## Generic trace semantics via coinduction (2007)

Venue: | Logical Methods in Comp. Sci |

Citations: | 17 - 6 self |

### BibTeX

@INPROCEEDINGS{Hasuo07generictrace,

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

title = {Generic trace semantics via coinduction},

booktitle = {Logical Methods in Comp. Sci},

year = {2007},

pages = {2007}

}

### OpenURL

### Abstract

Abstract. Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these “trace

### Citations

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Citation Context ...ces, for example in order to abstract away internal branching structures. One of such coarser semantics, which has been extensively studied, is trace equivalence. For example, the process algebra CSP =-=[21]-=- has trace semantics as its operational model. Trace equivalence is coarser than bisimilarity, as the following classic example of “trace-equivalent but not bisimilar” systems illustrates. a x a • b •... |

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Citation Context ...ural transformation µ : T 2 ⇒ T, consisting of functions T 2 X µX → TX with X ranging over sets. The unit and multiplication are required to satisfy the following compatibility conditions. TX ηTX See =-=[42, 3]-=- for the details. id T 2 X TηX TX T 3 X TµX T 2 X µX id µTX µX TX T 2X µX TX 4 Non-examples include LTSs with unbounded branching degree. They are modeled as coalgebras for FX = P(Σ × X). Lambek’s Lem... |

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Citation Context ...s. First we motivate our contribution through examples of various forms of “trace semantics”. Think of the following three state-based, branching systems. x a y b � a[ 1 3 ] x ′ z ′ a[ 1 3 ] 1 3 y ′ a=-=[1]-=- � 1 2 a[ 1 2 ] A context-free grammar (for Peano Arithmetic) Terminal symbols: 0,s Non-terminal symbol: T Generation rules: T → 0 T → sT (1.1) • The first one is a non-deterministic system with a spe... |

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Citation Context ...VA This is in fact an instance of a more general notion of V-enriched categories where V is the category Cppo of pointed (i.e. with ⊥) cpo’s and continuous (but not necessarily strict) functions. See =-=[38, 29, 7]-=- for more details on enriched category theory, and [1] on cpo’s and domain theory. Lemma 2.4. For our three examples L, P and D of a monad T, the Kleisli category Kℓ(T) is Cppo-enriched. Moreover, com... |

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Citation Context ... infinite trace a ω ↦→ 1/3.sGENERIC TRACE SEMANTICS VIA COINDUCTION 3 1.2. Coalgebras and coinduction. In recent years the theory of coalgebras has emerged as the “mathematics of state-based systems” =-=[25, 47, 26]-=-. In the categorical theory of coalgebras, an important definition/reasoning principle is coinduction: a system (identified with a coalgebra c : X → FX) is assigned a unique morphism behc into the fin... |

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Citation Context ... but do not yet include systems with both non-deterministic and probabilistic branching. The importance of having both of these branchings in system verification has been claimed by many authors e.g. =-=[60, 48]-=-, with an intuition that probabilistic branching models the choices “made by the system, i.e. on our side”, while (coarser) nondeterministic choices are “made by the (unknown) environment of the syste... |

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Citation Context ... infinite trace a ω ↦→ 1/3.sGENERIC TRACE SEMANTICS VIA COINDUCTION 3 1.2. Coalgebras and coinduction. In recent years the theory of coalgebras has emerged as the “mathematics of state-based systems” =-=[25, 47, 26]-=-. In the categorical theory of coalgebras, an important definition/reasoning principle is coinduction: a system (identified with a coalgebra c : X → FX) is assigned a unique morphism behc into the fin... |

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Citation Context ... but do not yet include systems with both non-deterministic and probabilistic branching. The importance of having both of these branchings in system verification has been claimed by many authors e.g. =-=[60, 48]-=-, with an intuition that probabilistic branching models the choices “made by the system, i.e. on our side”, while (coarser) nondeterministic choices are “made by the (unknown) environment of the syste... |

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Citation Context ...ural transformation µ : T 2 ⇒ T, consisting of functions T 2 X µX → TX with X ranging over sets. The unit and multiplication are required to satisfy the following compatibility conditions. TX ηTX See =-=[42, 3]-=- for the details. id T 2 X TηX TX T 3 X TµX T 2 X µX id µTX µX TX T 2X µX TX 4 Non-examples include LTSs with unbounded branching degree. They are modeled as coalgebras for FX = P(Σ × X). Lambek’s Lem... |

