Canonical Model Construction Research Articles

Many-valued coalgebraic modal logic: One-step completeness and finite model property

In this paper, we investigate the many-valued version of coalgebraic modal logic through the predicate lifting approach. Coalgebras, understood as generic transition systems, can serve as semantic structures for various kinds of modal logics. A well-known result in coalgebraic modal logic is that its completeness can be determined at the one-step level. We generalize the result to the finitely many-valued case by using the canonical model construction. We prove the result for coalgebraic modal logics based on three different many-valued algebraic structures, namely the finitely-valued Łukasiewicz algebra, the commutative integral Full-Lambek algebra (FLew-algebra) expanded with canonical constants and Baaz Delta, and the FLew-algebra expanded with valuation operations. In addition, we also prove the finite model property of the many-valued coalgebraic modal logic by using the filtration technique.

Practical output regulation for nonlinear multiagent systems with event‐triggered control protocol under directed switching networks

SummaryIn this paper, the event‐triggered‐based cooperative global robust practical output regulation problem is studied for nonlinear multiagent systems with directed switching networks. Firstly, the considered problem can be resolved with a prerequisite where the problem is solvable by an input‐to‐state stabilizing controller with measurement feedback. Specifically, the considered problem can be converted into a stabilization one for the so‐called augmented system through the canonical internal model construction. Then by combining the multiple Lyapunov functions (MLF) with the average dwell time (ADT) method, it is demonstrated that this stabilization problem can be solved by output feedback with measurement disturbances, which indicates a solution to the initial problem. At last, a suitable event‐triggering mechanism is proposed to save communication resources and reduce computational burden among all the agents. Finally, a group of FitzHugh‐Nagumo multiagent systems is provided to verify our theoretical results.

GALOIS CONJUGATES OF SPECIAL POINTS AND SPECIAL SUBVARIETIES IN SHIMURA VARIETIES

AbstractLet $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\text{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih’s construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.

FIRST-ORDER POSSIBILITY MODELS AND FINITARY COMPLETENESS PROOFS

AbstractThis article builds on Humberstone’s idea of defining models of propositional modal logic where total possible worlds are replaced by partial possibilities. We follow a suggestion of Humberstone by introducing possibility models for quantified modal logic. We show that a simple quantified modal logic is sound and complete for our semantics. Although Holliday showed that for many propositional modal logics, it is possible to give a completeness proof using a canonical model construction where every possibility consists of finitely many formulas, we show that this is impossible to do in the first-order case. However, one can still construct a canonical model where every possibility consists of a computable set of formulas and thus still of finitely much information.

Reasoning in the Defeasible Description Logic —computing standard inferences under rational and relevant semantics

Defeasible Description Logics (DDLs) extend Description Logics with defeasible concept inclusions. Reasoning in DDLs often employs rational closure according to the (propositional) KLM postulates. A well-known approach to lift this closure to DDLs is by so-called materialisation. Previously investigated algorithms for materialisation-based reasoning employ reductions to classical reasoning using all Boolean connectors. As a first result in this paper, we present a materialisation-based algorithm for the sub-Boolean DDL ▪, using a reduction to reasoning in classical ▪, rendering materialisation-based defeasible reasoning tractable.The main contribution of this article is a kind of canonical model construction, which can be used to decide defeasible subsumption and instance queries in ▪ under rational and the stronger relevant entailment. Our so-called typicality models can reproduce the entailments obtained from materialisation-based rational and relevant closure and, more importantly, obtain stronger versions of rational and relevant entailment. These do not suffer from neglecting defeasible information for concepts appearing nested inside quantifications, which all materialisation-based approaches do. We also show the computational complexity of defeasible subsumption and instance checking in our stronger rational and relevant semantics.

A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

RETRACTED ARTICLE: A completeness theorem for continuous predicate modal logic

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.

A Canonical Model Construction for Iteration-Free PDL with Intersection

We study the axiomatisability of the iteration-free fragment of Propositional Dynamic Logic with Intersection and Tests. The combination of program composition, intersection and tests makes its proof-theory rather difficult. We develop a normal form for formulae which minimises the interaction between these operators, as well as a refined canonical model construction. From these we derive an axiom system and a proof of its strong completeness.

HOW TO EXPRESS SELF-REFERENTIAL PROBABILITY. A KRIPKEAN PROPOSAL

AbstractWe present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of theω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent.

Epistemic Closure and Epistemic Logic I: Relevant Alternatives and Subjunctivism

Epistemic closure has been a central issue in epistemology over the last forty years. According to versions of the relevant alternatives and subjunctivist theories of knowledge, epistemic closure can fail: an agent who knows some propositions can fail to know a logical consequence of those propositions, even if the agent explicitly believes the consequence (having “competently deduced” it from the known propositions). In this sense, the claim that epistemic closure can fail must be distinguished from the fact that agents do not always believe, let alone know, the consequences of what they know—a fact that raises the “problem of logical omniscience” that has been central in epistemic logic. This paper, part I of II, is a study of epistemic closure from the perspective of epistemic logic. First, I introduce models for epistemic logic, based on Lewis’s models for counterfactuals, that correspond closely to the pictures of the relevant alternatives and subjunctivist theories of knowledge in epistemology. Second, I give an exact characterization of the closure properties of knowledge according to these theories, as formalized. Finally, I consider the relation between closure and higher-order knowledge. The philosophical repercussions of these results and results from part II, which prompt a reassessment of the issue of closure in epistemology, are discussed further in companion papers. As a contribution to modal logic, this paper demonstrates an alternative approach to proving modal completeness theorems, without the standard canonical model construction. By “modal decomposition” I obtain completeness and other results for two non-normal modal logics with respect to new semantics. One of these logics, dubbed the logic of ranked relevant alternatives, appears not to have been previously identified in the modal logic literature. More broadly, the paper presents epistemology as a rich area for logical study.

