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Guillermo Badia

Publications and source records attributed to Guillermo Badia.

At least 19 recordsLinked to original sources
arXiv2026

Definable Classes of Models and Frames in Bi-intuitionistic Logic

The question of the expressive power of a given logical language with Kripke relational semantics has at least two dimensions: (1) what the language can say about frames, and (2) what it can say about models. The Goldblatt-Thomason theorem provides a model-theoretic characterisation of modal axiomatisability for elementary classes of frames in terms of closure under taking generated subframes, disjoint unions, bounded morphic images, and reflection of ultrafilter extensions. Goldblatt also provides a similar characterisation for axiomatisability in intuitionistic logic of classes of models rather than frames. In this article we provide analogous results for bi-intuitionistic logic, a natural expressive extension of intuitionistic logic obtained by adding a binary connective dual to the intuitionistic implication, introduced in the 1970s independently by Dieter Klemke and Cecylia Rauszer. Together with previous results, such as a van Benthem bisimulation characterisation theorem and a Lindstrom theorem, this provides a complete picture of the expressive power of propositional bi-intuitionistic logic.

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arXiv2025

Fagin's Theorem for Semiring Turing Machines

In recent years, quantitative complexity over semirings has been intensively investigated. In this context, Eiter and Kiesel (Semiring Reasoning Frameworks in AI and Their Computational Complexity, J. Artif. Intell. Res., 2023) introduced non-deterministic Turing Machines with semiring-weighted transitions (SRTMs) to capture the complexity of a manifold of semiring frameworks. Beyond computational complexity, they posed the question of how we can relate the computational power of SRTMs to logical expressiveness. While this question was partially addressed for a more limited machine model by Badia et al.\ (Logical characterizations of weighted complexity classes, MFCS, 2024), the full question remained open. To answer it, we present an improved version of Eiter and Kiesel's SRTM model of computation. First and foremost, this enables us to prove a Fagin Theorem for the SRTM model, i.e., we show that the quantitative complexity class $\text{NP}_\infty(R)$, which comprises non-deterministic polynomial time computability in the improved SRTM model over a commutative semiring $R$, is captured by a version of weighted existential second-order logic that allows for predicates interpreted as semiring-annotated relations over $R$. Furthermore, we argue that the new SRTM model is preferable over the original one and show that it reclaims some important results from Eiter and Kiesel (2023) that were flawed with respect to the latter.

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arXiv2025

Codd's Theorem for Databases over Semirings

Codd's Theorem, a fundamental result of database theory, asserts that relational algebra and relational calculus have the same expressive power on relational databases. We explore Codd's Theorem for databases over semirings and establish two different versions of this result for such databases: the first version involves the five basic operations of relational algebra, while in the second version the division operation is added to the five basic operations of relational algebra. In both versions, the difference operation of relations is given semantics using semirings with monus, while on the side of relational calculus a limited form of negation is used. The reason for considering these two different versions of Codd's theorem is that, unlike the case of ordinary relational databases, the division operation need not be expressible in terms of the five basic operations of relational algebra for databases over an arbitrary positive semiring; in fact, we show that this inexpressibility result holds even for bag databases.

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arXiv2024

Hybrid-Dynamic Ehrenfeucht-Fraisse Games

Ehrenfeucht-Fraisse games provide means to characterize elementary equivalence for first-order logic, and by standard translation also for modal logics. We propose a novel generalization of Ehrenfeucht- Fraisse games to hybrid-dynamic logics which is direct and fully modular: parameterized by the features of the hybrid language we wish to include, for instance, the modal and hybrid language operators as well as first-order existential quantification. We use these games to establish a new modular Fraisse-Hintikka Theorem for hybrid-dynamic propositional logic and its various fragments. We study the relationship between countable game equivalence (determined by countable Ehrenfeucht- Fraisse games) and bisimulation (determined by countable back-and-forth systems). In general, the former turns out to be weaker than the latter, but under certain conditions on the language, the two coincide. We also use games to prove that for reachable image-finite Kripke structures elementary equivalence implies isomorphism.