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Citation Context ...gebras X → FX in a Kleisli category Kℓ(T). For example, • LTSs with explicit termination, with T = P and F = 1 + Σ × ; • probabilistic LTSs (also called generative probabilistic transition systems in =-=[58, 52]-=-) with explicit termination, with T = D and F = 1 + Σ × ; • context-free grammars with T = P and F = (Σ + ) ∗ . The main observation underlying this work is the following. If we instantiate the parame... |

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Citation Context ....P � a.Q) and a structural operational semantics (SOS) rule determines its dynamics, that is, its coalgebraic structure. This is where “algebra meets coalgebra” and the interaction is studied e.g. in =-=[55, 4, 31]-=-. In our recent work [18] we claim the importance of the microcosm principle in this context and provide a “general compositionality theorem”: under suitable assumptions, the final coalgebra semantics... |

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Citation Context ...VA This is in fact an instance of a more general notion of V-enriched categories where V is the category Cppo of pointed (i.e. with ⊥) cpo’s and continuous (but not necessarily strict) functions. See =-=[38, 29, 7]-=- for more details on enriched category theory, and [1] on cpo’s and domain theory. Lemma 2.4. For our three examples L, P and D of a monad T, the Kleisli category Kℓ(T) is Cppo-enriched. Moreover, com... |

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Citation Context ...s is a commonly used semantic relation for reasoning about state-based systems. Trace semantics for labeled transition systems is found on the coarsest edge of the linear time-branching time spectrum =-=[57]-=-. Moreover, trace semantics is defined for a variety of systems, among which are probabilistic systems [49]. In this paper we claim that these various forms of “trace semantics” are instances of a gen... |

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Citation Context .../limit of the initial/final sequence. 3.3. Related work: axiomatic domain theory. The initial algebra-final coalgebra coincidence is heavily exploited in the field of axiomatic domain theory, e.g. in =-=[13, 12, 9, 50]-=-. There, categories which have coinciding initial algebra and final coalgebra for each endofunctor are called algebraically compact categories. They draw special attention as suitable “categories of d... |

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Citation Context .../limit of the initial/final sequence. 3.3. Related work: axiomatic domain theory. The initial algebra-final coalgebra coincidence is heavily exploited in the field of axiomatic domain theory, e.g. in =-=[13, 12, 9, 50]-=-. There, categories which have coinciding initial algebra and final coalgebra for each endofunctor are called algebraically compact categories. They draw special attention as suitable “categories of d... |

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Citation Context .../limit of the initial/final sequence. 3.3. Related work: axiomatic domain theory. The initial algebra-final coalgebra coincidence is heavily exploited in the field of axiomatic domain theory, e.g. in =-=[13, 12, 9, 50]-=-. There, categories which have coinciding initial algebra and final coalgebra for each endofunctor are called algebraically compact categories. They draw special attention as suitable “categories of d... |

60 | A semantics for shape
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Citation Context ...wing double strengths. dst L � (u, v) if u ∈ X and v ∈ Y, X,Y (u, v) = ⊥ if u = ⊥ or v = ⊥, dst P (2.5) X,Y (u, v) = u × v , (u, v) = λ(x, y). u(x) · v(y) . dst D X,Y • The family of shapely functors =-=[27]-=- 6 on Sets is defined inductively by the following BNF notation: F ::= id | Σ | F1 × F2 | � i∈IFi , where Σ denotes the constant functor into an arbitrary set Σ. Notice that taking infinite product is... |

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Citation Context ...nd trace semantics. Since the emergence of the theory of coalgebras, the significance of modal logics as specification languages has been noticed by many authors. This is exemplified by the slogan in =-=[36]-=-: ‘modal logic is to coalgebras what equational logic is to algebras’. Inspired by coalgebras on Stone spaces and the corresponding modal logic, recent developments [35, 5, 6, 37, 45, 33, 32] have ide... |

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Citation Context ...ult. However, an order structure is not the only one that can yield such coincidence: other examples include metric, quasi-metric and quantale-enriched structures (in increasing generality). See e.g. =-=[56, 10]-=- for the potential use of such enriched structures in a coalgebraic setting. The relation of the current work to such structures is yet to be investigated. In the discipline of process algebra, a syst... |

43 |
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Citation Context ...by identifying lax/oplax morphisms of coalgebras in a Kleisli category as forward/backward simulations. Use of traces and simulations is a common technique in formal verification of systems (see e.g. =-=[41]-=-): a desirable property is expressed in terms of traces; and then a system is shown to satisfy the property by finding a suitable simulation. Therefore this paper, together with [14], forms an essenti... |

