## Generic trace theory (2006)

Venue: | International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci |

Citations: | 10 - 5 self |

### BibTeX

@INPROCEEDINGS{Hasuo06generictrace,

author = {Ichiro Hasuo and Bart Jacobs and Ana Sokolova},

title = {Generic trace theory},

booktitle = {International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci},

year = {2006},

pages = {47--65},

publisher = {Elsevier}

}

### OpenURL

### Abstract

Trace semantics has been defined for various non-deterministic systems with different input/output types, or with different types of “non-determinism ” such as classical non-determinism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of “trace semantics” are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in

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Citation Context ...on. • The subdistribution monad D. It models probabilistic systems, or systems with probabilistic non-determinism: see Example 5.3. Its action is: for a set X and a function f : X → Y , DX = {d : X → =-=[0, 1]-=- | � d(x) ≤ 1} , (Df)(d) = λy. � d(x) , x∈X x∈f −1 ({y}) where d ∈ DX. Hence the set DX consists of probability distributions on X, with sum ≤ 1, instead of = 1. Its unit and multiplication is as foll... |

393 | Basic concepts of enriched category theory
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Citation Context ... the reader is referred to [1]. Throughout this section we assume that our base category C—later instantiated with Kℓ(T)—is DCpo-enriched. Spelling out the definition of enriched categories (see e.g. =-=[14,5]-=-), this means that each homset C(X, Y ) carries a partial order ≤ in such a way that each directed collection (fi)i∈I of maps fi: X → Y in C has a join � i∈I fi: X → Y . Additionally, composition pres... |

325 | Universal coalgebra: a theory of systems
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Citation Context ... that those various forms of “trace semantics” are instances of a general construction, namely coinduction in a Kleisli category. Our point of view here is categorical, coalgebraic in particular: see =-=[12,19]-=- for preliminaries. Hence this paper demonstrates the abstraction power of categorical/coalgebraic methods in computer science, uncovering basic mathematical structures underlying various concrete exa... |

256 |
Modeling and Verification of Randomized Distributed Real-Time Systems
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Citation Context ...s approach is to apply it to combined monads, producing trace semantics for suitably combined computational behaviours. An interesting example is combining classical and probabilistic non-determinism =-=[27,21]-=-. It has been shown (see [26]) that the simple composition PD has no monad structure: to make it a monad the authors propose to take the so-called indexed-valuation monad instead of the subdistributio... |

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Citation Context ...s approach is to apply it to combined monads, producing trace semantics for suitably combined computational behaviours. An interesting example is combining classical and probabilistic non-determinism =-=[27,21]-=-. It has been shown (see [26]) that the simple composition PD has no monad structure: to make it a monad the authors propose to take the so-called indexed-valuation monad instead of the subdistributio... |

228 | Derivatives of regular expressions
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Citation Context ...f all possible interleavings, such that: 〈〉 ∈ u � v a · w ∈ u � v def ⇐⇒ 〈〉 ∈ u and 〈〉 ∈ v , def ⇐⇒ w ∈ ∂au � v or w ∈ u � ∂av . Here ∂au = {w ∈ Σ ∗ | a ·w ∈ u} is the so-called Brzozowski derivative =-=[6]-=-. For example, {a, ab} � {〈〉, c} = {a, ab, ac, ca, cab, acb, abc}. Then the operation � is a map P(Σ ∗ ) × P(Σ ∗ ) P(Σ ∗ ) in Sets, i.e. P(Σ ∗ ) × P(Σ ∗ ) Σ ∗ in Kℓ(P). We obtain the map � via coinduc... |

163 | Reactive, generative and stratified models of probabilistic processes
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Citation Context ... to which we can apply our finality result. For example, • LTS’s with explicit termination (see e.g. [4,3]) are TF-coalgebras for T = P and F = 1 + Σ × ; • generative probabilistic transition systems =-=[25,23]-=- are TF-coalgebras for T = D and F = 1 + Σ × . In this section we take a step further ahead from the previous section to instantiate a shapely functor F, principally with 1 + Σ × . Then we observe tha... |

69 |
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Citation Context ...d ω op -limits: hence we can use the construction in Propositions 2.1 and 2.2. Lemma 2.6 Every shapely functor F : Sets → Sets has both an initial algebra and a final coalgebra. ✷ We recall (see e.g. =-=[10]-=-) that each monad T on Sets is strong, i.e. it comes with a natural transformation st: X × TY → T(X × Y ) that commutes appropriately with the monad’s unit and multiplication. Then there are two “obvi... |