Continuous Markovian Logics - Axiomatization and Quantified Metatheory

Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-time labelled Markov processes with arbitrary (analytic) state-spaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions of a CMP. In this paper we present weak and strong complete axiomatizations for CML and prove a series of metaproperties, including the finite model property and the construction of canonical models. CML characterizes stochastic bisimilarity and it supports the definition of a quantified extension of the satisfiability relation that measures the "compatibility" between a model and a property. In this context, the metaproperties allows us to prove two robustness theorems for the logic stating that one can perturb formulas and maintain "approximate satisfaction".

An Example of Non-Homeomorphic Conjugate Varieties

(C) arenot homeomorphic.Recently, Milne and Suh [MS], gave further examples in the context of connectedShimura varieties. Their method is to find a conjugate such that the reductive groupunderlying the Shimura datum is different, and then apply the super-rigidity resultsof Margulis.Our examples are in the same context as that of Milne and Suh, but we work withShimura’s construction of canonical models ([Sh]). Shimura’s construction allowsus to identify the adelic congruence subgroup defining the conjugate variety as aconjugate by an element of the adjoint group. We then appeal to Mostow rigidityand the failure of strong approximation (or non-triviality of class number) for theadjoint group to get at the desired examples. In our example, the congruent latticesdefining the variety and it’s conjugate are commensurable. Earlier in [R], we observedusing Shimura’s construction coupled with the theorems of Labesse and Langlandson the mulitplicity of cusp forms for SL(1,D), that a Galois twist of these spacesattached to SL(1,D) over the reflex field preserves the spectrum of the Laplacian;this provides examples of locally symmetric spaces attached to a quaternion divisionalgebra over a number field which are isospectral but not isometric,

Generalized Kripke Frames

Algebraic work [9] shows that the deep theory of possible world semantics is available in the more general setting of substructural logics, at least in an algebraic guise. The question is whether it is also available in a relational form.This article seeks to set the stage for answering this question. Guided by the algebraic theory, but purely relationally we introduce a new type of frames. These structures generalize Kripke structures but are two-sorted, containing both worlds and co-worlds. These latter points may be viewed as modelling irreducible increases in information where worlds model irreducible decreases in information. Based on these structures, a purely model theoretic and uniform account of completeness for the implication-fusion fragment of various substructural logics is given. Completeness is obtained via a generalization of the standard canonical model construction in combination with correspondence results.

Coalgebraic modal logic of finite rank

This paper studies coalgebras from the perspective of finite observations. We introduce the notion of finite step equivalence and a corresponding category with finite step equivalence-preserving morphisms. This category always has a final object, which generalises the canonical model construction from Kripke models to coalgebras. We then turn to logics whose formulae are invariant under finite step equivalence, which we call logics of rank $\omega$. For these logics, we use topological methods and give a characterisation of compact logics and definable classes of models.

Coalgebraic modal logic: soundness, completeness and decidability of local consequence

This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations.

Kripke Semantics for Modal Substructural Logics

We introduce Kripke semantics for modal substructural logics, and prove the completeness theorems with respect to the semantics. The completeness theorems are proved using an extended Ishihara's method of canonical model construction (Ishihara, 2000). The framework presented can deal with a broad range of modal substructural logics, including a fragment of modal intuitionistic linear logic, and modal versions of Corsi's logics, Visser's logic, Mendez's logics and relevant logics.

On the completeness of first degree weakly aggregative modal logics

This paper extends David Lewis’ result that all first degree modal logics are complete to weakly aggregative modal logic by providing a filtration-theoretic version of the canonical model construction of Apostoli and Brown. The completeness and decidability of all first-degree weakly aggregative modal logics is obtained, with Lewis’s result for Kripkean logics recovered in the case k=1.

A Completeness Proof for a Logic with an Alternative Necessity Operator

We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.

Representation of Residuated Semigroups in Some Algebras of Relations (The Method of Canonical Models)

We prove some theorems on representation of residuated semigroups and monoids in algebras of binary relations. One of them has been proved in Andreka and Mikulas [3], and the others are new, though also closely related to some results in [3]. The main novelty of this paper is a simple method of proof, based upon a construction of canonical models for Lambek calculi by means of labeled formulas, whereas [3] uses graph-theoretic constructions. We handle labeled formulas in a way similar to Kurtonina [15] and the second author [14], but we make no use of Labeled Deductive Systems, which is an essential simplification.

To be and not to be: Dialectical tense logic

The paper concerns time, change and contradiction, and is in three parts. The first is an analysis of the problem of the instant of change. It is argued that some changes are such that at the instant of change the system is in both the prior and the posterior state. In particular there are some changes from p being true to ℸp being true where a contradiction is realized. The second part of the paper specifies a formal logic which accommodates this possibility. It is a tense logic based on an underlying paraconsistent prepositional logic, the logic of paradox. (See the author's article of the same name Journal of Philosophical Logic 8 (1979).) Soundness and completeness are established, the latter by the canonical model construction, and extensions of the basic system briefly considered. The final part of the paper discusses Leibniz's principle of continuity: “Whatever holds up to the limit holds at the limit”. It argues that in the context of physical changes this is a very plausible principle. When it is built into the logic of the previous part, it allows a rigorous proof that change entails contradictions. Finally the relation of this to remarks on dialectics by Hegel and Engels is briefly discussed.