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Citation Context ...mutative and a functor F is shapely, we can provide a canonical distributive law. The class of such monads and functors is wide and all the examples in this paper are contained. • A commutative monad =-=[34]-=- is intuitively a monad whose corresponding algebraic theory has only commutative operators. We exploit the fact that a commutative monad is equipped with an arrow called double strength dstX,Y : TX ×... |

38 | On generalised coinduction and probabilistic specification formats
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Citation Context ....P � a.Q) and a structural operational semantics (SOS) rule determines its dynamics, that is, its coalgebraic structure. This is where “algebra meets coalgebra” and the interaction is studied e.g. in =-=[55, 4, 31]-=-. In our recent work [18] we claim the importance of the microcosm principle in this context and provide a “general compositionality theorem”: under suitable assumptions, the final coalgebra semantics... |

37 | A coinduction principle for recursive data types based on bisimulation
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Citation Context ...ult. However, an order structure is not the only one that can yield such coincidence: other examples include metric, quasi-metric and quantale-enriched structures (in increasing generality). See e.g. =-=[56, 10]-=- for the potential use of such enriched structures in a coalgebraic setting. The relation of the current work to such structures is yet to be investigated. In the discipline of process algebra, a syst... |

36 |
Compositional trace–based semantics for probabilistic automata
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Citation Context ... transition systems is found on the coarsest edge of the linear time-branching time spectrum [57]. Moreover, trace semantics is defined for a variety of systems, among which are probabilistic systems =-=[49]-=-. In this paper we claim that these various forms of “trace semantics” are instances of a general construction, namely coinduction in a Kleisli category. Our point of view here is categorical, coalgeb... |

34 |
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Citation Context ...ble adjunctionlifting result;s4 I. HASUO, B. JACOBS, AND A. SOKOLOVA – in a Kleisli category we have initial algebra-final coalgebra coincidence. Here we use the classical result by Smyth and Plotkin =-=[51]-=-, namely limit-colimit coincidence which is applicable in a suitably order-enriched category. Note the presence of two parameters in (1.5): a monad T and an endofunctor F, both on Sets. The monad T sp... |

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Citation Context ...s exemplified by the slogan in [36]: ‘modal logic is to coalgebras what equational logic is to algebras’. Inspired by coalgebras on Stone spaces and the corresponding modal logic, recent developments =-=[35, 5, 6, 37, 45, 33, 32]-=- have identified the following situation as the essential mathematical structure underlying modal logics for coalgebras. F op C op P ⊤ A M S op together with MP δ =⇒ PF op In fact, it is noticed in [4... |

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Citation Context ...s exemplified by the slogan in [36]: ‘modal logic is to coalgebras what equational logic is to algebras’. Inspired by coalgebras on Stone spaces and the corresponding modal logic, recent developments =-=[35, 5, 6, 37, 45, 33, 32]-=- have identified the following situation as the essential mathematical structure underlying modal logics for coalgebras. F op C op P ⊤ A M S op together with MP δ =⇒ PF op In fact, it is noticed in [4... |

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Citation Context ... infinite trace a ω ↦→ 1/3.sGENERIC TRACE SEMANTICS VIA COINDUCTION 3 1.2. Coalgebras and coinduction. In recent years the theory of coalgebras has emerged as the “mathematics of state-based systems” =-=[25, 47, 26]-=-. In the categorical theory of coalgebras, an important definition/reasoning principle is coinduction: a system (identified with a coalgebra c : X → FX) is assigned a unique morphism behc into the fin... |

24 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
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Citation Context ...Kℓ(T). A functor F is said to be a lifting of F if the following diagram commutes. Here J is the left adjoint in (2.2). Kℓ(T) J Sets F F Kℓ(T) J Sets The following fact is presented in [43]; see also =-=[39, 40]-=-. Its proof is straightforward. . (2.3) Lemma 2.1. A lifting F of F is in bijective correspondence with a distributive law λ : FT⇒TF. A distributive law λ is a natural transformation which is compatib... |

19 | Monadic Maps and Folds for Arbitrary Datatypes
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Citation Context ...e initial algebra in Sets is the final coalgebra in Kℓ(T). First, it is standard that an initial algebra in Sets is lifted to an initial algebra in Kℓ(T). Such a phenomenon is studied for instance in =-=[11, 44]-=- in the context of combining datatypes (modeled by an initial algebra) and effectful computations (modeled by a Kleisli category). For this result we do not need an order structure. Proposition 3.1. L... |

