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## Modal Logics are Coalgebraic (2008)

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3192 | Model Checking
- Clarke, Grumberg, et al.
- 1996
(Show Context)
Citation Context ...cience where one investigates the ongoing and temporal behavior of nonterminating programs such as operating systems. As an example we mention the automata-based verification method of model checking =-=[11]-=- and reasoning with fixpoint logics [17] for example in the context of common knowledge or temporal logics. This research also has a long and strong theoretical tradition, in which an extensive body o... |

612 |
Communicating and mobile systems: the π-calculus
- Milner
- 1999
(Show Context)
Citation Context ...sive domain equations [25, 29]. We now proceed to illustrate how coalgebraic techniques and a suitable choice of base category can be combined to derive in a systematic way a logic for the π-calculus =-=[30]-=- which characterises strong late bisimilarity and accounts for name binding; see [23] for a full treatment. This showcases yet another orthogonal aspect of genericity in the coalgebraic framework. To ... |

594 | Algebraic laws for nondeterminism and concurrency
- Hennessy, Milner
- 1985
(Show Context)
Citation Context ...ding hybrid and description logics that also axiomatise the notion of individuals, have been studied in a variety of application contexts. Apart from classical applications in the field of concurrent =-=[1]-=-, mobile [2], and probabilistic systems [3], modal logics play a central role in artificial intelligence, e.g. in the context of reasoning with uncertainty [4], non-monotonic reasoning [5], and – in p... |

535 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...made. A famous example, and indeed the ancestor of this approach, is Abramsky’s programme of Domain Theory in Logical Form, which extends Stone dualities to the solution of recursive domain equations =-=[25, 29]-=-. We now proceed to illustrate how coalgebraic techniques and a suitable choice of base category can be combined to derive in a systematic way a logic for the π-calculus [30] which characterises stron... |

521 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...o axiomatise the notion of individuals, have been studied in a variety of application contexts. Apart from classical applications in the field of concurrent [1], mobile [2], and probabilistic systems =-=[3]-=-, modal logics play a central role in artificial intelligence, e.g. in the context of reasoning with uncertainty [4], non-monotonic reasoning [5], and – in particular in their description logic incarn... |

352 |
Decidability of second-order theories and automata on infinite trees
- Rabin
- 1969
(Show Context)
Citation Context ...of landmark results. Many of these link the field to neighbouring areas such as logic and game theory, see [35] for an overview. The outstanding example here is of course Rabin’s decidability theorem =-=[36]-=- for the monadic second order logic of trees; to mention a more recent result, Janin and Walukiewicz [37] identified the modal µ-calculus as the bisimulation invariant fragment of the monadic second o... |

351 | Practical reasoning for expressive description logics
- Horrocks, Sattler, et al.
- 1999
(Show Context)
Citation Context ... across several layers of decomposition while keeping decidability, even in NEXPTIME; this contrasts sharply with existing approaches using transitive parthood, which lead quickly into undecidability =-=[21]-=-. Within the realm of rank-1 logics, the generic exponential time bounds of [39] generally do not match the actual complexity of individual logics, which is typically PSPACE. It does however turn out ... |

346 |
Reasoning About Uncertainty
- Halpern
- 2003
(Show Context)
Citation Context ... applications in the field of concurrent [1], mobile [2], and probabilistic systems [3], modal logics play a central role in artificial intelligence, e.g. in the context of reasoning with uncertainty =-=[4]-=-, non-monotonic reasoning [5], and – in particular in their description logic incarnation – in the field of knowledge representation and ontologies [6]. Modal logics are employed to reason about games... |

248 | Domain theory in logical form
- Abramsky
- 1991
(Show Context)
Citation Context ...ach comes into play by associating a logical description to every component that can be composed to obtain a description of combined systems, in the spirit of Abramsky’s Domain Theory in Logical Form =-=[25]-=-. In more detail, we have for every type functor T on sets: • a one-step syntax, consisting of a set of modal operators with arities that are used to describe possible next-state behaviours; • a one-s... |

225 |
Semantical considerations on modal logic
- Kripke
- 1963
(Show Context)
Citation Context ...every one of the current challenges involves modal logic in some form or other, variously referring e.g. to knowledge representation, logics of agents, or logics for concurrency. Today, Kripke frames =-=[12]-=- constitute the most popular semantics of modal logics, and the class of logics that can be interpreted over Kripke frames forms the class of so-called normal modal logics. However, many modal logics ... |

