A coalgebraic view on decorated traces

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories. Monads and Algebraic TheoriesIn this paper, on the algebraic side, we deal with Eilenberg-Moore algebras of a monad on the category Sets of sets and functions, for which we also give presentations in terms of operations and equations, i.e., algebraic theories. MonadsA monad on Sets is a functor M : Sets → Sets together with two natural transformations: a unit η : Id ⇒ M and multiplication µ :We next introduce several monads on Sets, relevant to this paper. Each monad can be seen as giving side-effects.Nondeterminism. The finite powerset monad P maps a set X to its finite powerset PX = {U | U ⊆ X, U is finite} and a function f :The unit η of P is given by singleton, i.e., η(x) = {x} and the multiplication µ is given by union, i.e., µ(S) = U∈S U for S ∈ PPX. Of particular interest to us in this paper is the submonad P ne of non-empty finite subsets, that acts on functions just like the (finite) powerset monad, and has the same unit and multiplication. We rarely mention the unrestricted (not necessarily finite) powerset monad, which we denote by P u . We sometimes write f for P u f in this paper.Probability. The finitely supported probability distribution monad D is defined, for a set X and a function f : X → Y , asThe support set of a distribution ϕ ∈ DX is supp(ϕ) = {x ∈ X | ϕ(x) = 0}. The unit of D is given by a Dirac distribution η(x) = δ x = (x → 1) for x ∈ X and the multiplication by µ(Φ)(x) = ϕ∈supp(Φ) Φ(ϕ) · ϕ(x) for Φ ∈ DDX. We sometimes write i∈I p i x i for a distribution ϕ with supp(ϕ) = {x i | i ∈ I} and ϕ(x i ) = p i . 1. Personal communication with Gordon Plotkin.

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“…and determinise the NA o, t : X → 2 × (PX) A . The induced semantics ≡ LT S on the state space X is the standard trace semantics for LTS [59].…”

Section: From Systems To Automatamentioning

confidence: 99%

The Theory of Traces for Systems with Nondeterminism and Probability

Bonchi

Sokolova

Vignudelli

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…The original functor P ↓ ω (A × (−)) then transformed into G by mapping S to {(π 1 [S], a, x) | (a, x) ∈ S} for S = ∅, where π 1 denotes the first projection, and to the unique element of 1 otherwise. This is similar to decoration-oriented coalgebraic approaches to trace semantics [4].…”

Section: Behavioural Preorders Via Graded Monadsmentioning

confidence: 60%

“…Equivalences may be defined by distinguishing suitable modal logics [18], while we pursue the opposite approach of designing characteristic logics for given equivalences or preorders. Variations within trace-type equivalences, such as complete or ready trace semantics, can be captured via so-called decorations [4]. In recent work [19], Kupke and Rot present a highly general fibrational framework for coinductive predicates and their characteristic logics.…”

Section: Related Workmentioning

confidence: 99%

Behavioural Preorders via Graded Monads

Ford¹,

Milius²,

Schröder³

2020

Preprint

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Like notions of process equivalence, behavioural preorders on processes come in many flavours, ranging from fine-grained comparisons such as ready simulation to coarse-grained ones such as trace inclusion. Often, such behavioural preorders are characterized in terms of theory inclusion in dedicated characteristic logics; e.g. simulation is characterized by theory inclusion in the positive fragment of Hennessy-Milner logic. We introduce a unified semantic framework for behavioural preorders and their characteristic logics in which we parametrize the system type in the coalgebraic paradigm while behavioural preorders are captured as graded monads on the category Pos of partially ordered sets, in generalization of a previous approach to notions of process equivalence. We show the equivalence of graded monads on Pos with theories in a form of graded ordered algebra that we introduce here. Moreover, we provide a general notion of modal logic compatible with a given graded behavioural preorder, along with a criterion for expressiveness. ⋆ Supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the Research and Training Group 2475 "Cybercrime and Forensic Computing" (393541319/GRK2475/1-2019) ⋆⋆ Supported by Deutsche Forschungsgemeinschaft (DFG) under projects MI 717/5-2 and MI 717/7-1

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“…Behavioural equivalence in EM(T ) then amounts to desired notions of equivalence. For instance: language equivalence of non-deterministic automata; weighted automata [7]; Mealy and Moore machines with side-effects [43]; or various types of trace equivalence of transition systems [8]. An illustrative non-example of this general determinisation procedure is in a natural construction of belief-state transformers from probabilistic automata, which feature both non-determinism and probabilities.…”

Section: Introductionmentioning

confidence: 99%

Preservation and Reflection of Bisimilarity via Invertible Steps

Turkenburg¹,

Kupke²,

Rot³

et al. 2023

Lecture Notes in Computer Science

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In the theory of coalgebras, distributive laws give a general perspective on determinisation and other automata constructions. This perspective has recently been extended to include so-called weak distributive laws, covering several constructions on state-based systems that are not captured by regular distributive laws, such as the construction of a belief-state transformer from a probabilistic automaton, and ultrafilter extensions of Kripke frames.In this paper we first observe that weak distributive laws give rise to the more general notion of what we call an invertible step: a pair of natural transformations that allows to move coalgebras along an adjunction. Our main result is that part of the construction induced by an invertible step preserves and reflects bisimilarity. This covers results that have previously been shown by hand for the instances of ultrafilter extensions and belief-state transformers.

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A coalgebraic view on decorated traces

Cited by 5 publications

References 20 publications

The Theory of Traces for Systems with Nondeterminism and Probability

The Theory of Traces for Systems with Nondeterminism and Probability

Behavioural Preorders via Graded Monads

Preservation and Reflection of Bisimilarity via Invertible Steps

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