## A Finite Model Construction For Coalgebraic Modal Logic

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@MISC{Schröder_afinite,

author = {Lutz Schröder},

title = {A Finite Model Construction For Coalgebraic Modal Logic},

year = {}

}

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### Abstract

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy-Milner logic, graded modal logic and probabilistic modal logic.

### Citations

1459 |
Theory of linear and integer programming
- Schrijver
- 1986
(Show Context)
Citation Context ...ings λk for the multiset functor of Example 7.3. Then the one-step satisfiability problem amounts to deciding the solvability of systems of linear inequations over the naturals; this problem is in NP =-=[22]-=-. By Theorem 32, we obtain that the satisfiability problem for graded modal logic is in NEXPTIME. (In fact, this problem is in PSPACE [25].) 2. By the same line of reasoning, the satisfiability proble... |

493 | Algebraic laws for nondeterminism and concurrency
- Hennessy, Milner
- 1985
(Show Context)
Citation Context ...and). An expressive set of modal operators is then obtained by indexing modal operators over a ∈ I in the form [ ]a. In the case T = Pω, this leads to the usual operators � a of Hennessy-Milner Logic =-=[9]-=-. In the probabilistic case, the meaning of 〈p〉a φ in reactive probabilistic automata is that on input a, φ holds in the next step with probability at least p, and in generative probabilistic automata... |

443 |
Y.: Modal Logic
- Blackburn, Rijke, et al.
- 2002
(Show Context)
Citation Context ...le 7) means that canonical models, based on the set of all maximally consistent sets w.r.t. a finitary deduction system, do not in general exist. An alternative is to use filtration methods (cf. e.g. =-=[4,2]-=-), in the variant that uses maximally consistent subsets of finite sublanguages. In more detail, we construct a satisfying model for a consistent formula φ by taking as the carrier set the set S of al... |

403 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...nal symbols (cf. Example 2) with the predicate liftings λp , p ∈ [0, 1] ∩ Q, defined by λ p (A) = {P ∈ DωX | P A ≥ p}. These induce the modal operators 〈p〉 = [λ p ] of probabilistic modal logic (PML) =-=[13, 8]-=-, where 〈p〉 φ reads ‘φ holds in the next step with probability at least p’. Note that [p], defined as ¬〈p〉¬, is monotone, but not normal. 6. For a field k, the linear space functor k · takes a set X t... |

298 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...c and probabilistic modal logic. Introduction Coalgebra has recently had increasing success as a generic theory of reactive systems, providing a unifying perspective on a wide variety of system types =-=[21]-=-. Many concepts of concurrency theory can be cast in the coalgebraic framework; this includes general notions of bisimulation, coinduction, and corecursion, as well as a generic modal logic [10, 19, 1... |

155 | Reasoning about knowledge and probability
- Fagin, Halpern
- 1994
(Show Context)
Citation Context ...ems over the reals (with strict inequalities arising from negated literals ¬Lpa), which by Theorem 41 is in P . By Theorem 36, it follows that probabilistic modal logic is in NEXPTIME. (By results of =-=[8]-=-, probabilistic modal logic is in fact in PSPACE.) (4) By the same reasoning, the modal logic for linear automata of Example 5.6 is decidable in NEXPTIME. (5) It is straightforward to extend the above... |

148 |
Modal Logic
- Chellas
- 1980
(Show Context)
Citation Context ...le 9) means that canonical models, based on the set of all maximally consistent sets w.r.t. a finitary deduction system, do not in general exist. An alternative is to use filtration methods (cf. e.g. =-=[4, 2]-=-), in the variant that uses consistent subsets of closed sublanguages. We recall a few basic definitions: Definition 19. Given a set Ax of axioms, a finite set {φ1, . . . , φn} of formulae is called c... |

89 | The Logic of
- Baltag, Moss, et al.
- 1998
(Show Context)
Citation Context ...an be cast in the coalgebraic framework; these include general notions of bisimulation, coinduction, and corecursion, as well as generic modal logics. The latter include the seminal coalgebraic logic =-=[16]-=- and the more recent coalgebraic modal logic [12,22,14,19,21]; here, we focus on coalgebraic modal logic, as it stays close to the traditional syntax and semantics of modal logic. The role of coalgebr... |

61 | Specifying Coalgebras with Modal Logic, Theoretical Computer Science 260 - Kurz - 2001 |

