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Expressiveness and Succinctness of First-Order Logic on Finite Words

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dc.contributor.advisorNeil Immerman
dc.contributor.advisorDavid A. Mix Barrington
dc.contributor.advisorRajesh Bhatt
dc.contributor.authorWeis, Philipp P
dc.contributor.departmentUniversity of Massachusetts Amherst
dc.date2023-09-23T04:35:50.000
dc.date.accessioned2024-04-26T19:49:14Z
dc.date.available2024-04-26T19:49:14Z
dc.date.issued2011-05-13
dc.description.abstractExpressiveness, 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.
dc.description.degreeDoctor of Philosophy (PhD)
dc.description.departmentComputer Science
dc.identifier.doihttps://doi.org/10.7275/2177022
dc.identifier.urihttps://hdl.handle.net/20.500.14394/38850
dc.relation.urlhttps://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1419&context=open_access_dissertations&unstamped=1
dc.source.statuspublished
dc.subjectEhrenfeucht-Fraïssé game
dc.subjectfinite model theory
dc.subjectfirst-order logic
dc.subjectgeneralized star-height
dc.subjectsatisfiability
dc.subjectsuccinctness
dc.subjectComputer Sciences
dc.titleExpressiveness and Succinctness of First-Order Logic on Finite Words
dc.typedissertation
dc.typearticle
dc.typedissertation
digcom.contributor.authorisAuthorOfPublication|email:pweis@cs.umass.edu|institution:University of Massachusetts Amherst|Weis, Philipp P
digcom.identifieropen_access_dissertations/407
digcom.identifier.contextkey2177022
digcom.identifier.submissionpathopen_access_dissertations/407
dspace.entity.typePublication
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