192 | A modal logic for coalitional power in games
- Pauly
(Show Context)
Citation Context ... in their description logic incarnation – in the field of knowledge representation and ontologies [6]. Modal logics are employed to reason about games [7] and coalitional power in multi-agent systems =-=[8]-=-. In economics, they have been used to describe probabilistic information of economic agents [9], whereas e.g. deontic logic, the logic of obligation and permission originally studied in philosophy [1... |

190 | Reasoning about knowledge and probability
- Fagin, Halpern
- 1988
(Show Context)
Citation Context ...totypic Coalgebraic Logic Satisfiability Solver CoLoSS [6]. Alternative semanticsbased algorithms [44] cover e.g. complex logics such as Presburger modal logic [12] and the modal logic of probability =-=[14]-=-, and have been used to establish new PSPACE upper bounds e.g. for Elgesem’s logic of agency [13]. Not only does this already cover an impressive collection of modal logics. What is more is that – fai... |

173 |
Modal Logic
- Chellas
- 1980
(Show Context)
Citation Context ...ture of a probabilistic modal operator manifests itself as providing a passage from properties of states to properties of successor distributions. Conditional Logic. The language of conditional logic =-=[13]-=- extends propositional logic with a binary connective The Computer Journal Vol. 00 No. 0, 0000Modal Logics are Coalgebraic 3 that we write as ⇒ using infix notation. The operator ⇒ represents a non-m... |

134 |
Deontic logic
- Wright
- 1951
(Show Context)
Citation Context ...8]. In economics, they have been used to describe probabilistic information of economic agents [9], whereas e.g. deontic logic, the logic of obligation and permission originally studied in philosophy =-=[10]-=-, is being used to model contracts in multi-agent systems. While we do not pretend to work specifically on one of the UKCRC Grand Challenges [11], it is interesting to note that every one of the curre... |

129 | Modal logics for mobile processes
- Milner, Parrow, et al.
- 1993
(Show Context)
Citation Context ...and description logics that also axiomatise the notion of individuals, have been studied in a variety of application contexts. Apart from classical applications in the field of concurrent [1], mobile =-=[2]-=-, and probabilistic systems [3], modal logics play a central role in artificial intelligence, e.g. in the context of reasoning with uncertainty [4], non-monotonic reasoning [5], and – in particular in... |

123 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...One has to describe the logical structure of the semantical base category, in this case N, and of the functor Pi. This is not hard: The case of the basic functors P, +, × has been treated in Abramsky =-=[25, 33]-=- and the axiomatisations can be reused without further modification (providing e.g. the usual modalities ✷ and ✸); allocation of new names gives rise to a modal operator [νb] for name creation. The re... |

117 |
eds). The Description Logic Handbook: Theory, Implementation, and Applications
- Baader, Calvanese, et al.
- 2003
(Show Context)
Citation Context .... in the context of reasoning with uncertainty [18], non-monotonic reasoning [16], and – in particular in their description logic incarnation – in the field of knowledge representation and ontologies =-=[4]-=-. Modal logics are employed to reason about games [32] and coalitional power in multi-agent systems [36]. In economics, they have been used to describe 1Partially supported by EPSRC grant number EP/F0... |

87 | The logic of games and its applications
- Parikh
- 1985
(Show Context)
Citation Context ... non-monotonic reasoning [5], and – in particular in their description logic incarnation – in the field of knowledge representation and ontologies [6]. Modal logics are employed to reason about games =-=[7]-=- and coalitional power in multi-agent systems [8]. In economics, they have been used to describe probabilistic information of economic agents [9], whereas e.g. deontic logic, the logic of obligation a... |

80 | Probabilistic extensions of process algebras
- Jonsson, Larsen, et al.
- 2001
(Show Context)
Citation Context ...C)) – a position on the game board and choice of strategies yields a distribution over positions on the game board. In a similar way, simple Segala systems (called probabilistic transition systems in =-=[24]-=-) can be modelled as coalgebras of type C → (P(DC)) A where (−) A represents function space: to every state c ∈ C and every label a ∈ A, one associates the non-deterministic choice of a probability di... |

71 | Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoretical Computer Science 309
- Pattinson
- 2003
(Show Context)
Citation Context ...ate. Indeed the scope of the established meta-theory of coalgebraic modal logic reaches surprisingly far. Besides basic meta-logical properties such as generic criteria for soundness and completeness =-=[14]-=-, it includes e.g. some now classical results of modal logic such as duality and ultrafilter extensions [15], but also computational aspects such as generic finite and shallow model constructions with... |

58 | Experience building a large, re-usable medical ontology using a description logic with transitivity and concept inclusions
- Rector, Horrocks
- 1997
(Show Context)
Citation Context ...ansport patterns is only one example, and we are faced with similar tasks, most prominently in the area of medicine (with a comparatively large body of knowledge formalised e.g. in the GALEN ontology =-=[38]-=-), but also in other areas like civil engineering, law, and life sciences. The diversity of the form of knowledge to be formalised, which stems from the different application areas, calls for modular ... |

