## Finite Model Theory and Its ApplicationsErich Grädel, Phokion G. Kolaitis, Leonid Libkin, Maarten Marx, Joel Spencer, Moshe Y. Vardi, Yde Venema, Scott Weinstein Finite model theory,as understoodhere, is an areaof mathematicallogic that has developed in close connection with applications to computer science, in particular the theory of computational complexity and database theory. One of the fundamental insights of mathematical logic is that our understanding of mathematical phenomena is enriched by elevating the languages we use to describe mathematical structures to objects of explicit study. If mathematics is the science of patterns, then the media through which we discern patterns, as well as the structures in which we discern them, command our attention. It isthis aspect oflogicwhichis mostprominentin model theory,“thebranchof mathematical logic which deals with the relation between a formal language and its interpretations”. No wonder, then, that mathematical logic, and ?nite model theory in particular, should ?nd manifold applications in computer science: from specifying programs to querying databases, computer science is rife with phenomena whose understanding requires close attention to the interaction between language and structure. This volume gives a broadoverviewof some central themes of ?nite model theory: expressive power, descriptive complexity, and zero–one laws, together with selected applications to database theory and arti?cial intelligence, es- cially constraint databases and constraint satisfaction problems. The ?nal chapter provides a concise modern introduction to modal logic,which emp- sizes the continuity in spirit and technique with ?nite model theory. |

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### Contents

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### Common terms and phrases

adom algorithm arity basic modal binary relation bisimulation bounded captures PTIME class of finite complexity classes Computer Science constraint databases constraint satisfaction CSP(B Datalog Datalog program Definition denote domain Duplicator wins Ehrenfeucht–Fra¨ıssé game elements encoding equivalent example existential expressive power finite graphs finite model theory finite structures first-order definable first-order formula first-order logic first-order sentence fixed-point logic FO(SC,M FOact fragment function graph G greatest fixed points homomorphism induction infinitary logic infinite input isomorphism k-pebble game language least fixed point Lemma linear order metafinite modal logic modal system model-checking monadic monotone mosaic nodes NP-complete o-minimal operator Player Poly polynomial polynomial-time positive integer predicate proof Proposition PSPACE PTIME quantifier elimination query Q recursive relation symbol result satisfiability problem SC-structure second-order logic Spoiler subformula Theorem tuple unary Vardi variables vocabulary winning strategy

### Popular passages

Page 367 - References [1] S. Abiteboul, R. Hull and V. Vianu, Foundations of Databases, Addison- Wesley, Reading, MA, 1995.

Page 367 - Part of this work was done while the second author was visiting the Department of Computer and Information Sciences, University of California, Santa-Cruz, supported by NSF grant IRI-9123692.

Page 429 - M. Vardi. Why is modal logic so robustly decidable? In DIM ACS Series in Discrete Mathematics and Theoretical Computer Science 31, pages 149-184. American Math. Society, 1997.