Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota, we define the Euler characteristic of a formula in Gödel logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e., coincides with the analogous notion defined over Boolean logic. Building on this, we define k-valued versions of the Euler characteristic of a formula φ, for each integer k>=2, and prove that they indeed provide information about the logical status of φ in Gödel k-valued logic. Specifically, our main result shows that the k-valued Euler characteristic is an invariant that separates k-valued tautologies from non-tautologies.

The Euler characteristic of a formula in Gödel logic / P. Codara, O. M. D'Antona, V. Marra - In: Proceedings [of the] 40th IEEE International Symposium on Multiple-Valued Logic —— ISMVL 2010 —— : 26–28 May 2010, Barcelona, SpainLos Alamitos : IEEE Computer Society, 2010. - ISBN 9780769540245. - pp. 108-112 (( Intervento presentato al 40th. convegno IEEE International Symposium on Multiple-Valued Logic tenutosi a Barcelona, Spain nel 2010 [10.1109/ISMVL.2010.28].

### The Euler characteristic of a formula in Gödel logic

#### Abstract

Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota, we define the Euler characteristic of a formula in Gödel logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e., coincides with the analogous notion defined over Boolean logic. Building on this, we define k-valued versions of the Euler characteristic of a formula φ, for each integer k>=2, and prove that they indeed provide information about the logical status of φ in Gödel k-valued logic. Specifically, our main result shows that the k-valued Euler characteristic is an invariant that separates k-valued tautologies from non-tautologies.
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Gödel Logic ; Euler Characteristic
Settore INF/01 - Informatica
Settore MAT/01 - Logica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/2434/142162`
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