Fortresses constitute a class of language descriptors completely based only on two fundamental concepts of propositional logic: the notion of logical consequence and the notion of substitution. Fortresses based on classical propositional logic precisely recognise the class of regular languages. In this paper, we characterise the formal languages obtained by replacing, in fortresses, the notion of logical consequence in classical logic with the one in Gödel infinitely-valued logic. We prove that fortresses in Gödel logic exactly recognise finite chains of inclusions of regular languages. To prove this, we make use of a Stone’s type categorical duality between the algebraic semantics of Gödel propositional logic and the category of finite forests and open maps.

Product logic is considered one of the major truth-functional fuzzy propositional logics. Its semantics is given by the variety of Product algebras ${\mathbb{P}}$. In the hierarchy of fuzzy logics based on left-continuous t-norms there are a few logics whose algebraic semantics are varieties categorically equivalent with ${\mathbb{P}}$. For these logics we shall describe finitely generated free algebras and their group of automorphisms, that is, invertible substitutions.

The algebraic semantics of Gödel propositional logic is given by the variety of Gödel algebras, which in turns form a category dually equivalent to the pro-finite completion of the category of finite forests and order-preserving open maps. Forests provide a sound and complete semantics for propositional infinite-valued Gödel logic, while propositional k-valued Gödel logic is sound and complete for forests of height at most k-1. In this work we shall mainly deal with three-valued Gödel logic. We shall show that the subcategory of forests of height at most 2 (bushes) forms an elementary topos, thus providing naturally a generalisation to bushes of all classical first-order set concepts, suitable for developing a first-order three-valued Gödel logic semantics based on bush concepts instead of sets.

We characterize the structure of the automorphism groups of finitely generated free algebras in locally finite varieties constituting the algebraic semantics of well-known many-valued propositional logics, such as Gödel logic and the logic of Gödel hoops, Nilpotent Minimum logic, n-valued Łukasiewicz logic, Drastic product logic. We introduce the subalgebras of automorphism invariant elements of the free algebras, and study their structure in the case of Gödel algebras.

We show that finite IUML-algebras, which are residuated lattices arising from an idempotent uninorm, can be interpreted as algebras of sequences of orthopairs whose main operation is defined starting from the three-valued Sobociński operator between rough sets. Our main tool is the representation of finite IUML-algebras by means of finite forests. 1

We give a [0, l]-functional representation of the finitely generated free algebras in the variety generated by Chain's MV-algebra Sω2, and in the variety generated by the left continuous t-norm arising as Jenei's rotation JII of the product t-norm. We generalise the construction of JII from Sω2by building a family Tnof involutive t-norm algebras such that the MV-algebras in the variety generated by Tnform the variety generated by Sωnand LN+1.

We present an equivalence between the category of Nelson Paraconsistent lattices (NPc-lattices) and a category of pairs of Brouwerian algebras and regular filters. Specializing such category of pairs to Gödel hoops, we get the subvariety of Gödel NPc-lattices and, using the dual equivalence of finite Gödel hoops with finite trees, we obtain a duality for finite Gödel NPc-lattices. This duality is used to describe finitely generated free Gödel NPc-lattices.

This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely |$\mathbb{SBP}_0$|⁠. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of triplets whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP|$_0$|-algebras. Finally, we will provide an explicit representation of SBP|$_0$|-algebras in terms of (weak) Boolean products.

Gödel propositional logic is one of the major t-norm based fuzzy logics. Axiomatically, Gödel propositional logic G arises by extending intuitionistic propositional logic with the prelinearity axiom scheme (φ → ψ) V (ψ → φ). In [9], Fiorino introduces a tableaux calculus yielding a decision procedure for the theoremhood problem of G that has space complexity O(n log n), where n is the length of the input formula. In this paper we provide a decision procedure that has deterministic space complexity O(n).

Gödel logic and its algebraic semantics, namely, the variety of Gödel algebras, play a major rôle in mathematical fuzzy logic. The category of finite Gödel algebras and their homomorphisms is dually equivalent to the category FF of finite forests and order-preserving open maps. The combinatorial nature of FF allows to reduce the usually difficult problem of computing coproducts of algebras and their cardinalities to the combinatorial problem of computing products of finite forests. In this paper we propose a neat, purely combinatorial, recursive formula to compute the product objects. Further, we formulate a dual equivalence between finite GödelΔ-algebras and a category of finite multisets of finite chains, and we provide recursive formulas to compute coproducts, and their cardinalities, in the categories of finite Gödel hoops and of finite GödelΔ-algebras.

