## Restriction Operator Error: Technical Note

The implementation of the restriction operator,
=>, in the Book Version of xfst is not
correct for restrictions that involve alternative contexts. For
example, the expression

```
a => b _ c , d
_ e
```

describes a language in which any instance of a occurs either in
the context b_c or in the
context d_e. The section on
Restriction
says correctly on page 65 that the above expression

is
equivalent to

```
[ [a => b _ c
]
| [ a => d _ e ] ]*
```

These two expressions are indeed equivalent in the Book version of xfst
and the text suggests
that the same would be true for more complex cases. But the equivalence
fails in more complicated cases where the language being constrained
partially overlaps with a context. For example, in the Book
version of xfst

(a) b (c) => x
(a) _ , _ (c) y

is not equivalent to

[
[(a) b (c) => x (a) _ ] | [(a) b (c) => _ (c)
y] ]*

due to an error in the compilation algorithm for the restriction
operator. The mistake was pointed out by Anssi Yli-Jyrä
of the University of Helsinki, who generously provided us his own
algorithm for compiling restriction expressions with multiple contexts.
Yli-Jyrä's elegant and correct compilation algorithm for =>
expressions is implemented in xfst-8.3.3 and subsequent
versions. The last two expressions are in fact equivalent and now
compile into identical networks. But this is a fortuitous coincidence.
The equivalence is due to the fact that a, b, c, d, e, x, and y in these examples are single
symbols. In the general case, where the symbols may denote arbitrary
languages with longer strings, the corresponding pairs of expressions
are not equivalent. Correct handling of restrictions with multiple
contexts requires a more complicated algorithm that xfst
uses internally. See "Compiling Contextual Restrictions on Strings
into Finite-State Automata" by Anssi Yli-Jyrä and Kimmo
Koskenniemi, in the Post-proceedings of Eindhoven Fastar Days,
September 3-4, 2004 (forthcoming).

*Last Modified: Thursday, 28-Oct-2004 18:01:51 PDT
*