Bakhtiari, Zeinab (author)

Hansen, H.H. (author)

Kurz, A (author)

Hansen, H.H.

Bakhtiari, Zeinab

Kurz, A

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A (Co)algebraic Approach to Hennessy-Milner Theorems for Weakly Expressive Logics

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Coalgebraic modal logic, as in [9, 6], is a framework in which modal logics for specifying coalgebras can be developed parametric in the signature of the modal language and the coalgebra type functor T. Given a base logic (usually classical propositional logic), modalities are interpreted via so-called predicate liftings for the functor T. These are natural transformations that turn a predicate over the state space X into a predicate over TX. Given that T-coalgebras come with general notions of T-bisimilarity [11] and behavioral equivalence [7], coalgebraic modal logics are designed to respect those. In particular, if two states are behaviourally equivalent then they satisfy the same formulas. If the converse holds, then the logic is said to be expressive. and we have a generalisation of the classic Hennessy-Milner theorem [5] which states that over the class of image-fjnite Kripke models, two states are Kripke bisimilar if and only if they satisfy the same formulas in Hennessy-Milner logic

Hansen, Helle Hvid

Kurz, Alexander

Chapman University Digital Commons

Chapman UniversityChapman University Digital CommonsEngineering Faculty Articles and Research Fowler School of Engineering1-2019A (Co)algebraic Approach to Hennessy-MilnerTheorems for Weakly Expressive LogicsZeinab BakhtiariUniversité de LorraineHelle Hvid HansenDelft University of TechnologyAlexander KurzChapman University, akurz@chapman.eduFollow this and additional works at: https://digitalcommons.chapman.edu/engineering_articlesPart of the Algebra CommonsThis Article is brought to you for free and open access by the Fowler School of Engineering at Chapman University Digital Commons. It has beenaccepted for inclusion in Engineering Faculty Articles and Research by an authorized administrator of Chapman University Digital Commons. Formore information, please contact laughtin@chapman.edu.Recommended CitationZ. Bakhtiari, H. H. Hansen, and A. Kurz. A (co)algebraic approach to Hennessy-Milner theorems for weakly expressive logics.Presented at SYSMICS, Amsterdam, 2019.A (Co)algebraic Approach to Hennessy-Milner Theorems for WeaklyExpressive LogicsCommentsThis paper was presented at SYSMICS 2019, held in January 2019 at the University of Amsterdam.This article is available at Chapman University Digital Commons: https://digitalcommons.chapman.edu/engineering_articles/69A (Co)algebraic Approach to Hennessy-Milner Theoremsfor Weakly Expressive LogicsZeinab Bakhtiari1∗, Helle Hvid Hansen2, and Alexander Kurz31 LORIA, CNRS-Universite´ de Lorraine, France, bakhtiarizeinab@gmail.com2 Delft University of Technology, Delft, The Netherlands, h.h.hansen@tudelft.nl3 Department of Mathematics and Computer Science, Chapman University, USA. axhkrz@gmail.com1 IntroductionCoalgebraic modal logic, as in [9, 6], is a framework in which modal logics for specifying coalgebrascan be developed parametric in the signature of the modal language and the coalgebra typefunctor T . Given a base logic (usually classical propositional logic), modalities are interpretedvia so-called predicate liftings for the functor T . These are natural transformations that turn apredicate over the state space X into a predicate over TX. Given that T -coalgebras come withgeneral notions of T -bisimilarity [11] and behavioral equivalence [7], coalgebraic modal logicsare designed to respect those. In particular, if two states are behaviourally equivalent then theysatisfy the same formulas. If the converse holds, then the logic is said to be expressive. andwe have a generalisation of the classic Hennessy-Milner theorem [5] which states that over theclass of image-finite Kripke models, two states are Kripke bisimilar if and only if they satisfythe same formulas in Hennessy-Milner logic.General conditions for when an expressive coalgebraic modal logic for T -coalgebras existshave been identified in [10, 2, 12]. A condition that ensures that a coalgebraic logic is expressive iswhen the set of predicate liftings chosen to interpret the modalities is separating [10]. Informally,a collection of predicate liftings is separating if they are able to distinguish non-identical elementsfrom TX. This line of research in coalgebraic modal logic has thus taken as starting point thesemantic