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0-1 law holds Ackermann class arities asymptotic probabilities atom(<p atomic formulas Bernays-Schonfinkel binomial distribution bounded vocabularies class E}(Ackermann class without equality clause composite symbols configuration consistent with t0 contain t0 containing the restriction countable random structure decision problem denote the formula encode existential second-order existential type existentially quantified exponential exponential hierarchy extending the type Fagin fi(P finite graphs finite satisfiability problem finite structures first-order logic first-order quantifier first-order sentences follows fragments Godel class infinitary logic input head input tape input(y Lemma lower bound NEXPTIME nondeterministic nondeterministic Turing machine nonrejecting computation NP-complete number of elements partial types predicate symbols prefix classes probability space Proof properly extending Proposition PSPACE-complete quantified variable quantifier-free formula restriction of t0 S)-existential type S)-type S)-witness second-order logic second-order sentences string subset tape of length Theorem truth values type t'(x unary relation underlying vocabulary universal quantifiers upper bounds wm,wm+i wm+i worktape xjt,y