Universal quantification makes automatic structures hard to decide
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Automatic structures are structures whose universe and relations can be represented as regular languages. It follows from the standard closure properties of regular languages that the first-order theory of an automatic structure is decidable. While existential quantifiers can be eliminated in linear time by application of a homomorphism, universal quantifiers are commonly eliminated via the identity ∀ x . Φ≡ (∃ x . Φ). If Φ is represented in the standard way as an NFA, a priori this approach results in a doubly exponential blow-up. However, the recent literature has shown that there are classes of automatic structures for which universal quantifiers can be eliminated by different means without this blow-up by treating them as first-class citizens and not resorting to double complementation. While existing lower bounds for some classes of automatic structures show that a singly exponential blow-up is unavoidable when eliminating a universal quantifier, it is not known whether there may be better approaches that avoid the naïve doubly exponential blow-up, perhaps at least in restricted settings. In this paper, we answer this question negatively and show that there is a family of NFA representing automatic relations for which the minimal NFA recognising the language after eliminating a single universal quantifier is doubly exponential, and deciding whether this language is empty is ExpSpace-complete.
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