# Taming Multirelations

@article{Furusawa2016TamingM, title={Taming Multirelations}, author={H. Furusawa and G. Struth}, journal={ACM Transactions on Computational Logic (TOCL)}, year={2016}, volume={17}, pages={1 - 34} }

Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential, and parallel composition, as well as finite and infinite iteration. Starting from a set-theoretic investigation, we propose axiom systems for multirelations in contexts ranging from bi-monoids to bi-quantales.

#### 7 Citations

Binary Multirelations

- Computer Science
- Arch. Formal Proofs
- 2015

This proof document supports an arXiv article that formalises the basic algebra of multirelations and proposes axiom systems for them, ranging from strong bi-monoids to weak bi-quantales. Expand

Relational Formalisations of Compositions and Liftings of Multirelations

- Mathematics, Computer Science
- RAMICS
- 2015

This paper presents relational formalisations of Kleisli, Parikh and Peleg’s compositions and liftings of multirelations. Expand

Kleisli, Parikh and Peleg compositions and liftings for multirelations

- Mathematics, Computer Science
- J. Log. Algebraic Methods Program.
- 2017

Rel relational formalisations of Kleisli, Parikh and Peleg compositions and liftings of multirelations are presented and it is shown that Kleislu composition ofMultirelations is associative, but need not have units, and Parikh composition may neither be associative nor have units. Expand

An algebraic approach to multirelations and their properties

- Mathematics, Computer Science
- J. Log. Algebraic Methods Program.
- 2017

The algebraic properties of a new composition operation based on the correspondence to predicate transformers, different ways to express reflexive–transitive closures of multirelations, numerous equational properties, how these properties are connected and their preservation by multirelational operations are investigated. Expand

On the construction of multi-valued concurrent dynamic logic

- Computer Science, Mathematics
- DaLí
- 2019

This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations, called multi-valued concurrent propositional dynamic logics (CGDL(A), parametric on an action lattice A specifying a notion of "weight" assigned to program execution. Expand

Differential Hoare Logics and Refinement Calculi for Hybrid Systems with Isabelle/HOL

- Computer Science
- RAMiCS
- 2020

This work presents simple new Hoare logics and refinement calculi for hybrid systems in the style of differential dynamic logic in a generic algebraic way, formalised with the Isabelle/HOL proof assistant. Expand

Relational and Algebraic Methods in Computer Science: 18th International Conference, RAMiCS 2020, Palaiseau, France, October 26–29, 2020, Proceedings

- Computer Science
- RAMiCS
- 2020

This paper presents results from a study of probabilistic Bisimulation with Silent Moves on the basis of the responses of invited speakers and the response of non-invited speakers. Expand

#### References

SHOWING 1-10 OF 55 REFERENCES

The Cube of Kleene Algebras and the Triangular Prism of Multirelations

- Mathematics, Computer Science
- RelMiCS
- 2009

This work refine and extend the known results that the set of ordinary binary relations forms a Kleene algebra, and introduces a notion of type of multirelations. Expand

Monotone Predicate Transformers as Up-Closed Multirelations

- Computer Science
- RelMiCS
- 2006

The two-level nature of multirelations is exploited to provide a factorisation of up-closedMultirelations which clarifies exactly how multirelation model nondeterminism. Expand

Modelling angelic and demonic nondeterminism with multirelations

- Computer Science
- Sci. Comput. Program.
- 2007

This paper presents an introduction to a calculus of binary multirelations, which can model both angelic and demonic kinds of non-determinism. The isomorphism between up-closed multirelations and… Expand

Concurrent Dynamic Algebra

- Computer Science, Mathematics
- ACM Trans. Comput. Log.
- 2015

Algebraic variants of Peleg’s axioms are shown to be derivable in these algebras, and their soundness is proved relative to the multirelational model. Expand

Domain and Antidomain Semigroups

- Mathematics, Computer Science
- RelMiCS
- 2009

We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation… Expand

Concurrent Kleene Algebra and its Foundations

- Mathematics, Computer Science
- J. Log. Algebraic Methods Program.
- 2011

A Concurrent Kleene Algebra is investigated in terms of a primitive independence relation between the traces, and a series of richer algebras are developed; the richest validates a proof calculus for programs similar to that of a Jones style rely/guarantee calculus. Expand

Domain Axioms for a Family of Near-Semirings

- Computer Science
- AMAST
- 2008

Axioms for domain operations in several variants of Kleene algebras and their semiring reducts are presented. They provide abstract enabledness conditions for algebras designed for the verification… Expand

Concurrency and Automata on Infinite Sequences

- Computer Science
- Theoretical Computer Science
- 1981

A general method for proving/deciding equivalences between omega-regular languages, whose recognizers are modified forms of Buchi or Muller-McNaughton automata, derived from Milner's notion of “simulation” is obtained. Expand

The Equational Theory of Pomsets

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 1988

It is found that the equational theory of sets, pomsets under concatenation, parallel composition and union is finitely axiomatizable, whereas the theory of languages under the analogous operations is not. Expand

Free Shuffle Algebras in Language Varieties

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 1996

These are simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” of theFree monoid Σ∗. Expand