Abstract
An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.
Highlights
We will use formulas F initeAntichainSubset(X, Y ) and F initeChoice(X, x) to define a choice function by an Monadic Second-Order Logic (MSO)-formula Choice(X, x) which is the conjunction of the following conditions: (1) ∃Z : “Z is the set of ≤-minimal elements in X” (2) ∃Y : F initeAntichainSubset(Z, Y ) (3) F initeChoice(Y, x)
We proved that the ambiguity hierarchy is strict for regular languages over infinite trees
For each level of the ambiguity hierarchy we provided a language which occupies this level
Summary
A recent paper [HSS17] surveys works on the degree of ambiguity and on various nondeterminism measures for finite automata on words. Every regular language is accepted by a deterministic automaton. Every regular language is accepted by a deterministic bottom-up tree automaton and by an unambiguous top-down tree automaton. Over ω-words every regular language is accepted by an unambiguous Buchi automaton [Arn83] and by a deterministic parity automaton. (2) L∃a1 := {t ∈ TΣω1 | there exists an a1-labeled node in t} This is a countably ambiguous language that is not finitely ambiguous (see Section 4). (3) Lno−max −a1 := {t ∈ TΣω1 | above every a1-labeled node in t there is an a1-labeled node} This is an uncountably ambiguous language that is not countably ambiguous (see Section 7). We added the proofs that were sketched or missing in [RT20], presented natural examples of uncountably ambiguous languages (in Section 7), and added Section 8 in which we prove that countable languages are unambiguous using Niwinski’s Representation for Countable Languages
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