63 | A semantics for shape
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Citation Context ... there. However our main result may still hold for monads that are not commutative and functors that are not shapely—we just require existence of a distributive law. Definition 2.5 (Shapely functors, =-=[13]-=-) The family of shapely functors on Sets is defined inductively by the following BNF notation: F, G, Fi ::= id | Σ | F × G | � i∈I Fi , where Σ denotes the constant functor into an arbitrary set Σ. No... |

56 | Initial algebra and final coalgebra semantics for concurrency
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Citation Context ...m an embedding-projection pair. Therein we use the monotonicity of Kℓ(F)’s action on arrows. ✷ Remark 3.6 The limit-colimit coincidence result of [22] is often applied to a (co)algebraic setting (see =-=[20]-=-). There it is common to assume the local continuity of a functor, such as Kℓ(F)( � i fi) = � � � i Kℓ(F)fi . For our main Theorem 3.1 we do not need that local continuity: the principal reason is tha... |

45 |
Monads on symmetric monoidal closed categories
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Citation Context ...he monad T is called commutative if these two maps are identical. In that case we call the resulting map the double strength of T and denote by dstX,Y : TX × TY → T(X × Y ). This definition is due to =-=[15]-=-. Lemma 2.7 Let T : Sets → Sets be a commutative monad, and F : Sets → Sets a shapely functor. Then there is a distributive law π: FT ⇒ TF. Proof. By induction on the structure of F. • If F is the ide... |

37 |
The category theoretic solution of recursive domain equations
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(Show Context)
Citation Context ...een reported in [8]. Here we generalize those results to monads with an order structure. The coincidence of initial algebra and final coalgebra—surprising at first sight—follows from the classic work =-=[22]-=- on limit-colimit coincidence. Here it is adapted to the setting of DCpo-enriched Kleisli categories. Many known non-deterministic systems are actually modelled as TF-coalgebras in Sets, with such T a... |

26 | Introduction to coalgebra: Towards mathematics of states and observations. http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro. pdf. Draft book
- Jacobs
- 2007
(Show Context)
Citation Context ... that those various forms of “trace semantics” are instances of a general construction, namely coinduction in a Kleisli category. Our point of view here is categorical, coalgebraic in particular: see =-=[12,19]-=- for preliminaries. Hence this paper demonstrates the abstraction power of categorical/coalgebraic methods in computer science, uncovering basic mathematical structures underlying various concrete exa... |

24 |
The linear time–branching time spectrum (extended abstract
- Glabbeek
(Show Context)
Citation Context ...antics, linear time semantics, monad, Kleisli category, non-determinism, probability 1 Introduction Trace semantics is a commonly used semantic relation for reasoning about nondeterministic 1 systems =-=[24]-=-. The notion of traces has been defined for various kinds of systems: for different input/output types, and more fundamentally for different types of “non-determinism” such as classical non-determinis... |

21 | Trace semantics for coalgebras
- Jacobs
- 2004
(Show Context)
Citation Context ... F Sets We shall now investigate the condition under which this distributive law π : FT ⇒ TF is available. For the case T = P, we have the following construction via relation lifting. Lemma 2.4 (From =-=[11]-=-) Let F : Sets → Sets be a functor that preserves weak pullbacks. Then there exists a “power law” π: F P ⇒ PF that forms a distributive law between F and the powerset monad P. The map πX : F(PX) → P(F... |

20 | Axiomatizing GSOS with termination
- Baeten, Vink
- 2002
(Show Context)
Citation Context ...ples Many known concrete dynamic systems are in fact TF-coalgebras for F shapely and T ∈ {L, P, D}, to which we can apply our finality result. For example, • LTS’s with explicit termination (see e.g. =-=[4,3]-=-) are TF-coalgebras for T = P and F = 1 + Σ × ; • generative probabilistic transition systems [25,23] are TF-coalgebras for T = D and F = 1 + Σ × . In this section we take a step further ahead from th... |

19 | Monadic Maps and Folds for Arbitrary Datatypes
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- 1994
(Show Context)
Citation Context ... is just a (functor-)coalgebra X → Kℓ(F)X. The following diagram of coinduction, now in Kℓ(T) for Kℓ(F)-coalgebras, captures trace semantics. Kℓ(F)X Kℓ(F)(trc) Kℓ(F)A c X trc It is standard (see e.g. =-=[7,17]-=-) that in such a situation—where we have a distributive law FT ⇒ TF—an initial F-algebra in Sets yields an initial Kℓ(F)-algebra in Kℓ(T). Our interest is in a final Kℓ(F)-coalgebra: in fact it coinci... |