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arXiv2024

Logical Characterizations of Weighted Complexity Classes

Fagin's seminal result characterizing $\mathsf{NP}$ in terms of existential second-order logic started the fruitful field of descriptive complexity theory. In recent years, there has been much interest in the investigation of quantitative (weighted) models of computations. In this paper, we start the study of descriptive complexity based on weighted Turing machines over arbitrary semirings. We provide machine-independent characterizations (over ordered structures) of the weighted complexity classes $\mathsf{NP}[\mathcal{S}], \mathsf{FP}[\mathcal{S}]$, $\mathsf{FPLOG}[\mathcal{S}]$, $\mathsf{FPSPACE}[\mathcal{S}]$, and $\mathsf{FPSPACE}_{poly}[\mathcal{S}]$ in terms of definability in suitable weighted logics for an arbitrary semiring $\mathcal{S}$. In particular, we prove weighted versions of Fagin's theorem (even for arbitrary structures, not necessarily ordered, provided that the semiring is idempotent and commutative), the Immerman--Vardi's theorem (originally for $\mathsf{P}$) and the Abiteboul--Vianu--Vardi's theorem (originally for $\mathsf{PSPACE}$). We also address a recent open problem proposed by Eiter and Kiesel.

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arXiv2023

A modular bisimulation characterisation for fragments of hybrid logic

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\store, \jump\}$ with or without nominals (Areces, Blackburn, Marx), $\jump$ with or without nominals (ten Cate), and $\store$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation theorem for all standard fragments of hybrid logic, in particular for the case with $\store$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense.

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arXiv2023

A parametrised axiomatization for a large number of restricted second-order logics

By limiting the range of the predicate variables in a second-order language one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras.

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arXiv2023

Asymptotic truth-value laws in many-valued logics

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. We obtain generalizations of Fagin's classical zero-one law for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including \L ukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp's generalization of Fagin's result. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete, and for some logics we may describe completely the set of truth-values that can be taken by sentences almost surely.

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arXiv2022

Maximality of logic without identity

Lindström theorem obviously fails as a characterization of $\mathcal{L}_{ωω}^{-} $, first-order logic without identity. In this note we provide a fix: we show that $\mathcal{L}_{ωω}^{-} $ is \emph{maximal} among abstract logics satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in \cite{Casa}), the Löwenheim--Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs we use a form of strong upwards Löwenheim--Skolem theorem not available in the framework with identity.

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arXiv2022

First-order friendliness

In this note we study a counterpart in predicate logic of the notion of 'logical friendliness', introduced into propositional logic in Makinson (2007). The result is a new consequence relation for predicate languages using first-order models. Although compactness and interpolation fail dramatically, other properties are preserved from the propositional case.

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arXiv2022

Relevant Consequence Relations: An Invitation

We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the \emph{use criterion}, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be \emph{used} in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining between \emph{multisets}. We motivate and state basic definitions of relevant consequence relations, both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases).

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arXiv2022

New foundations of reasoning via real-valued first-order logics

Many-valued logics in general, and fuzzy logics in particular, usually focus on a notion of consequence based on preservation of full truth, typical represented by the value 1 in the semantics given the real unit interval [0,1]. In a recent paper (\emph{Foundations of Reasoning with Uncertainty via Real-valued Logics}, arXiv:2008.02429v2, 2021), Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on multi-dimensional sentences that allow to prescribe any truth-values, not just 1, for the premises and conclusion of a given entailment. In this paper, we extend their work to the first-order (as well as modal) logic of multi-dimensional sentences. We give axiomatic systems and prove corresponding completeness theorems, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a 0-1 law for finitely-valued versions of these logics.

