# Coinductive Proof Principles for Stochastic Processes

@article{Kozen2006CoinductivePP, title={Coinductive Proof Principles for Stochastic Processes}, author={Dexter Kozen}, journal={21st Annual IEEE Symposium on Logic in Computer Science (LICS'06)}, year={2006}, pages={359-366} }

We give an explicit coinduction principle for recursively-defined stochastic processes. The principle applies to any closed property, not just equality, and works even when solutions are not unique. The rule encapsulates low-level analytic arguments, allowing reasoning about such processes at a higher algebraic level. We illustrate the use of the rule in deriving properties of a simple coin-flip processĀ

## 9 Citations

Applications of Metric Coinduction

- Computer ScienceCALCO
- 2007

This paper examines the application of the coinduction principle in a variety of areas, including infinite streams, Markov chains,Markov decision processes, and non-well-founded sets, and points to the usefulness of coinductions as a general proof technique.

Applications of Metric Coinduction

- Computer Science, MathematicsLog. Methods Comput. Sci.
- 2009

This paper examines the application of the coinduction principle in a variety of areas, including infinite streams, Markov chains,Markov decision processes, and non-well-founded sets, and points to the usefulness of coinductions as a general proof technique.

Fixed-Points for Quantitative Equational Logics

- Computer Science2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

The result is a novel theory of fixed points which can not only provide solutions to the traditional fixed-point equations but can also define the rate of convergence to the fixed point.

Tiered Objects

- Computer ScienceFundam. Informaticae
- 2016

This work focuses on a natural notion of tiering involving an operation of restriction of elements to levels forming a complete Heyting algebra, and derives a general proof principle for tiered objects.

Optimal Coin Flipping

- Mathematics, Computer ScienceHorizons of the Mind
- 2014

A lower bound that is strictly greater than the information-theoretic bound is established and it is shown that as a function of q, it is an everywhere-discontinuous self-similar fractal.

Introduction to Coalgebra: Towards Mathematics of States and Observation

- Computer ScienceCambridge Tracts in Theoretical Computer Science
- 2016

This book acts as the first mature and accessible introduction to coalgebra and provides clear mathematical explanations, with many examples and exercises involving deterministic and non-deterministic automata, transition systems, streams, Markov chains and weighted automata.

Dexter Kozen's Influence on the Theory of Labelled Markov Processes

- Computer ScienceLogic and Program Semantics
- 2012

In the Fall of 1985 Dexter and I both started at Cornell as new faculty members in the celebrated Computer Science Department, home to luminaries such as Juris Hartmanis, John Hopcroft, David Griesā¦

CertRL: formalizing convergence proofs for value and policy iteration in Coq

- Computer ScienceCPP
- 2021

A Coq formalization of two canonical reinforcement learning algorithms: value and policy iteration for finite state Markov decision processes and a contraction property of Bellman optimality operator to establish that a sequence converges in the infinite horizon limit.

MICo: Improved representations via sampling-based state similarity for Markov decision processes

- Computer Science
- 2021

A new behavioural distance over the state space of a Markov decision process is presented, and empirical evidence that learning this distance alongside the value function yields structured and informative representations, including strong results on the Arcade Learning Environment benchmark.

## References

SHOWING 1-10 OF 16 REFERENCES

Coinductive Proof Principles for Stochastic Processes

- Mathematics, Computer ScienceLICS
- 2006

We give an explicit coinduction principle for recursively-defined stochastic processes. The principle applies to any closed property, not just equality, and works even when solutions are not unique.ā¦

Applications of Metric Coinduction

- Computer ScienceCALCO
- 2007

This paper examines the application of the coinduction principle in a variety of areas, including infinite streams, Markov chains,Markov decision processes, and non-well-founded sets, and points to the usefulness of coinductions as a general proof technique.

The metric analogue of weak bisimulation for probabilistic processes

- Computer Science, MathematicsProceedings 17th Annual IEEE Symposium on Logic in Computer Science
- 2002

A metric analogue of weak bisimulation is developed and it is shown that quantitative properties of interest are continuous with respect to the metric, which says that if two processes are close in the metric then observable quantitative propertiesof interest are indeed close.

Behavioural differential equations: a coinductive calculus of streams, automata, and power series

- Computer Science, MathematicsTheor. Comput. Sci.
- 2003

We present a theory of streams (infinite sequences), automata and languages, and formal power series, in terms of the notions of homomorphism and bisimulation, which are the cornerstones of theā¦

Approximate reasoning for real-time probabilistic processes

- Computer Science, MathematicsFirst International Conference on the Quantitative Evaluation of Systems, 2004. QEST 2004. Proceedings.
- 2004

This work develops a pseudo-metric analogue of bisimulation for generalized semiMarkov processes that is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity and provides a logical characterization of this uniformity using a real-valued modal logic.

Stochastic processes as concurrent constraint programs

- Computer SciencePOPL '99
- 1999

This paper describes a stochastic concurrent constraint language for the description and programming of concurrent probabilistic systems, and uses the semantic study to illustrate a novel use of probability to analyze a problem stated without reference to probability.

A Tutorial on Co-induction and Functional Programming

- Mathematics, Computer ScienceFunctional Programming
- 1994

It is shown how to prove properties of lazy streams by co-induction and derive Bird and Wadlerās Take Lemma, a well-known proof technique for lazy streams.

Semantics of probabilistic programs

- Computer Science, Mathematics20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
- 1979

Two complementary but equivalent semantic interpretations of a high level probabilistic programming language are given and how the ordered domains of Scott and others are embedded naturally into these spaces.

Universal coalgebra: a theory of systems

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2000

The three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to: coalgebra, homomorphicism of coalgebrAs, and bisimulation, respectively, which are taken as the basic ingredients of a theory called universal coalgebra.

Recursive subtyping revealed: (functional pearl)

- Computer ScienceICFP '00
- 2000

An "end-to-end" introduction to recursive types and subtyping algorithms, from basic theory to efficient implementation, set in the unifying mathematical framework of coinduction.