The Euler characteristic of a formula in many-valued logic
Pietro Codara · 2010
Some decades ago, V. Klee and G.-C. Rota introduced a lattice-theoretic analogue of the Euler characteristic, the celebrated topological invariant of polyhedra. Using the Klee-Rota definition, we introduce the Euler characteristic of a formula in Godel logic (an extension of intuitionistic logic, or, equivalently, a many-valued logic). We then prove that the Euler characteristic of a formula φ coincides with the number of Boolean assignments that satisfy φ. Building on this, we generalise this notion to other invariants of φ that provide additional information about the satisfiability of φ in Godel logic. Specifically, while the Euler characteristic does not determine non-classical tautologies, these new invariants do. Finally, if time allows, we sketch a similar analysis for other many-valued logics.