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arXiv2022

Frame definability in finitely-valued modal logics

In this paper we study frame definability in finitely-valued modal logics and establish two main results via suitable translations: (1) in finitely-valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf.~\citep[Thm.~8]{tho}), and (2) a large family of finitely-valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and \MV-algebras, or even \BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt--Thomason theorem applies immediately to these logics. In particular, we obtain the central result from~\citep{te} with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely-valued modal logics.

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arXiv2022

Craig interpolation theorem fails in bi-intuitionistic predicate logic

In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains (G. Mints, G. K. Olkhovikov and A. Urquhart. Failure of Interpolation in Constant Domain Intuitionistic Logic. Journal of Symbolic Logic, 78: 937--950 (2013)). More precisely, we show that there is a valid implication $ϕ\rightarrow ψ$ with no interpolant (i.e. a formula $θ$ in the intersection of the vocabularies of $ϕ$ and $ψ$ such that both $ϕ\rightarrow θ$ and $θ\rightarrow ψ$ are valid). Importantly, this result does not contradict the unfortunately named `Craig interpolation' theorem established by Rauszer in (Cecylia Rauszer. Craig Interpolation Theorem for an Extention of Intuitionistic Logic. Bull. Ac. Pol. Sc., 25(4), 337--341 (1977)) since that article is about the property more correctly named `deductive interpolation' (see Galatos, Jipsen, Kowalski and Ono's use of this term in N. Galatos, P. Jipsen, T. Kowalski, \& H. Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, Vol. 151. Amsterdam: Elsevier B. V. (2007)) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.

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arXiv2022

Omitting Types Theorem in hybrid-dynamic first-order logic with rigid symbols

In the the present contribution, we prove an Omitting Types Theorem (OTT) for an arbitrary fragment of hybriddynamic first-order logic with rigid symbols (i.e. symbols with fixed interpretations across worlds) closed under negation and retrieve. The logical framework can be regarded as a parameter and it is instantiated by some well-known hybrid and/or dynamic logics from the literature. We develop a forcing technique and then we study a forcing property based on local satisfiability, which lead to a refined proof of the OTT. For uncountable signatures, the result requires compactness, while for countable signatures, compactness is not necessary. We apply the OTT to obtain upwards and downwards Löwenheim-Skolem theorems for our logic, as well as a completeness theorem for its constructor-based variant. The main result of this paper can easily be recast in the institutional model theory framework, giving it a higher level of generality.

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arXiv2022

Axiomatization via translation: Hiz's warning for predicate logic

The problems of logical translation of axiomatizations and the choice of primitive operators have surfaced several times over the years. An early issue was raised by H. Hi{\. z} in the 1950s on the incompleteness of translated calculi. Further pertinent work, some of it touched on here, was done in the 1970s by W. Frank and S. Shapiro, as well as by others in subsequent decades. As we shall see, overlooking such possibilities has led to incorrect claims of completeness being made (e.g. by J. L. Bell and A. B. Slomson as well as J. N. Crossley) for axiomatizations of classical predicate logic obtained by translation from axiomatizations suited to differently chosen logical primitives. In this note we begin by discussing some problematic aspects of an early article by W. Frank on the difficulties of obtaining completeness theorems for translated calculi. Shapiro had established the incompleteness of Crossley's axiomatization by exhibiting a propositional tautology that was not provable. In contrast, to deal with Bell and Slomson's system which is complete for propositional tautologies, we go on to show that taking a formal system for classical predicate calculus with the primitive $ \exists$, setting $\forall x ϕ(x) \stackrel{\text{def}}{=}\neg \exists x \neg ϕ(x)$, and writing down a set of axioms and rules complete for the calculus with $\forall $ instead of $ \exists$ as primitive, does not guarantee completeness of the resulting system. In particular, instances of the valid schema $\exists x ϕ(x) \rightarrow \exists x \neg \negϕ(x)$ are not provable, which is analogous to what occurs in modal logic with $\Box$ and $\Diamond$.

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arXiv2021

Maximality of bi-intuitionistic propositional logic

In the style of Lindström's theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of previous work done by the authors for intuitionistic logic in: G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020).

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