57 |
First Steps in Modal Logic
- Popkorn
- 1994
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Citation Context ... are present, i.e. it may be the case that φ is locally satisfiable in some model globally satisfying a formula ψ, but not in any finite such model. Examples are found already in standard modal logic =-=[18]-=-. 4 Decidability Unlike in the classical case, the finite model construction of the preceding section does not immediately imply decidability, even though it gives a computable bound on the size of th... |

51 | Coalgebraic modal logic: Soundness, completeness and decidability of local consequence
- Pattinson
(Show Context)
Citation Context ...ther hand, coalgebraic modal logics frequently correspond to known modal logics such as graded modal logic or probabilistic modal logic, and thus provide these logics with a coalgebraic semantics. In =-=[16]-=- and subsequent work [5, 11], a (necessarily weak) completeness result for coalgebraic modal logic has been established stating that a deductive system consisting of propositional entailment, a congru... |

39 | Expressivity of coalgebraic modal logic: The limits and beyond
- Schröder
(Show Context)
Citation Context ...gic is twofold: on the one hand, one obtains a suitable generic reactive specification language, which respects encapsulation of the state space, i.e. relates well to behavioral equivalence of states =-=[17, 23]-=-, and is sufficiently intuitive for use in actual software specification languages, including object-oriented specification [20, 12, 14]. On the other hand, coalgebraic modal logics frequently corresp... |

37 | A hierarchy of probabilistic system types. Theoretical Computer Science
- Bartels, Sokolova, et al.
- 2004
(Show Context)
Citation Context ...hing degrees require a more elaborate measure theoretic treatment [27].) A separating set for T is obtained by combining the propositional symbols (cf. Example 2) with the predicate liftings λp , p ∈ =-=[0, 1]-=- ∩ Q, defined by λ p (A) = {P ∈ DωX | P A ≥ p}. These induce the modal operators 〈p〉 = [λ p ] of probabilistic modal logic (PML) [13, 8], where 〈p〉 φ reads ‘φ holds in the next step with probability a... |

36 | The Coalgebraic Class Specification Language CCSL
- Rothe, Tews, et al.
(Show Context)
Citation Context ... state space, i.e. relates well to behavioral equivalence of states [17, 23], and is sufficiently intuitive for use in actual software specification languages, including object-oriented specification =-=[20, 12, 14]-=-. On the other hand, coalgebraic modal logics frequently correspond to known modal logics such as graded modal logic or probabilistic modal logic, and thus provide these logics with a coalgebraic sema... |

33 | Coalgebras and modal logic
- Rößiger
(Show Context)
Citation Context ...types [21]. Many concepts of concurrency theory can be cast in the coalgebraic framework; this includes general notions of bisimulation, coinduction, and corecursion, as well as a generic modal logic =-=[10, 19, 12, 15, 17]-=-. The role of this coalgebraic modal logic is twofold: on the one hand, one obtains a suitable generic reactive specification language, which respects encapsulation of the state space, i.e. relates we... |

30 | Semantical principles in the modal logic of coalgebras
- Pattinson
- 2001
(Show Context)
Citation Context ...types [21]. Many concepts of concurrency theory can be cast in the coalgebraic framework; this includes general notions of bisimulation, coinduction, and corecursion, as well as a generic modal logic =-=[10, 19, 12, 15, 17]-=-. The role of this coalgebraic modal logic is twofold: on the one hand, one obtains a suitable generic reactive specification language, which respects encapsulation of the state space, i.e. relates we... |

28 |
Probability Logic for Type Spaces
- Heifetz, Mongin
(Show Context)
Citation Context ...utions on X with finite support. Coalgebras for the functor T = Dω × P(V ), where V �= ∅ is a set of propositional symbols, are probabilistic transition systems (also called probabilistic type spaces =-=[8]-=-) with finite branching degree. (The example is easily extended to countable branching by considering instead the functor Dω1 of probability distributions with countable support, while higher branchin... |

27 | PSPACE reasoning for graded modal logics
- Tobies
(Show Context)
Citation Context ... with probability at least p, the input is a and φ holds in the next step. Remark 8. Graded modal logic is more standardly interpreted over Kripke models by just counting successor states (as e.g. in =-=[25]-=-), rather than in multigraphs as in the above example and e.g. in [6]. One can regard Kripke models as multigraph models by just regarding sets as multisets where all elements have multiplicity 1; con... |

