Algebraic Semantics Research Articles

Algebraic Semantics for a Mixed Type Fragment of IPC

Algebraic Semantics for a Mixed Type Fragment of IPC

GADTs are not (Even partial) functors

Abstract Generalized Algebraic Data Types (GADTs) are a syntactic generalization of the usual algebraic data types (ADTs), such as lists, trees, etc. ADTs’ standard initial algebra semantics (IAS) in the category $\mathit{Set}$ of sets justify critical syntactic constructs – such as recursion, pattern matching, and fold – for programming with them. In this paper, we show that semantics for GADTs that specialize to the IAS for ADTs are necessarily unsatisfactory. First, we show that the functorial nature of such semantics for GADTs in $\mathit{Set}$ introduces ghost elements, i.e., elements not writable in syntax. Next, we show how such ghost elements break parametricity. We observe that the situation for GADTs contrasts dramatically with that for ADTs, whose IAS coincides with the parametric model constructed via their Church encodings in System F. Our analysis reveals that the fundamental obstacle to giving a functorial IAS for GADTs is the inherently partial nature of their map functions. We show that this obstacle cannot be overcome by replacing $\mathit{Set}$ with other categories that account for this partiality.

Free Constructions in Hoops via $$\ell $$-Groups

AbstractLattice-ordered abelian groups, or abelian$$\ell $$ ℓ -groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call $$\textsf{DLMV}$$ DLMV . In this work we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete.

ON THE STRUCTURE OF BOCHVAR ALGEBRAS

Abstract Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$ . Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP).

Algebraic Completeness of Connexive and Bi-Intuitionistic Multilattice Logics

In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing’s connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and complete with respect to the presented algebraic structures.

An algebraic approach to simulation and verification for cyber-physical systems with shared-variable concurrency

Cyber-Physical systems (CPS), containing discrete behaviors of the cyber and continuous behaviors of the physical, have gained wide applications in many fields. Since CPS subsume the intersection of cyber systems and physical processes, the traditional modeling languages which merely include discrete variables are no longer applicable to CPS. Accordingly, a shared variable language called CPSLsc was proposed to specify CPS. In this paper, we elaborate the algebraic semantics for this language, so that every program of CPSLsc can be converted into a unified form called guarded choice form and the sequentialization of parallel programs is achieved. Additionally, we formalize the algebraic semantics in the rewriting engine Real-Time Maude. With the algebraic laws constructed, for every program specified with CPSLsc, we can simulate its execution step by step. Furthermore, automatic transformation and execution are attained. As a consequence, if the program and its initial data state are provided, the corresponding trace of data states during execution can be generated. In the light of the generated trace, automatic verification can be carried out as well.

Semiconic idempotent logic I: Structure and local deduction theorems

Semiconic idempotent logic is a common generalization of intuitionistic logic, relevance logic with mingle, and semilinear idempotent logic. It is an algebraizable logic and it admits a cut-free hypersequent calculus. We give a structural decomposition of its characteristic algebraic semantics, conic idempotent residuated lattices, showing that each of these is an ordinal sum of simpler partially ordered structures. This ordinal sum is indexed by a totally ordered residuated lattice, which serves as its skeleton and is both a subalgebra and nuclear image. We equationally characterize the totally ordered residuated lattices appearing as such skeletons. Further, we describe both congruence and subalgebra generation in conic idempotent residuated lattices, proving that every variety generated by these enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices. Based on this analysis, we obtain a local deduction theorem for semiconic idempotent logic, which also specializes to semilinear idempotent logic.

Congruence filter pairs, equational filter pairs and adjoints

Abstract Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.

An Algebraic View of the Mares-Goldblatt Semantics

An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered.

Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

Wansing's bi-intuitionistic logic: semantics, extension and unilateralisation

ABSTRACT The well-known algebraic semantics and topological semantics for intuitionistic logic ( Int ) is here extended to Wansing's bi-intuitionistic logic ( 2 Int ). The logic 2 Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2 Int . Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2 Int is extended with a ‘Fregean negation’ connective ∼ , obtaining 2 In t ∼ , and it is showed that the logic N 4 ⋆ (an extension of Nelson's paraconsistent logic) results to be the unilateralisation of 2 In t ∼ via ∼ . The logic 2 In t ∼ is also characterised by a Kripke-style semantics, a twist structure semantics and a topological semantics.

Linear Abelian Modal Logic

A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization is investigated.

Some Remarks on the Logic of Probabilistic Relevance

In this paper we deepen some aspects of the statistical approach to relevance by providing logics for the syntactical treatment of probabilistic relevance relations. Specifically, we define conservative expansions of Classical Logic endowed with a ternary connective ⇝ - indeed, a constrained material implication - whose intuitive reading is “x materially implies y and it is relevant to y under the evidence z”. In turn, this ensures the definability of a formula in three-variables R(x, z, y) which is the representative of relevance in the object language. We outline the algebraic semantics of such logics, and we apply the acquired machinery to investigate some termdefined weakly connexive implications with some intuitive appeal. As a consequence, a further motivation of (weakly) connexive principles in terms of relevance and background assumptions obtains.

RoboWorld: Verification of Robotic Systems with Environment in the Loop

A robot affects and is affected by its environment, so that typically its behaviour depends on properties of that environment. For verification, we need to formalise those properties. Modelling the environment is very challenging, if not impossible, but we can capture assumptions. Here, we present RoboWorld, a domain-specific controlled natural language with a process algebraic semantics that can be used to define (a) operational requirements, and (b) environment interactions of a robot. RoboWorld is part of the RoboStar framework for verification of robotic systems. In this article, we define RoboWorld’s syntax and hybrid semantics, and illustrate its use for capturing operational requirements, for automatic test generation, and for proof. We also present a tool that supports the writing of RoboWorld documents. Since RoboWorld is a controlled natural language, it complements the other RoboStar notations in being accessible to roboticists, while at the same time benefitting from a formal semantics to support rigorous verification (via testing and proof).

Modal expansions of ririgs

Abstract In this paper, we introduce the variety of $I$-modal ririgs. We characterize the congruence lattice of its members by means of $I$-filters, and we provide a description of $I$-filter generation. We also provide an axiomatic presentation for the variety generated by chains of the subvariety of contractive $I$-modal ririgs. Finally, we introduce a Hilbert-style calculus for a logic with $I$-modal ririgs as an equivalent algebraic semantics and we prove that such a logic has the parametrized local deduction-detachment theorem.

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

Carnap’s Problem for Intuitionistic Propositional Logic

We show that intuitionistic propositional logic is Carnap categorical: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds with respect to the most well-known semantics relative to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, topological semantics, and algebraic semantics. These facts turn out to be consequences of an observation about interpretations in Heyting algebras.

A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices

We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.

On subreducts of subresiduated lattices and some related logics

Abstract Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subintuitionistic logics, i.e. logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose equivalent algebraic semantics is given by these classes of algebras. Several expansions of these calculi are also studied together with some interesting properties of them.

Polyatomic logics and generalized Blok–Esakia theory

Abstract This paper presents a novel concept of a polyatomic logic and initiates its systematic study. This approach, inspired by inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalized Blok–Esakia theory, which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of polyatomic logics and the general Blok–Esakia theory for a wide class of interesting translations.