18 |
Handbook of Categorical Algebra, volume 50, 51 and 52 of Encyclopedia of Mathematics. Cambridge Univ
- Borceux
- 1994
(Show Context)
Citation Context ... the reader is referred to [1]. Throughout this section we assume that our base category C—later instantiated with Kℓ(T)—is DCpo-enriched. Spelling out the definition of enriched categories (see e.g. =-=[14,5]-=-), this means that each homset C(X, Y ) carries a partial order ≤ in such a way that each directed collection (fi)i∈I of maps fi: X → Y in C has a join � i∈I fi: X → Y . Additionally, composition pres... |

15 | Fusion of recursive programs with computational effects
- Pardo
(Show Context)
Citation Context ... is just a (functor-)coalgebra X → Kℓ(F)X. The following diagram of coinduction, now in Kℓ(T) for Kℓ(F)-coalgebras, captures trace semantics. Kℓ(F)X Kℓ(F)(trc) Kℓ(F)A c X trc It is standard (see e.g. =-=[7,17]-=-) that in such a situation—where we have a distributive law FT ⇒ TF—an initial F-algebra in Sets yields an initial Kℓ(F)-algebra in Kℓ(T). Our interest is in a final Kℓ(F)-coalgebra: in fact it coinci... |

14 | A coalgebraic foundation for linear time semantics
- Power, Turi
- 1999
(Show Context)
Citation Context ...some basic facts on monads, Kleisli categories and distributive laws. A distributive law allows us to move our base category from Sets to Kℓ(T), by lifting a functor F. This shift, first exploited in =-=[18]-=-, plays a central role in this paper’s study about trace semantics for non-deterministic systems. Although some material applies to more general settings, here we restrict our base category to Sets fo... |

13 |
Least fixed point of a functor
- Adámek, Koubek
- 1979
(Show Context)
Citation Context ...er look at its construction. Finally, in Section 5 we instantiate the general result and present concrete examples. 2 Preliminaries 2.1 Initial/final sequence Here we recall the standard construction =-=[2]-=- of initial algebras (or final coalgebras) via the initial (or final) sequence. The construction will be heavily utilized throughout the paper: notice that the base category need not be Sets. Let C be... |

12 | Context-free languages via coalgebraic trace semantics
- Hasuo, Jacobs
(Show Context)
Citation Context ...)-algebra for a wide variety of a functor F and a monad T equipped with a suitable order structure. This is our main result. A special case of this result for the powerset monad has been presented in =-=[9]-=- and preliminary investigations for the probability subdistribution monad have been reported in [8]. Here we generalize those results to monads with an order structure. The coincidence of initial alge... |

12 | Closure properties of coalgebra automata
- Kupke, Venema
(Show Context)
Citation Context ...le composition PD has no monad structure: to make it a monad the authors propose to take the so-called indexed-valuation monad instead of the subdistribution monad. Another example are the F-automata =-=[16]-=- where the combination of type PP is used. Describing finite traces of such combined monads is a non-trivial matter which we postpone to a follow-up paper. Acknowledgements Thanks are due to Jiˇrí Adá... |

12 |
Coalgebraic Analysis of Probabilistic Systems
- Sokolova
- 2005
(Show Context)
Citation Context ... to which we can apply our finality result. For example, • LTS’s with explicit termination (see e.g. [4,3]) are TF-coalgebras for T = P and F = 1 + Σ × ; • generative probabilistic transition systems =-=[25,23]-=- are TF-coalgebras for T = D and F = 1 + Σ × . In this section we take a step further ahead from the previous section to instantiate a shapely functor F, principally with 1 + Σ × . Then we observe tha... |

8 |
G.: Distributing probabililty over nondeterminism
- Varacca, Winskel
- 2006
(Show Context)
Citation Context ...bined monads, producing trace semantics for suitably combined computational behaviours. An interesting example is combining classical and probabilistic non-determinism [27,21]. It has been shown (see =-=[26]-=-) that the simple composition PD has no monad structure: to make it a monad the authors propose to take the so-called indexed-valuation monad instead of the subdistribution monad. Another example are ... |

5 | Coalgebraic trace semantics for probabilistic systems, 2005
- Hasuo, Jacobs
(Show Context)
Citation Context ... This is our main result. A special case of this result for the powerset monad has been presented in [9] and preliminary investigations for the probability subdistribution monad have been reported in =-=[8]-=-. Here we generalize those results to monads with an order structure. The coincidence of initial algebra and final coalgebra—surprising at first sight—follows from the classic work [22] on limit-colim... |