Recent zbMATH articles in MSC 03G https://zbmath.org/atom/cc/03G 2021-05-28T16:06:00+00:00 Werkzeug A short overview of hidden logic. https://zbmath.org/1459.03101 2021-05-28T16:06:00+00:00 "Ferreirim, Isabel" https://zbmath.org/authors/?q=ai:ferreirim.isabel-m-a "Martins, Manuel A." https://zbmath.org/authors/?q=ai:martins.manuel-a Summary: In this paper we review a hidden (sorted) generalization of $$k$$-deductive systems -- hidden $$k$$-logics. They encompass deductive systems as well as hidden equational logics and inequational logics. The special case of hidden equational logics has been used to specify and to verify properties in program development of behavioral systems within the dichotomy visible vs. hidden data. We recall one of the main applications of this work -- the study of behavioral equivalence. Related results are obtained through combinatorial properties of the Leibniz congruence relation. For the entire collection see [Zbl 1390.03005]. The equationally-defined commutator in quasivarieties generated by two-element algebras. https://zbmath.org/1459.03035 2021-05-28T16:06:00+00:00 "Czelakowski, Janusz" https://zbmath.org/authors/?q=ai:czelakowski.janusz Summary: The notion of the equationally-defined commutator was introduced and thoroughly investigated in [the authors, The equationally-defined commutator. A study in equational logic and algebra. Cham: Birkhäuser/Springer (2015; Zbl 1352.08001)]. In this work the properties of the equationally-defined commutator in quasivarieties generated by two-element algebras are examined. It is proved: If a quasivariety $$Q$$ is generated by a finite set of two-element algebras, then the equationally-defined commutator of $$Q$$ is additive (Theorem 3.1) Moreover it satisfies the associativity law (Theorem 3.6). The second result is strengthened if the quasivariety is generated by a single two-element algebra 2: If $$Q = \mathrm{SP}(2)$$, then the equationally-defined commutator of $$Q$$ universally validates one of the following laws: $$[x,y] = x\wedge y$$ or $$[x,y] = 0$$ (Theorem 3.9). In other words, any quasivariety generated by a single two-element algebra is either relatively congruence-distributive or Abelian. A syntactical characterization of all quasivarieties generated by finite sets of two-element algebras is also presented (Theorems 2.2--2.3). For the entire collection see [Zbl 1390.03005]. Introducing Boolean semilattices. https://zbmath.org/1459.03097 2021-05-28T16:06:00+00:00 "Bergman, Clifford" https://zbmath.org/authors/?q=ai:bergman.clifford Summary: We present and discuss a variety of Boolean algebras with operators that is closely related to the variety generated by all complex algebras of semilattices. We consider the problem of finding a generating set for the variety, representation questions, and axiomatizability. Several interesting subvarieties are presented. We contrast our results with those obtained for a number of other varieties generated by complex algebras of groupoids. For the entire collection see [Zbl 1390.03005]. Deduction-detachment theorem and Gentzen-style deductive systems. https://zbmath.org/1459.03100 2021-05-28T16:06:00+00:00 "Babenyshev, Sergey" https://zbmath.org/authors/?q=ai:babenyshev.sergey Summary: Logical implication is an attempt to catch the essence of causeeffect relationships of the real world in the context of formal deductive systems. The Deduction-Detachment Theorem (DDT) being, in its turn, a statement about essential logical properties of classical implication, was therefore of great interest for logicians. Although a statement about a Hilbert-style deductive system, DDT can be formulated by means of Gentzen-style rules, and as such seems to be a statement about the metatheoretical properties of Hilbert-style deductive systems. As is often the case with metatheoretical properties, DDT leaves the question about its meaning and scope a bit vague or at least requires a higher order abstraction layer to formalize them. Closure relations, discussed in this paper, present a convenient context to give a precise formal statement of the DDT and its connection with Hilbert- and pertinent Gentzen-style deductive systems. For the entire collection see [Zbl 1390.03005]. Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic. https://zbmath.org/1459.03099 2021-05-28T16:06:00+00:00 "Albuquerque, Hugo" https://zbmath.org/authors/?q=ai:albuquerque.hugo "Font, Josep Maria" https://zbmath.org/authors/?q=ai:font.josep-maria "Jansana, Ramon" https://zbmath.org/authors/?q=ai:jansana.ramon "Moraschini, Tommaso" https://zbmath.org/authors/?q=ai:moraschini.tommaso Summary: We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract algebraic logic, and open a new one. For the entire collection see [Zbl 1390.03005]. $$L$$-effect algebras. https://zbmath.org/1459.08003 2021-05-28T16:06:00+00:00 "Rump, Wolfgang" https://zbmath.org/authors/?q=ai:rump.wolfgang "Zhang, Xia" https://zbmath.org/authors/?q=ai:zhang.xia In this paper, $$L$$-effect algebras are introduced as a class of $$L$$-algebras which specialize to all known generalizations of effect algebras with a $$\wedge$$-semilattice structure. Moreover, $$L$$-effect algebras $$X$$ arise in connection with quantum sets and Frobenius algebras. The translates of $$X$$ in the self-similar closure $$S(X)$$ form a covering, and the structure of $$X$$ is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an $$L$$-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right $$\ell$$-group, the structure group of the corresponding $$L$$-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description. Reviewer: Jafar Pashazadeh (Bonab) Coalgebraic trace semantics via forgetful logics. https://zbmath.org/1459.68114 2021-05-28T16:06:00+00:00 "Klin, Bartek" https://zbmath.org/authors/?q=ai:klin.bartek "Rot, Jurriaan" https://zbmath.org/authors/?q=ai:rot.jurriaan Summary: We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete examples such as the language semantics of weighted, alternating and tree automata. We provide a sufficient condition under which a logical semantics coincides with the trace semantics obtained via a given determinization construction. Finally, we consider a condition that guarantees the existence of a canonical determinization procedure that is correct with respect to a given logical semantics. That procedure is closely related to Brzozowski's minimization algorithm. For the entire collection see [Zbl 1320.68027]. Localization of tetravalent modal algebras. https://zbmath.org/1459.06009 2021-05-28T16:06:00+00:00 "Figallo, Aldo V." https://zbmath.org/authors/?q=ai:figallo.aldo-victorio "Pelaitay, Gustavo" https://zbmath.org/authors/?q=ai:pelaitay.gustavo A classical realizability model arising from a stable model of untyped lambda calculus. https://zbmath.org/1459.03017 2021-05-28T16:06:00+00:00 "Streicher, Thomas" https://zbmath.org/authors/?q=ai:streicher.thomas Summary: In [\textit{T. Streicher} and \textit{B. Reus}, J. Funct. Program. 8, No. 6, 543--572 (1998; Zbl 0928.68074)] it has been shown that $$\lambda$$-calculus with control can be interpreted in any domain $$D$$ which is isomorphic to the domain of functions from $$D^\omega$$ to the 2-element (Sierpiński) lattice in $$\Sigma$$. By a theorem of A. Pitts there exists a unique subset $$P$$ of $$D$$ such that $$f\in P$$ iff $$f(\overset{\rightarrow}{d})=\bot$$ for all $$\overset{\rightarrow}{d}\in P^\omega$$. The domain $$D$$ gives rise to a \textit{realizability structure} in the sense of [\textit{J.-L. Krivine}, Log. Methods Comput. Sci. 7, No. 3, Paper No. 2, 47 p. (2011; Zbl 1237.03012)] where the set of proof-like terms is given by $$P$$. When working in Scott domains the ensuing realizability model coincides with the ground model \textbf{Set} but when taking $$D$$ within the coherence spaces we obtain a classical realizability model of set theory different from any forcing model. We will show that this model validates countable and dependent choice since an appropriate form of bar recursion is available in stable domains. Model theory and proof theory of coalgebraic predicate logic. https://zbmath.org/1459.03104 2021-05-28T16:06:00+00:00 "Litak, Tadeusz" https://zbmath.org/authors/?q=ai:litak.tadeusz "Pattinson, Dirk" https://zbmath.org/authors/?q=ai:pattinson.dirk "Sano, Katsuhiko" https://zbmath.org/authors/?q=ai:sano.katsuhiko "Schröder, Lutz" https://zbmath.org/authors/?q=ai:schroder.lutz Summary: We propose a generalization of first-order logic originating in a neglected work by \textit{C. C. Chang} [Lect. Notes Math. 337, 599--617 (1973; Zbl 0276.02012)]: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness -- and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result. Focusing in orthologic. https://zbmath.org/1459.03096 2021-05-28T16:06:00+00:00 "Laurent, Olivier" https://zbmath.org/authors/?q=ai:laurent.olivier Summary: We propose new sequent calculus systems for orthologic (also known as minimal quantum logic) which satisfy the cut elimination property. The first one is a simple system relying on the involutive status of negation. The second one incorporates the notion of focusing (coming from linear logic) to add constraints on proofs and to optimise proof search. We demonstrate how to take benefits from the new systems in automatic proof search for orthologic. Algebraic perspectives on substructural logics. Selected papers based on the presentations at the workshop AsubL (Algebra \& Substructural Logics -- Take 6), Cagliari, Italy, June 11--13, 