What is the key result of the paper?

Their key results are:• a description of a new class of algebras that capture in algebraic form the probabilistic modal logics used for continuous-state Markov processes;• a version of duality for countable algebras and a certain class of countablygenerated Markov processes; and• a complete axiomatization that does not involve infinitary axiom schemes with uncountably many instances and avoids postulating Lindenbaum’s lemma as a meta-axioms.

What is the axiom scheme used in the strong completeness theorems?

Follow-ing the pioneering work of Goldblatt [14], the strong completeness theorems use a powerful infinitary axiom scheme called the countably additivity rule in [14].

Open AccessProceedings Article10.1109/LICS.2013.38

Stone Duality for Markov Processes

Q: What is the meaning of the term?

Jonsson and Tarski [3] extended this to modal logic and showed a duality between Boolean algebraswith additional operators and certain topological spaces with additional operators.

Dexter Kozen, +3 more

- 25 Jun 2013

- pp 321-330

TL;DR: A Stone-type duality theorem is proved between countable Aumann algebras and countably-generated continuous-space Markov processes and the results subsume existing results on completeness of probabilistic modal logics for Markov process.

Abstract: We define Aumann algebras, an algebraic analog of probabilistic modal logic. An Aumann algebra consists of a Boolean algebra with operators modeling probabilistic transitions. We prove a Stone-type duality theorem between countable Aumann algebras and countably-generated continuous-space Markov processes. Our results subsume existing results on completeness of probabilistic modal logics for Markov processes.

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Most frequently asked questions

1. What kind of algebras have the authors named?

In this paper, the authors develop a Stone-type duality for continuous-space probabilistic transitions systems and a certain kind of algebra that the authors have named Aumann algebras.

2. What is the meaning of the word Stone?

Plotkin [4] and Smyth [5] emphasized that the duality between state-transformer semantics and predicate-transformer semantics is an instance of a Stone-type duality.

3. What is the meaning of the term Stone?

This includes both smaller algebras, which could be finite and generate finite structures, or larger algebras, which could be uncountable, thus not syntactically generated.

4. What is the meaning of the term Stone duality?

Stone duality states that the category of Boolean algebras and Boolean algebra homomorphisms and the category of Stone spaces and continuous maps are contravariantly equivalent.

Citations

Brzozowski's algorithm (co)algebraically

Filippo Bonchi, +3 more

- 01 Dec 2011

TL;DR: In this paper, a new presentation of Brzozowski's algorithm to minimize finite automata, using elementary facts from universal algebra and coalgebra, was given. But it was not shown to be correct.

...read moreread less

•Book Chapter•10.1007/978-3-642-40313-2_58

Strong Completeness for Markovian Logics

Dexter Kozen, +2 more

- 26 Aug 2013

TL;DR: Hilary-style axiomatizations for three logics for reasoning about continuous-space Markov processes, defined for probability distributions on measurable state spaces, sub-probability distributions and a logic defined for arbitrary distributions are presented.

...read moreread less

•Posted Content•10.23638/LMCS-16(1:6)2020

Probabilistic logics based on Riesz spaces

Robert Furber, +2 more

- 22 Mar 2019

- arXiv: Logic in Computer Science

TL;DR: The duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence is developed, which has a number of consequences including a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems.

...read moreread less

Journal Article•10.1007/S11225-014-9592-X

Duality for the Logic of Quantum Actions

Jort M. Bergfeld, +3 more

- 01 Aug 2015

- Studia Logica

TL;DR: This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice.

...read moreread less

•Journal Article

Review: M. H. Stone, The theory of Representations for Boolean Algebras

Alonzo Church

- 01 Sep 1936

- Journal of Symbolic Logic

References

•Book

Probability and Measure

Patrick Billingsley

- 01 Jan 1979

TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.

...read moreread less

7.5K

•Book

Real Analysis and Probability

Dudley

- 01 Jan 2006

•Journal Article•10.1016/0890-5401(91)90030-6

Bisimulation through probabilistic testing

Kim Guldstrand Larsen, +1 more

- 01 Sep 1991

- Information & Computation

TL;DR: By using probabilistic transition systems as the underlying semantic model, it is shown how a testing algorithm can distinguish, with a probability arbitrarily close to one, between processes that are not bisimulation equivalent.

...read moreread less

1.3K

•Journal Article•10.1090/S0002-9947-1936-1501865-8

The theory of representations for Boolean algebras

M. H. Stone

- 01 Jan 1936

- Transactions of the American Mathematica...

TL;DR: Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methods and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington as mentioned in this paper.

...read moreread less

963

Bisimulation Through Probabilistic Testing.

Kim Guldstrand Larsen, +1 more

- 01 Jan 1989

TL;DR: In this paper, a probabilistic transition system-based language for testing concurrent processes is proposed, and the strength of the testing language is examined in terms of the processes that are distinguished by a test.

...read moreread less

881

Stone Duality for Markov Processes

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Most frequently asked questions

1. What kind of algebras have the authors named?

2. What is the meaning of the word Stone?

3. What is the meaning of the term Stone?

4. What is the meaning of the term Stone duality?

5. What is the key result of the paper?

6. What is the meaning of the term?

7. What is the axiom scheme used in the strong completeness theorems?

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Citations

Brzozowski's algorithm (co)algebraically

Strong Completeness for Markovian Logics

Probabilistic logics based on Riesz spaces

Duality for the Logic of Quantum Actions

Review: M. H. Stone, The theory of Representations for Boolean Algebras

References

Probability and Measure

Real Analysis and Probability

Bisimulation through probabilistic testing

The theory of representations for Boolean algebras

Bisimulation Through Probabilistic Testing.

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