Abstract
We investigate the expressive power of existential second-order formulas whose second-order quantifiers range over bijective unary functions. We show that as long as interpretations are taken over structures with a built-in linear order relation and an addition function, then quantifying over bijections is as expressive as quantifying over arbitrary unary functions. The originality of our result is that it remains true even if the first-order part of a formula contains exactly one variable (which is universally quantified). Our result immediately provides a new characterization of non-deterministic linear time on RAMs. It also permits us to derive a corollary on the Skolem normal form of first-order formulas over weakly arithmetized structures.
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Ailloud, É., Durand, A. The Expressive Power of Bijections over Weakly Arithmetized Structures. Theory Comput Syst 39, 297–309 (2006). https://doi.org/10.1007/s00224-005-1165-y
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DOI: https://doi.org/10.1007/s00224-005-1165-y