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On valuations in Gödel and nilpotent minimum logics
No full textSome decades ago, V. Klee and G.-C. Rota introduced a lattice-theoretic analogue of the Euler characteristic, the celebrated topological invariant of polyhedra. In [1], using the Klee-Rota defi nition, we introduce the Euler characteristic of a formula in Goedel logic, the extension of intuitionistic logic via the prelinearity axiom (A->B)V(B->A). We then prove that the Euler characteristic of a formula A over n propositional variables coincides with the number of Boolean assignments to these n variables that satisfy A. Building on this, we generalise this notion to other invariants of A that provide additional information about the satis ability of A in Goedel logic. Specifically, the Euler characteristic does not determine non-classical tautologies: the maximum value of the characteristic of A(X1,...,Xn) is 2^n, and this can be attained even when A is not a tautology in Goedel logic. By contrast, we prove that these new invariants do.
In this talk, we present the aforementioned results and compare what has been obtained for Goedel logic with analogous results for a diff erent many-valued logic, namely, the logic of Nilpotent Minimum. This logic can also be described as the extension of Nelson logic by the prelinearity axiom. The latter results are joint work with D. Valota. [1] P. Codara, O. M. D'Antona, V. Marra, Valuations in Goedel Logic, and the Euler Characteristic, Journal of Multiple-Valued Logic and Soft Computing 19(1-3) (2012), 71-84
The Euler characteristic of a formula in many-valued logic
No full textSome 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 de nition, we introduce the Euler characteristic
of a formula in G odel logic (an extension of intuitionistic logic, or,
equivalently, a many-valued logic). We then prove that the Euler characteristic
of a formula \phi coincides with the number of Boolean assignments
that satisfy the formula \phi. Building on this, we generalise this notion to other invariants
of \phi that provide additional information about the satis ability of \phi in
G odel 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
Propositional Gödel logic, Delannoy paths and ordered partitions
No full textGoedel propositional logic is the logic of the minimum t-norm, and can be axiomatised as propositional intuitionistic logic augmented by the prelinearity axiom. Thus, its Tarski-Lindenbaum algebras are Heyting algebras satisfying prelinearity; we shall call them Goedel algebras. A Delannoy path is a lattice path in Z^2 that only uses northward, westward, and northwestward steps. We give a representation theorem for n-generated free Goedel algebras based upon the boolean unit n-cube {0,1}^n enriched by families of Delannoy paths (more precisely, their straightforward multidimensional generalisation). If time allows, we report on work in progress that, using ordered partitions of finite sets, aims at building a new combinatorial semantics for Goedel propositional logic
Profinite Heyting algebras, and partitions of image-finite posets under open maps
No full textWe are interested in developing a theory of quotient objects (i.e. partitions) in
sufficiently general categories related to non-classical propositional logics.
Our objective, is to obtain a notion of partition for an image-finite poset
P that (i) dualises subobjects in the category of profinite Heyting algebras and their complete homomorphisms, and (ii) can be directly described in terms
of 'parts' of P, without reference to epic arrows in the category of image-finite posets and open maps
A theory of partitions of partially ordered sets
No full textA partition of a set A is a set of nonempty pairwise disjoint subsets of A whose union is A. An equivalent definition of a partition can be given using functions. In these terms, a partition of a set is the set of fibres of a surjective function. The latter definition allows us to introduce the notion of partition for finite partially ordered sets. Analyzing the category Poset of partially ordered sets, or posets, and order-preserving maps, we see that two kinds of surjections have to be taken into account. Therefore, we must deal with two different classes of partitions, namely, monotone and regular partitions of a poset. These two notions form the basis for a theory of partitions of posets. In analogy with the set-theoretic case, our first step is to obtain characterizations of monotone and regular partitions. Then, in Chapter 4, we study the collection of all monotone and regular partitions of a poset. We endow these classes with a lattice structure, obtaining the monotone and regular partition lattices of a poset. A bijection between monotone and regular partitions and certain classes of quasiorders is established. This result generalizes the usual correspondence between partitions and equivalence relations. In Chapter 5, we present the well-known Birkhoff duality for Poset. We thus investigate the duals of monotone and regular partitions via this duality. In the last chapter we discuss enumeration problems within the theory of partitions of finite posets
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