42 |
Stone Spaces. Cambridge
- Johnstone
- 1982
(Show Context)
Citation Context ...ting, we wish to find a systematic and modular way of linking logics to structured models. The semantics of the logics we want to use as the basic building blocks are best described via Stone duality =-=[28]-=- and its relatives. The idea is to describe a logic (syntax and proof system) as a category of algebras (such as Boolean algebras, Heyting algebras, distributive lattices) and the models as topologica... |

39 | The Modal Logic of Agency
- Elgesem
- 1997
(Show Context)
Citation Context ... [44] cover e.g. complex logics such as Presburger modal logic [12] and the modal logic of probability [14], and have been used to establish new PSPACE upper bounds e.g. for Elgesem’s logic of agency =-=[13]-=-. Not only does this already cover an impressive collection of modal logics. What is more is that – faithful to the coalgebraic paradigm – the reasoning principles for individual logics can be combine... |

34 |
Probabilistic logic for type spaces
- Heifetz, Mongin
(Show Context)
Citation Context ...s [6]. Modal logics are employed to reason about games [7] and coalitional power in multi-agent systems [8]. In economics, they have been used to describe probabilistic information of economic agents =-=[9]-=-, whereas e.g. deontic logic, the logic of obligation and permission originally studied in philosophy [10], is being used to model contracts in multi-agent systems. While we do not pretend to work spe... |

34 | PSPACE bounds for rank-1 modal logics
- Schröder, Pattinson
(Show Context)
Citation Context ... of modal logic such as duality and ultrafilter extensions [15], but also computational aspects such as generic finite and shallow model constructions with ensuing decidability and complexity results =-=[16, 17]-=-, as well as proof-theoretic results such as cut elimination and interpolation [18]. The parametricity over system types afforded by the coalgebraic approach may be combined with reasoning principles ... |

33 |
Automata for the modal µ-calculus and related results
- Janin, Walukiewicz
- 1995
(Show Context)
Citation Context ...e [35] for an overview. The outstanding example here is of course Rabin’s decidability theorem [36] for the monadic second order logic of trees; to mention a more recent result, Janin and Walukiewicz =-=[37]-=- identified the modal µ-calculus as the bisimulation invariant fragment of the monadic second order logic of labelled transition systems. The automata that we refer to come in many kinds and shapes, a... |

31 | A finite model construction for coalgebraic modal logic
- Schröder
(Show Context)
Citation Context ... of modal logic such as duality and ultrafilter extensions [15], but also computational aspects such as generic finite and shallow model constructions with ensuing decidability and complexity results =-=[16, 17]-=-, as well as proof-theoretic results such as cut elimination and interpolation [18]. The parametricity over system types afforded by the coalgebraic approach may be combined with reasoning principles ... |

25 | Automata and fixed point logic: a coalgebraic perspective
- Venema
(Show Context)
Citation Context ...ance game is exactly the standard one that we know from classical automata theory. The coalgebraic perspective on automata theory that we just described, applies to set coalgebras of arbitrary type T =-=[46]-=-. In addition, in order to specify and reason about ongoing coalgebraic behavior, one may extend the coalgebraic logics mentioned in the earlier parts of this note, with fixpoint operators, obtaining ... |

24 |
A fully abstract domain model for the π-calculus
- Stark
- 1996
(Show Context)
Citation Context ...cesses is captured by a type functor, but now taking sets with names, i.e. the category N as the semantic base. For the π-calculus, the following functor on N was introduced independently in [31] and =-=[32]-=-: Pi(X) = P(X + N × X N + N × (N × X) + N × δX) As before P is for non-determinism, + is binary choice; N is a constant for the set of names, δ allows creating a fresh name, and (·) N inputs a (possib... |

20 | Majority logic
- Pacuit, Salame
- 2004
(Show Context)
Citation Context ...or such diverse logics as K (or KD), coalition logic, graded modal logic, and probabilistic modal logic, and moreover has led (simultaneously with [12]) to a new PSPACE upper bound for majority logic =-=[31]-=-. This algorithm, in modularised form, is implemented in the prototypic Coalgebraic Logic Satisfiability Solver CoLoSS [6]. Alternative semanticsbased algorithms [44] cover e.g. complex logics such as... |