26 | Expressive logics for coalgebras via terminal sequence induction
- Pattinson
(Show Context)
Citation Context ...ightforward. ✷ Remark 12 The converse of the previous proposition holds under mild additional assumptions (namely when T is ω-accessible, the set Λ of predicate liftings is separating in the sense of =-=[21]-=-, and the final T -coalgebra is infinite); this is a straightforward generalisation of [26], Theorem 30. Proof systems for coalgebraic modal logic are parametrised over a given axiomatisation, i.e. a ... |

25 | PSPACE bounds for rank-1 modal logics - Schröder, Pattinson - 1985 |

22 | Modular construction of modal logics
- Cîrstea, Pattinson
- 2004
(Show Context)
Citation Context ...dal logics frequently correspond to known modal logics such as graded modal logic or probabilistic modal logic, and thus provide these logics with a coalgebraic semantics. In [16] and subsequent work =-=[5, 11]-=-, a (necessarily weak) completeness result for coalgebraic modal logic has been established stating that a deductive system consisting of propositional entailment, a congruence rule, and a given set o... |

21 | Finality regained: A coalgebraic study of Scott-sets and multisets - D’Agostino, Visser |

21 | Towards a duality result in the modal logic of coalgebras
- Jacobs
(Show Context)
Citation Context ...types [21]. Many concepts of concurrency theory can be cast in the coalgebraic framework; this includes general notions of bisimulation, coinduction, and corecursion, as well as a generic modal logic =-=[10, 19, 12, 15, 17]-=-. The role of this coalgebraic modal logic is twofold: on the one hand, one obtains a suitable generic reactive specification language, which respects encapsulation of the state space, i.e. relates we... |

19 | Algebraiccoalgebraic specification in CoCasl
- Mossakowski, Reichel, et al.
- 2003
(Show Context)
Citation Context ... state space, i.e. relates well to behavioral equivalence of states [17, 23], and is sufficiently intuitive for use in actual software specification languages, including object-oriented specification =-=[20, 12, 14]-=-. On the other hand, coalgebraic modal logics frequently correspond to known modal logics such as graded modal logic or probabilistic modal logic, and thus provide these logics with a coalgebraic sema... |

17 | Majority logic
- Pacuit, Salame
- 2004
(Show Context)
Citation Context ...e known PSPACE upper bounds for K, graded modal logic, and probabilistic modal logic (and, in the conference presentation of [31], established a presumably novel PSPACE upper bound for majority logic =-=[18]-=- simultaneously with [7]); this requires the construction of a suitable tractable axiomatisation in each case [31]. Adaptation of these upper bounds to generalised graded modal logic and the logic of ... |

16 |
Algebraic semantics for coalgebraic logics
- Kupke, Kurz, et al.
- 2004
(Show Context)
Citation Context ...dal logics frequently correspond to known modal logics such as graded modal logic or probabilistic modal logic, and thus provide these logics with a coalgebraic semantics. In [16] and subsequent work =-=[5, 11]-=-, a (necessarily weak) completeness result for coalgebraic modal logic has been established stating that a deductive system consisting of propositional entailment, a congruence rule, and a given set o... |

15 |
Realizations by stochastic finite automata
- Carlyle, Paz
- 1971
(Show Context)
Citation Context ...t normal. 6. For a field k, the linear space functor k · takes a set X to the free k-vector space k · X, i.e. the set of formal k-linear combinations, over X. A coalgebra for k· is a linear automaton =-=[3, 26]-=- (where one would in general also assume linear output in a vector space V , corresponding to the functor (k · ) × V ). In the case k = R, a separating set of predicate liftings can be constructed in ... |

15 |
PSPACE reasoning for rank-1 modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ...a single V -instance rather than several Prop(V )-instances of the given rules, and thus actually shows that the set of all one-step sound one-step rules is strictly one-step complete in the sense of =-=[31]-=-. Remark 21 Corollary 19 may also be viewed as a limitative result saying that coalgebraic modal logic covers only modal logics that are axiomatisable by rank-1 clauses (in fact, exactly the modal log... |

15 |
Modalities for Reasoning about Knowledge and Quantities
- Hoek
- 1992
(Show Context)
Citation Context ...a finite T -coalgebra of size at most 2 |φ| . 14Remark 33 By Remark 6, the above corollary reproves the fact that graded modal logic has the finite model property when interpreted over Kripke frames =-=[35]-=-, although of course not with the same size bound as for the interpretation over multigraphs (e.g. any Kripke frame in which φ ≡ ✸k⊤ is satisfiable will have at least k states, while |φ| = 2). 4 Decid... |

