### Local Normal Forms for First-Order Logic with Applications to Games and Automata

*Thomas Schwentick, Klaus Barthelmann*

#### Abstract

Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form &exists; x1,...,xl, ∀ y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y.

From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs.

As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs.

As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

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