18 | Coalgebraic automata theory: Basic results
- Kupke, Venema
(Show Context)
Citation Context ...dition on the functor T (namely, that it preserves so-called weak pullbacks), most of the important results in the theory of tree and graph automata, can in fact be proved at this level of generality =-=[27]-=-. As examples we mention the following results: • reduction of alternating to nondeterministic automata • various closure properties of recognizable languages • decidability and finite model property ... |

16 | Modular algorithms for heterogeneous modal logics
- Schröder, Pattinson
- 2007
(Show Context)
Citation Context ...s of varying strength, leading to e.g. generic fixed-point logics [19] and generic hybrid logics [20]. The coalgebraic framework is not only parametric, but also modular w.r.t. combinations of logics =-=[21, 22]-=-. Moreover, the parametricity extends also to the underlying form of propositional logics and thus includes e.g. logics over nominal frameworks [23]. Having thus emphasised the suitability of the coal... |

16 | Rank-1 modal logics are coalgebraic
- Schröder, Pattinson
- 2007
(Show Context)
Citation Context ...particular application domains. Coalgebraic modal logic caters for both semantics-centered approaches, where one needs to design a logic to describe given semantic phenomena, and syntax-centered ones =-=[42]-=-, where one needs semantic underpinnings for the analysis of given means of expression. As new and domain specific modal logics emerge steadily, for example in the fields of knowledge representation a... |

15 | Ultrafilter extensions of coalgebras
- Kupke, Kurz, et al.
- 2005
(Show Context)
Citation Context ...sides basic meta-logical properties such as generic criteria for soundness and completeness [14], it includes e.g. some now classical results of modal logic such as duality and ultrafilter extensions =-=[15]-=-, but also computational aspects such as generic finite and shallow model constructions with ensuing decidability and complexity results [16, 17], as well as proof-theoretic results such as cut elimin... |

15 |
PSPACE reasoning for rank-1 modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ... of modal logic such as duality and ultrafilter extensions [25], but also computational aspects such as generic finite and shallow model constructions with ensuing decidability and complexity results =-=[39, 40]-=- and decidability of generic fixed-point logics [26], as well as proof-theoretic results such as cut elimination and interpolation [34]. The coalgebraic framework is not only parametric, but also modu... |

13 | Conditional logics of belief change
- Friedman, Halpern
- 1994
(Show Context)
Citation Context ... concurrent [1], mobile [2], and probabilistic systems [3], modal logics play a central role in artificial intelligence, e.g. in the context of reasoning with uncertainty [4], non-monotonic reasoning =-=[5]-=-, and – in particular in their description logic incarnation – in the field of knowledge representation and ontologies [6]. Modal logics are employed to reason about games [7] and coalitional power in... |

13 | Pi-calculus in logical form
- Bonsangue, Kurz
- 2007
(Show Context)
Citation Context ..., but also modular w.r.t. combinations of logics [9, 41]. Moreover, the parametricity extends also to the underlying form of propositional logics and thus includes e.g. logics over nominal frameworks =-=[5]-=-. Having thus emphasised the suitability of the coalgebraic approach as a universal framework for modal logic, we set out to develop our vision of a unified description logic with universally applicab... |

13 |
Presburger modal logic is only PSPACE-complete
- Demri, Lugiez
- 2006
(Show Context)
Citation Context ... [40], captures the known tight PSPACE upper bounds for such diverse logics as K (or KD), coalition logic, graded modal logic, and probabilistic modal logic, and moreover has led (simultaneously with =-=[12]-=-) to a new PSPACE upper bound for majority logic [31]. This algorithm, in modularised form, is implemented in the prototypic Coalgebraic Logic Satisfiability Solver CoLoSS [6]. Alternative semanticsba... |

11 | Closure properties of coalgebra automata
- Kupke, Venema
(Show Context)
Citation Context ... interpolation [18]. The parametricity over system types afforded by the coalgebraic approach may be combined with reasoning principles of varying strength, leading to e.g. generic fixed-point logics =-=[19]-=- and generic hybrid logics [20]. The coalgebraic framework is not only parametric, but also modular w.r.t. combinations of logics [21, 22]. Moreover, the parametricity extends also to the underlying f... |

11 | Beyond rank 1: Algebraic semantics and finite models for coalgebraic logics
- Pattinson, Schröder
- 2008
(Show Context)
Citation Context ...e latter two have, to our knowledge, first been stated and proved in a coalgebraic setting [21, 22]. So far, only a limited amount of interaction between the different components can be accounted for =-=[26]-=- but more general results are anticipated. A variation of the main technique used to derive expressiveness results w.r.t. bisimilarity can be used to obtain logics that characterise (weaker) notions o... |