14 | Rank-1 modal logics are coalgebraic
- Schröder, Pattinson
(Show Context)
Citation Context ... viewed as a limitative result saying that coalgebraic modal logic covers only modal logics that are axiomatisable by rank-1 clauses (in fact, exactly the modal logics axiomatisable by rank-1 clauses =-=[32]-=-). However, modal logics axiomatised outside rank 1, such as the classical logic S4, may be modelled by restricting the semantics to a suitable subclass of the class of all coalgebras for the given si... |

4 |
The modal logic of probability
- Heifetz, Mongin
- 1998
(Show Context)
Citation Context ...is straightforward to extend the above results to include proposition symbols, where not already present, or inputs (cf. Examples 7.2 and 7.7). Remark 35. A decision algorithm for PML is announced in =-=[7]-=-, but not explicitly contained in the full version [8]. The latter proves the finite model propertysfor PML, from which decidability does follow by the argument sketched at the beginning of this secti... |

4 | Undecidability in integer weighted finite automata
- Halava, Harju
- 1999
(Show Context)
Citation Context ...ts of finitely supported maps X → Z, called generalised multisets (this set is also familiar as the free abelian group over X). Coalgebras for BZ appear in the literature as integer weighted automata =-=[9]-=-. One has predicate liftings λk , k ∈ Z, with induced modal operators ✸k defined analogously as for multisets. We refer to the arising logic as generalised graded modal logic. Note that in this case, ... |

2 |
Final sequences and final coalgebras for measurable spaces
- Viglizzo
- 2005
(Show Context)
Citation Context ... to countable branching by considering instead the functor Dω1 of probability distributions with countable support, while higher branching degrees require a more elaborate measure theoretic treatment =-=[27]-=-.) A separating set for T is obtained by combining the propositional symbols (cf. Example 2) with the predicate liftings λp , p ∈ [0, 1] ∩ Q, defined by λ p (A) = {P ∈ DωX | P A ≥ p}. These induce the... |

2 |
Compositional algorithms for heterogeneous modal logics
- Schröder, Pattinson
- 2006
(Show Context)
Citation Context ...or composition S ◦ T , as the application of T to finite sets may e.g. produce an exponential blowup or lead to infinite sets. This problem may possibly be resolved by moving to multisorted coalgebra =-=[30]-=-. Besides the examples whose decidability is already captured by the finite model result of [20], i.e. functors preserving finite sets, such as P, our results cover also more complex logics. The treat... |

1 |
PSPACE reasoning for coalgebraic modal logic
- Schröder, Pattinson
(Show Context)
Citation Context ...earch. The best general bound we can hope for is PSPACE, since the decision problem for K, i.e. the modal logic of P, is known to be PSPACE-complete [2]. The following further results are forthcoming =-=[24]-=-: (i) One can show by means of elimination of Hintikka sets (in the same manner as in known algorithms for PDL [2]) that satisfiability of L(Λ)-formulae is in EXPTIME if one-step satisfiability is in ... |

1 |
On the minimization of linear space automata
- Turakainen
- 1972
(Show Context)
Citation Context ...t normal. 6. For a field k, the linear space functor k · takes a set X to the free k-vector space k · X, i.e. the set of formal k-linear combinations, over X. A coalgebra for k· is a linear automaton =-=[3, 26]-=- (where one would in general also assume linear output in a vector space V , corresponding to the functor (k · ) × V ). In the case k = R, a separating set of predicate liftings can be constructed in ... |

1 | Realizations by stochastic finite automata - Blackburn, Rijke, et al. |

1 | 4] B. Chellas. Modal Logic - Sci - 1971 |

1 | Specifying coalgebras with modal logic. Theoret - Elsevier - 1991 |

1 | First Steps in Modal Logic - Logic - 1994 |

1 |
A semantic PSPACE-criterion for rank 1 modal logic
- Schröder
- 2006
(Show Context)
Citation Context ...an show that satisfiability of L(Λ)-formulas is in PSPACE by means of a shallow model construction [31]. (iii) A semantic criterion for a coalgebraic modal logic to be decidable in PSPACE is given in =-=[28]-=-; the relevant conditions here are a small model property and tractability of model checking at the one-step level. (iv) It is shown in [30] that the method of (ii) is compositional w.r.t. composition... |