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Expressiveness and Succinctness of First-Order Logic on Finite Words

Expressiveness, and more recently, succinctness, are two central concerns of finite model theory and descriptive complexity theory. Succinctness is particularly interesting because it is closely related to the complexity-theoretic trade-off between parallel time and the amount of hardware. We develop new bounds on the expressiveness and succinctness of first-order logic with two variables on finite words, present a related result about the complexity of the satisfiability problem for this logic, and explore a new approach to the generalized star-height problem from the perspective of logical expressiveness. We give a complete characterization of the expressive power of first-order logic with two variables on finite words. Our main tool for this investigation is the classical Ehrenfeucht-Fra¨ıss´e game. Using our new characterization, we prove that the quantifier alternation hierarchy for this logic is strict, settling the main remaining open question about the expressiveness of this logic. A second important question about first-order logic with two variables on finite words is about the complexity of the satisfiability problem for this logic. Previously it was only known that this problem is NP-hard and in NEXP. We prove a polynomialsize small-model property for this logic, leading to an NP algorithm and thus proving that the satisfiability problem for this logic is NP-complete. Finally, we investigate one of the most baffling open problems in formal language theory: the generalized star-height problem. As of today, we do not even know whether there exists a regular language that has generalized star-height larger than 1. This problem can be phrased as an expressiveness question for first-order logic with a restricted transitive closure operator, and thus allows us to use established tools from finite model theory to attack the generalized star-height problem. Besides our contribution to formalize this problem in a purely logical form, we have developed several example languages as candidates for languages of generalized star-height at least 2. While some of them still stand as promising candidates, for others we present new results that prove that they only have generalized star-height 1.
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