Riesz Modal logic for Markov processes

Mio, Matteo; Furber, Robert; Mardare, Radu

doi:10.1109/lics.2017.8005091

Cited by 7 publications

(17 citation statements)

References 17 publications

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“…In Sect. 2 we provide the necessary definitions about the Riesz modal logic from [MFM17,Mio18] and about the hypersequent calculus GA of [MOG05,MOG09]. In Sect.…”

Section: The Non-trivial Adaptation Of the Proof-technique Used In [Mmentioning

confidence: 99%

“…The Riesz modal logic R introduced in [MFM17] is a probabilistic logic for expressing properties of discrete or continuous Markov chains. We refer to [MFM17] for a detailed introduction. Here we just restrict ourselves to the purely syntactical aspects of this logic: its syntax and its axiomatisation.…”

Section: The Riesz Modal Logic and Its Scalar-free Fragmentmentioning

confidence: 99%

“…A main result of [MFM17] is that two formulas φ and ψ are semantically equivalent if and only if the identity φ = ψ holds in all modal Riesz spaces.…”

Section: The Riesz Modal Logic and Its Scalar-free Fragmentmentioning

confidence: 99%

“…These logical formalisms are, similarly to Kozen's modal μ-calculus, obtained by extending a base real-valued modal logic with (co)inductively defined operators. Recently, in [MFM17], a base real-valued modal logic called Riesz modal logic (R) has been defined and a sound and complete equational axiomatisation has been obtained (see Definition 2). Importantly, the logic R extended with (co)inductively defined operators is sufficiently expressive to interpret most other probabilistic logics, including pCTL [Mio12b,Mio18,MS13a].…”

Section: Introductionmentioning

confidence: 99%

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Towards a Structural Proof Theory of Probabilistic $$\mu $$ -Calculi

Lucas

Mio

2019

Lecture Notes in Computer Science

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We present a structural proof system, based on the machinery of hypersequent calculi, for a simple probabilistic modal logic underlying very expressive probabilistic µ-calculi. We prove the soundness and completeness of the proof system with respect to an equational axiomatisation and the fundamental cut-elimination theorem. 1 Example: the law of idempotence x ∨ x = x can be derived from the standard axioms of Boolean algebras (i.e., complemented distributive lattices) as: x∨x = (x∨x)∧ = (x ∨ x) ∧ (x ∨ ¬x) = x ∨ (x ∧ ¬x) = x ∨ ⊥ = x.

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“…In Sect. 2 we provide the necessary definitions about the Riesz modal logic from [MFM17,Mio18] and about the hypersequent calculus GA of [MOG05,MOG09]. In Sect.…”

Section: The Non-trivial Adaptation Of the Proof-technique Used In [Mmentioning

confidence: 99%

Section: The Riesz Modal Logic and Its Scalar-free Fragmentmentioning

confidence: 99%

“…A main result of [MFM17] is that two formulas φ and ψ are semantically equivalent if and only if the identity φ = ψ holds in all modal Riesz spaces.…”

Section: The Riesz Modal Logic and Its Scalar-free Fragmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Towards a Structural Proof Theory of Probabilistic $$\mu $$ -Calculi

Lucas

Mio

2019

Lecture Notes in Computer Science

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“…Importantly, once extended with fixed-point operators in the style of the modal µ-calculus [Koz83], this logic is sufficiently expressive to interpret other popular probabilistic logics for verification such as probabilistic CTL (see, e.g., chapter 8 in [BK08] for an introduction to this logic). One key contribution from [MFM17,FMM20] is a duality theory which provides a bridge between the probabilisitic transition system semantics of the Riesz modal logic and its algebraic semantics given in terms of so-called modal Riesz spaces.…”

Section: Introductionmentioning

confidence: 99%

Proof Theory of Riesz Spaces and Modal Riesz Spaces

Lucas¹,

Mio²

2022

Logical Methods in Computer Science

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We design hypersequent calculus proof systems for the theories of Riesz spaces and modal Riesz spaces and prove the key theorems: soundness, completeness and cut elimination. These are then used to obtain completely syntactic proofs of some interesting results concerning the two theories. Most notably, we prove a novel result: the theory of modal Riesz spaces is decidable. This work has applications in the field of logics of probabilistic programs since modal Riesz spaces provide the algebraic semantics of the Riesz modal logic underlying the probabilistic mu-calculus.

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Duality for Instantial Neighbourhood Logic via Coalgebra

Bezhanishvili¹,

Enqvist²,

Groot³

2020

Lecture Notes in Computer Science

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Instantial Neighbourhood Logic (INL) has been introduced recently as a language for neighbourhood frames where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. Apart from its semantics, its proof theory and bisimulation games have also been studied. However, conspicuously absent from the treatment of INL is the notion of descriptive frames. This is the gap that we are closing in this paper. We introduce descriptive frames for INL and we prove that these are dual to boolean algebras with instantial operators (BAIOs), which give the algebraic semantics of INL. Our methods for establishing this duality make essential use of coalgebra: we observe that BAIOs are algebras for a functor on the category of boolean algebras and show that this functor is dual to the double Vietoris functor (i.e. the composition of the Vietoris functor with itself), thus obtaining a dual equivalence between double Vietoris coalgebras and BAIOs. The proof of our main result is then completed by showing that double Vietoris coalgebras correspond precisely to descriptive frames. As a corollary we obtain that every extension of INL is sound and complete with respect to descriptive frames, that descriptive frames enjoy the Hennessy-Milner property, and as a result, that finite neighbourhood frames enjoy the Hennessy-Milner property.

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Riesz Modal logic for Markov processes

Cited by 7 publications

References 17 publications

Towards a Structural Proof Theory of Probabilistic $$\mu $$ -Calculi

Towards a Structural Proof Theory of Probabilistic $$\mu $$ -Calculi

Proof Theory of Riesz Spaces and Modal Riesz Spaces

Duality for Instantial Neighbourhood Logic via Coalgebra

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