9 |
A modular approach to defining and characterising notions of simulation
- Cı̂rstea
(Show Context)
Citation Context ...eral results are anticipated. A variation of the main technique used to derive expressiveness results w.r.t. bisimilarity can be used to obtain logics that characterise (weaker) notions of similarity =-=[27]-=-. In addition to the three ingredients mentioned earlier, a fourth one identifies a particular notion of similarity between coalgebras by specifying a one-step simulation relation between one-step beh... |

9 | CoLoSS: The Coalgebraic Logic Satisfiability Solver (system description
- Calin, Myers, et al.
- 2009
(Show Context)
Citation Context ...ed (simultaneously with [12]) to a new PSPACE upper bound for majority logic [31]. This algorithm, in modularised form, is implemented in the prototypic Coalgebraic Logic Satisfiability Solver CoLoSS =-=[6]-=-. Alternative semanticsbased algorithms [44] cover e.g. complex logics such as Presburger modal logic [12] and the modal logic of probability [14], and have been used to establish new PSPACE upper bou... |

8 | Admissibility of cut in coalgebraic logics
- Pattinson, Schröder
(Show Context)
Citation Context ... aspects such as generic finite and shallow model constructions with ensuing decidability and complexity results [16, 17], as well as proof-theoretic results such as cut elimination and interpolation =-=[18]-=-. The parametricity over system types afforded by the coalgebraic approach may be combined with reasoning principles of varying strength, leading to e.g. generic fixed-point logics [19] and generic hy... |

8 |
Modular construction of complete coalgebraic logics
- Cîrstea, Pattinson
- 2007
(Show Context)
Citation Context ...s of varying strength, leading to e.g. generic fixed-point logics [19] and generic hybrid logics [20]. The coalgebraic framework is not only parametric, but also modular w.r.t. combinations of logics =-=[21, 22]-=-. Moreover, the parametricity extends also to the underlying form of propositional logics and thus includes e.g. logics over nominal frameworks [23]. Having thus emphasised the suitability of the coal... |

6 | 2008a) How many toes do I have? Parthood and number restrictions in description logics
- Schröder, Pattinson
(Show Context)
Citation Context ...ectronic Workshops in Computing The British Computer Society 7Modal Logics are Coalgebraic on the underlying models. Advances into the generic study of logics outside rank 1 are being made currently =-=[35, 43]-=-. The first widely applicable criterion [39] made use of a filtration-based finite model construction to obtain a generic decidability criterion and ensuing EXPTIME or NEXPTIME upper bounds, establish... |

6 | 2008b) Shallow models for non-iterative modal logics
- Schröder, Pattinson
(Show Context)
Citation Context ...CE upper bound for majority logic [31]. This algorithm, in modularised form, is implemented in the prototypic Coalgebraic Logic Satisfiability Solver CoLoSS [6]. Alternative semanticsbased algorithms =-=[44]-=- cover e.g. complex logics such as Presburger modal logic [12] and the modal logic of probability [14], and have been used to establish new PSPACE upper bounds e.g. for Elgesem’s logic of agency [13].... |

5 |
A fully-abstract model for the pi-calculus (extended abstract
- Fiore, Moggi, et al.
- 1996
(Show Context)
Citation Context ...rs of processes is captured by a type functor, but now taking sets with names, i.e. the category N as the semantic base. For the π-calculus, the following functor on N was introduced independently in =-=[31]-=- and [32]: Pi(X) = P(X + N × X N + N × (N × X) + N × δX) As before P is for non-determinism, + is binary choice; N is a constant for the set of names, δ allows creating a fresh name, and (·) N inputs ... |

3 |
Modular Games for Coalgebraic Fixed Point Logics
- Ĉırstea, Sadrzadeh
- 2008
(Show Context)
Citation Context ...s of recognizable languages • decidability and finite model property of an associated coalgebraic fixpoint logic. It should be noted that all the above constructions are naturally of a modular nature =-=[10]-=- and so truly faithful to the coalgebraic paradigm. 1.4. Generic Algorithms and Reasoners for Modal Logics Given the extremely broad scope of coalgebraic modal logic, one of the most important aspects... |

1 |
Coalgebraic hybrid logic. Unpublished manuscript, available under http://www.informatik.uni-bremen.de/ ∼lschrode/papers/hybrid.pdf
- Myers, Pattinson, et al.
- 2008
(Show Context)
Citation Context ...tricity over system types afforded by the coalgebraic approach may be combined with reasoning principles of varying strength, leading to e.g. generic fixed-point logics [19] and generic hybrid logics =-=[20]-=-. The coalgebraic framework is not only parametric, but also modular w.r.t. combinations of logics [21, 22]. Moreover, the parametricity extends also to the underlying form of propositional logics and... |