As is well-known standard MTL-algebras in general do not define the t-conorm +*associated with their t-norm *. As +*is defined by x+*y = 1-((1-x) * (1-y)), we address the generalised problem of characterising those MTL-algebras with monoidal operation * that define the dual monoidal operation x+*y = ~(~x*~y) for some order-reversing involution ~. The barest requirement on such structures is clearly that they are subdirect products of order-anti-automorphic chains (o.a.a., for short). We deal with the case of BL-algebras, stating two properties of involutions and fully characterising those BL-algebras defining the dual monoidal operation when the involution satisfies both properties. We also exhibit a BL-chain defining the dual monoidal operation determined by an involution failing both properties. We further prove that all o.a.a. algebras in the variety generated by EMTL-algebras and IMTL-algebras define the dual monoidal operation uniformly with the same term. By contrast, we present a variety whose o.a.a. chains define the dual monoidal operation, but with distinct terms for distinct algebras, generally. If we require definability of the dual residual operation, too, we are left with IMTL-algebras as the only known examples.

In this paper we present, by way of case studies, a proof of concept, based on a prototype working on a automotive data set, aimed at showing the potential usefulness of using formulas of Łukasiewicz propositional logic to query databases in a fuzzy way. Our approach distinguishes itself for its stress on the purely linguistic, contraposed with numeric, formulations of queries. Our queries are expressed in the pure language of logic, and when we use (integer) numbers, these stand for shortenings of formulas on the syntactic level, and serve as linguistic hedges on the semantic one. Our case-study queries aim first at showing that each numeric-threshold fuzzy query is simulated by a Łukasiewicz formula. Then they focus on the expressing power of Łukasiewicz logic which easily allows for updating queries by clauses and for modifying them through a potentially infinite variety of linguistic hedges implemented with a uniform syntactic mechanism. Finally we shall hint how, already at propositional level, Łukasiewicz natural semantics enjoys a degree of reflection, allowing to write syntactically simple queries that semantically work as meta-queries weighing the contribution of simpler ones.

We solve the minimization problem for finitely axiomatizable theories in Gödel infinite-valued propositional logic. That is, we obtain an algorithm that when input a formula α(X1,…,Xn) outputs a formula β(X1,…,Xm) such that (i) the theories singly axiomatized by {α} and {β} have isomorphic algebraic semantics, and (ii) if β′(X1,…,Xm′) is any formula satisfying (i), then m′≥m.

Stefano Aguzzoli, Brunella Gerla, Vincenzo Marra; Applications of Topological Dualities to Measure Theory in Algebraic Many-valued Logic, Journal of Logic

We address the question, which MV-algebras have finite automorphism group. We prove that finitely presented MV-algebras whose maximal spectral space has topological dimension not exceeding 1 do have finite automorphism group. We give examples to show that finite presentability is an essential hypothesis. Our proof produces as an interesting by-product a complete graph–theoretic isomorphism invariant for the class of MV-algebras involved.

Godel logic is the fuzzy logic of the minimum triangular norm and its residuum. Using the functional representation of the Lindenbaum algebra of Godel logic, we analyze the interaction between the integral operator and the logical connectives. On these grounds, we put forth a notion of finitely additive probability measure for Godel logic. Our first main result shows that such measures precisely correspond to integrating the truth value functions induced by Godel formulas with respect to a Borel probability measure on the real unit cube [0,1]n. Our second main result shows that they also coincide with convex combinations of finitely many [0,1]-valued assignments.

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Gödel-Dummett logic), mutatis mutandis. We give several such applications.

In this paper we deal with the expressive power of some logics based on residuated left-continuous t-norms. We investigate the class of truth functions for Nilpotent Minimum, Godel and NMG logics counting the number of different elements and describing normal forms which generalize the classical Boolean sum of minterms and product of maxterms. It turns out that the logics considered in the paper have much greater expressive power than Boolean propositional logic, while the complexity of their normal forms remains almost as manageable as Boolean normal forms.

Systems which predict the items a user would like, on the basis of the preferences given by other users, are attracting growing attention as automated recommender services on the Internet. Collaborative filtering techniques are widely used to implement such recommender systems. We show how fuzzy sets and many-valued logics can be fruitfully applied in the description and design of collaborative filtering methods.

In this paper we define internal cut-free sequent calculi for any n-valued Lukasiewicz logic Ln. These calculi are based on a representation of formulas of Ln, by n - 1 many {0, 1}-valued formulas of Ln. They enjoy the usual properties of sequent systems like symmetry, subformula property and invertibility of the rules. Upon dualizing our calculi one obtains Hähnle's tableau systems. Then they provide a reformulation of Hähnle's approach to theorem proving that makes no use of nonlogical elements.