - 2017-07-05 - KyleGorman

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