equivalence notion of behavioral equivalence (or T -bisimilarity), and provided resultsfor how to obtain an expressive logic. However, for some applications, modal logics that arenot expressive are of independent interest. Such an example is given by contingency logic (seee.g. [3, 8]). We can now turn the question of expressiveness around and ask, given a modallanguage, what is a suitable notion of semantic equivalence?This abstract is a modest extension of [1] in which the first two authors proposed a notionof Λ-bisimulation which is parametric in a collection Λ of predicate liftings, and thereforetailored to the expressiveness of a given coalgebraic modal logic. The main result was a finitaryHennessy-Milner theorem (which does not assume Λ is separating): If T is finitary, then twostates are Λ-bisimilar if and only if they satisfy the same modal Λ-formulas. The definitionof Λ-bisimulation was formulated in terms of so-called Z-coherent pairs, where Z is the Λ-bisimulation relation. It was later observed by the third author that Λ-bisimulations can becharacterised as the relations Z between T -coalgebras for which the dual relation (consisting ofso-called Z-coherent pairs) is a congruence between the complex algebras. Here we collect thoseresults.∗Zeinab Bakthiari was funded by ERC grant EPS 313360.A (Co)algebraic Approach to Hennessy-Milner Theorems Bakhtiari, Hansen and Kurz2 Syntax and semantics of coalgebraic modal logic.Due to lack of space, we assume the reader is familiar with the basic theory of coalgebras andalgebras for a functor, and with coalgebraic modal logic. Here we only introduce a few basicconcepts and fix notation. We refer to [6, 11] for more details.A similarity type Λ is a set of modal operators with finite arities. Given such a Λ, the set LΛof modal formulas is defined in the usual inductive manner.We denote by Q the contravariant powerset functor on Set. A T -coalgebraic semanticsof LΛ-formulas is given by providing a Λ-structure (T, ([[♥]])♥∈Λ) where T is a functor onSet, and for each n-ary ♥ ∈ Λ, [[♥]] is an n-ary predicate lifting, i.e., [[♥]] : Qn ⇒ QT is anatural transformation. Different choices of predicate liftings yield different Λ-structures andconsequently different logics.Given a Λ-structure (T, ([[♥]])♥∈Λ), and a T -coalgebra X = (X, γ : X → TX), the truth ofLΛ-formulas in X is defined inductively in the usual manner for atoms (i.e., > and ⊥) and Booleanconnectives, and for modalities: (X, υ), x |= ♥(ϕ1, . . . , ϕn) iff γ(x) ∈ [[♥]]X([[ϕ1]]X, . . . , [[ϕn]]X).(Atomic propositions can be included in the usual way via a valuation.)In the remainder, we let T be a fixed but arbitrary endofunctor on the category Set of setsand functions, and X = (X, γ) and Y = (Y, δ) are T -coalgebras. We write X, x ≡Λ Y, y, if X, xand Y, y satisfy the same LΛ-formulas,3 Λ-bisimulationsLet R ⊆ X × Y be a relation with projections pil : R→ X and pir : R→ Y , and let U ⊆ X andV ⊆ Y . The pair (U, V ) is R-coherent if R[U ] ⊆ V and R−1[V ] ⊆ U . One easily verifies that(U, V ) is R-coherent iff (U, V ) is in the pullback of Qpil and Qpir.Definition 3.1 (Λ-bisimulation )A relation Z ⊆ X × Y is a Λ-bisimulation between X and Y, if whenever (x, y) ∈ Z, thenfor all ♥ ∈ Λ, n-ary, and all Z-coherent pairs (U1, V1), . . . , (Un, Vn), we have thatγ(x) ∈ [[♥]]X(U1, . . . , Un) iff δ(y) ∈ [[♥]]Y (V1, . . . , Vn). (Coherence)We write X, x ∼Λ Y, y, if there is a Λ-bisimulation between X and Y that contains (x, y). AΛ-bisimulation on a T -coalgebra X is a Λ-bisimulation between X and X.We have the following basic properties.Lemma 3.21. The set of Λ-bisimulations between two T -coalgebras forms a complete lattice.2. On a single T-coalgebra, the largest Λ-bisimulation is an equivalence relation.3. Λ-bisimulations are closed under converse, but not composition.The following proposition compares Λ-bisimulations with the coalgebraic notions of T -bisimulations [11] and the weaker notion of precocongruences [4]. Briefly stated, a relation is aprecocongruence of its pushout is a behavioural equivalence [7]).Proposition 3.3 Let X = (X, γ) and Y = (Y, δ) be T -coalgebras, and Z be a relation betweenX and Y .1. If Z is a T -bisimulation then Z is a Λ-bisimulation.2A (Co)algebraic Approach to Hennessy-Milner Theorems Bakhtiari, Hansen and Kurz2. If Z is a precocongruence then Z is a Λ-bisimulation.3. If Λ is separating then Z is a Λ-bisimulation iff Z is a precocongruence.It was shown in [4, Proposition 3.10] that, in general, T -bisimilarity implies precocongruenceequivalence which in turn implies behavioural equivalence [7]. This fact together with Proposi-tion 3.3 tells us that Λ-bisimilarity implies behavioural equivalence, whenever Λ is separating.Moreover, it is well known [11] that if T preserves weak pulbacks, then T -bisimilarity coincideswith behavioural equivalence. Hence in this case, by Proposition 3.3, it follows that Λ-bisimilaritycoincides with T -bisimilarity and behavioural equivalence.The main result in [1] is the following.Theorem 3.4 (Finitary Hennessy-Milner theorem) If T is a finitary functor, then1. For all states x, x′ ∈ X: X, x ≡Λ X, x′ iff X, x ∼Λ X, x.