2018. https://zbmath.org/1459.06001 2021-05-28T16:06:00+00:00 "Fazio, Davide (ed.)" https://zbmath.org/authors/?q=ai:fazio.davide "Ledda, Antonio (ed.)" https://zbmath.org/authors/?q=ai:ledda.antonio "Paoli, Francesco (ed.)" https://zbmath.org/authors/?q=ai:paoli.francesco Publisher's description: This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra \& Substructural Logics -- Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics. The articles of this volume will be reviewed individually. Indexed articles: \textit{Fazio, Davide; Ledda, Antonio; Paoli, Francesco}, Editorial introduction, 1-9 [Zbl 07326286] \textit{Aglianò, Paolo}, Distributivity and varlet distributivity, 11-20 [Zbl 07326287] \textit{Ertola-Biraben, Rodolfo C.; Esteva, Francesc; Godo, Lluís}, On distributive join semilattices, 21-40 [Zbl 07326288] \textit{Chajda, Ivan; Länger, Helmut}, Implication in weakly and dually weakly orthomodular lattices, 41-56 [Zbl 07326289] \textit{Chajda, Ivan; Länger, Helmut; Paseka, Jan}, Residuated operators and Dedekind-MacNeille completion, 57-72 [Zbl 07326290] \textit{Giuntini, Roberto; Mureşan, Claudia; Paoli, Francesco}, PBZ*-lattices: ordinal and horizontal sums, 73-105 [Zbl 07326291] \textit{Dvurečenskij, Anatolij; Zahiri, Omid}, EMV-algebras -- extended MV-algebras, 107-132 [Zbl 07326292] \textit{Rivieccio, Umberto; Spinks, Matthew}, Quasi-Nelson; or, non-involutive Nelson algebras, 133-168 [Zbl 07326293] \textit{Logan, Shay Allen}, Hyperdoctrines and the ontology of stratified semantics, 169-193 [Zbl 07326294] Categorical abstract algebraic logic: compatibility operators and correspondence theorems. https://zbmath.org/1459.03103 2021-05-28T16:06:00+00:00 "Voutsadakis, George" https://zbmath.org/authors/?q=ai:voutsadakis.george Summary: Very recently \textit{H. Albuquerque} et al. [J. Symb. Log. 81, No. 2, 417--462 (2016; Zbl 1402.03101)], based on preceding work of \textit{J. Czelakowski} [Stud. Log. 74, No. 1--2, 181--231 (2003; Zbl 1043.03050)] on compatibility operators, introduced coherent compatibility operators and used Galois connections, formed by these operators, to provide a unified framework for the study of the Leibniz, the Suszko and the Tarski operators of abstract algebraic logic. Based on this work, we present a unified treatment of the operator approach to the categorical abstract algebraic logic hierarchy of $$\pi$$-institutions. This approach encompasses previous work by the author in this area, started under Don Pigozzi's guidance, and provides resources for new results on the semantic, i.e., operator-based, side of the hierarchy. For the entire collection see [Zbl 1390.03005]. Diagrammatic duality. https://zbmath.org/1459.03105 2021-05-28T16:06:00+00:00 "Romanowska, Anna B." https://zbmath.org/authors/?q=ai:romanowska.anna-b "Smith, Jonathan D. H." https://zbmath.org/authors/?q=ai:smith.jonathan-d-h A new method, called diagrammatic duality, for obtaining new dualities between algebras and representation spaces founded on existing ones is introduced (Section 7). The method is illustrated by examples, including Nelson algebras, quasigroups and bilattices. For the entire collection see [Zbl 1390.03005]. Reviewer: Jiří Močkoř (Ostrava) Characterization of protoalgebraic $$k$$-deductive systems. https://zbmath.org/1459.03102 2021-05-28T16:06:00+00:00 "Pałasińska, Katarzyna" https://zbmath.org/authors/?q=ai:palasinska.katarzyna Summary: A sentential logic is protoalgebraic iff it has a finite system of equivalence formulas [\textit{W. J. Blok} and \textit{D. Pigozzi}, Stud. Log. 45, 337--369 (1986; Zbl 0622.03020)]. This can be generalized to the context of universal Horn logic without equality, [\textit{W. J. Blok} and \textit{D. Pigozzi}, in: Universal algebra and quasigroup theory. Lectures delivered at the conference, held in Jadwisin, Poland, 1989. Berlin: Heldermann Verlag. 1--56 (1992; Zbl 0768.03008)]. In this paper we revise this characterization. For the entire collection see [Zbl 1390.03005]. Obstinate, weak implicative and fantastic filters of non commutative residuated lattices. https://zbmath.org/1459.03098 2021-05-28T16:06:00+00:00 "Ghorbani, Shokoofeh" https://zbmath.org/authors/?q=ai:ghorbani.shokoofeh The notion of an obstinate filter in a residuated lattice was introduced in [\textit{A. Borumand Saeid} and \textit{M. Pourkhatoun}, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 55(103), No. 4, 413--422 (2012; Zbl 1274.03105)] and the notion of a fantastic filter was introduced in [\textit{Y. Zhu} and \textit{Y. Xu}, Inf. Sci. 180, No. 19, 3614--3632 (2010; Zbl 1228.03045)]. In this paper, the author generalizes these notions for non-commutative residuated lattices and investigates some of their properties. Reviewer: Jiří Močkoř (Ostrava)