′2. For all x ∈ X and y ∈ Y : X, x ≡Λ Y, y iff X+ Y, inl(x) ∼Λ X+ Y, inr(y).where inl, inr are the injections into the coproduct/disjoint union.4 Λ-Bisimulations as duals of congruencesWe now use the fact that the contravariant powerset functor Q can be viewed as one part of theduality between Set and CABA, the category of complete atomic Boolean algebras and theirhomomorphisms. By duality, Q turns a pushout in Set into a pullback in CABA. So given arelation Z ⊆ X × Y with projections pil, pir (forming a span in Set), and letting (P, pl, pr) be itspushout, we have that (QP,Qpl, Qpr) ∼= (pb(Qpil, Qpir), Qpil, Qpir).In the context of coalgebraic modal logic, we define complex algebras as follows. Thisdefinition coincides with the classic one.Definition 4.1 (Complex algebras)• Let L : CABA→ CABA be the functor L(A) =∐♥∈ΛAar(♥), and let σ : LQ =⇒ QT be thebundling up of [[Λ]] into one natural transformation. For example, if Λ consists of one unarymodality and one binary modality, then L(A) = A+A2 and σX : QX + (QX)2 =⇒ QTX.• The complex algebra of X = (X, γ : X → TX) is the L-algebra X∗ = (QX, γ∗) whereγ∗ = LQXσX //QTXQγ//QX .We can now reformulate the definition of Λ-bisimilarity in terms of the complex algebrasassociated with the coalgebras (by using (QP,Qpl, Qpr) ∼= (pb(Qpil, Qpir), Qpil, Qpir)).Lemma 4.2 Z is Λ-bisimulation if and only if the following diagram commutes:LQPLQX LQYQX QZ QYLQpl LQprγ∗ δ∗Qpil Qpir3A (Co)algebraic Approach to Hennessy-Milner Theorems Bakhtiari, Hansen and KurzProposition 4.3 Z is Λ-bisimulation between X and Y iff the dual of its pushout is a congruencebetween the complex algebras X∗ and Y∗ (i.e. a span in the category of L-algebras and L-algebrahomomorphisms).Proof. (⇒) Since (QP,Qpl, Qpr) is a pullback of (QZ,Qpil, Qpir), we get a map h : LQP → QPsuch that (QP,Qpl, Qpr) is a congruence:LQPLQX LQYQX QZ QYQPLQpl LQprγ∗ δ∗Qpil QpirQpl Qpr(⇐) Follows from commutativity of pullback square.References[1] Z. Bakhtiari and H.H. Hansen. Bisimulation for Weakly Expressive Coalgebraic Modal Logics. InFilippo Bonchi and Barbara Ko¨nig, editors, 7th Conference on Algebra and Coalgebra in ComputerScience (CALCO 2017), volume 72 of Leibniz International Proceedings in Informatics (LIPIcs),pages 4:1–4:16, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.[2] M. B´ılkova´ and M. Dosta´l. Expressivity of many-valued modal logics, coalgebraically. In WoLLIC2016, volume 9803 of LNCS, pages 109–124. Springer, 2016.[3] J. Fan, Y. Wang, and H. van Ditmarsch. Almost necessary. In AiML 2014, pages 178–196. CollegePublications, 2014.[4] H.H. Hansen, C. Kupke, and E. Pacuit. Neighbourhood structures: Bisimilarity and basic modeltheory. Logical Methods in Computer Science, 5(2:2), 2009.[5] M. Hennessy and R. Milner. Algebraic laws for non-determinism and concurrency. Journal of theACM, 32:137–161, 1985.[6] C. Kupke and D. Pattinson. Coalgebraic semantics of modal logics: an overview. TheoreticalComputer Science, 412(38):5070–5094, 2011.[7] A. Kurz. Logics for coalgebras and applications to computer science. PhD thesis, Ludwig-Maximilians-Universita¨t Mu¨nchen, 2000.[8] H. Montgomery and R. Routley. Contingency and non-contingency bases for normal modal logics.Logique et Analyse, 9:318–328, 1966.[9] D. Pattinson. Coalgebraic modal logic: Soundness, completeness and decidability of local conse-quence. Theoretical Computer Science, 309(1-3):177–193, 2003.[10] D. Pattinson et al. Expressive logics for coalgebras via terminal sequence induction. Notre DameJournal of Formal Logic, 45(1):19–33, 2004.[11] J.J.M.M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science,249(1):3–80, 2000.[12] L. Schro¨der. Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical ComputerScience, 390:230–247, 2008.4

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A (Co)algebraic Approach to Hennessy-Milner Theorems for Weakly Expressive Logics

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