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Citation Context ...VA This is in fact an instance of a more general notion of V-enriched categories where V is the category Cppo of pointed (i.e. with ⊥) cpo’s and continuous (but not necessarily strict) functions. See =-=[38, 29, 7]-=- for more details on enriched category theory, and [1] on cpo’s and domain theory. Lemma 2.4. For our three examples L, P and D of a monad T, the Kleisli category Kℓ(T) is Cppo-enriched. Moreover, com... |

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Citation Context ... It is first noticed in [46] that the Kleisli category for the powerset monad is an appropriate base category for trace semantics for non-deterministic systems. This observation is pursued further in =-=[24, 16, 17]-=-. In [15] it is recognized that the same is true for the subdistribution monad for probabilistic systems. The current paper provides a unified framework which yields those preceding results, in terms ... |

18 |
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Citation Context ...a functor F on Kℓ(T). A functor F is said to be a lifting of F if the following diagram commutes. Here J is the left adjoint in (2.2). Kℓ(T) J Sets F F Kℓ(T) J Sets The following fact is presented in =-=[43]-=-; see also [39, 40]. Its proof is straightforward. . (2.3) Lemma 2.1. A lifting F of F is in bijective correspondence with a distributive law λ : FT⇒TF. A distributive law λ is a natural transformatio... |

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Citation Context ...is given by probabilistic automata introduced by Segala [48]. In fact this combination of non-deterministic and probabilistic branching is a notoriously difficult one from a theoretical point of view =-=[8, 59, 54]-=-: many mathematical tools that are useful in a purely non-deterministic or probabilistic setting cease to work in the presence of both. For our framework of generic trace semantics, the problem is tha... |

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Citation Context ...y: Kℓ(P) op F op Kℓ(P) op Op Op Kℓ(P) F Kℓ(P) (3.6) which is because: FR = RelF(R) (see (2.4)); and taking relation liftings is compatible with opposite relations (i.e. RelF(R op ) = (RelFR) op , see =-=[22]-=-). Moreover the category Alg(F op ) is obviously isomorphic to (Coalg(F)) op . Therefore the initial object in Alg(F) is carried to that in (Coalg(F)) op , hence the final object in Coalg(F).s18 I. HA... |

15 |
Reconciling Nondeterministic and Probabilistic Choices
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Citation Context ...is given by probabilistic automata introduced by Segala [48]. In fact this combination of non-deterministic and probabilistic branching is a notoriously difficult one from a theoretical point of view =-=[8, 59, 54]-=-: many mathematical tools that are useful in a purely non-deterministic or probabilistic setting cease to work in the presence of both. For our framework of generic trace semantics, the problem is tha... |

14 |
Coalgebras and their logics
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(Show Context)
Citation Context ...s exemplified by the slogan in [36]: ‘modal logic is to coalgebras what equational logic is to algebras’. Inspired by coalgebras on Stone spaces and the corresponding modal logic, recent developments =-=[35, 5, 6, 37, 45, 33, 32]-=- have identified the following situation as the essential mathematical structure underlying modal logics for coalgebras. F op C op P ⊤ A M S op together with MP δ =⇒ PF op In fact, it is noticed in [4... |

14 | Fusion of Recursive Programs with Computational Effects
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Citation Context ...e initial algebra in Sets is the final coalgebra in Kℓ(T). First, it is standard that an initial algebra in Sets is lifted to an initial algebra in Kℓ(T). Such a phenomenon is studied for instance in =-=[11, 44]-=- in the context of combining datatypes (modeled by an initial algebra) and effectful computations (modeled by a Kleisli category). For this result we do not need an order structure. Proposition 3.1. L... |

14 | D.: A coalgebraic foundation for linear time semantics
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(Show Context)
Citation Context ...l. Trace equivalence is coarser than bisimilarity, as the following classic example of “trace-equivalent but not bisimilar” systems illustrates. a x a • b • c � � y a • b c � � It is first noticed in =-=[46]-=- that the Kleisli category for the powerset monad is an appropriate base category for trace semantics for non-deterministic systems. This observation is pursued further in [24, 16, 17]. In [15] it is ... |

13 | Least fixed point of a functor - Adámek, Koubek - 1979 |

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Recursive types in Kleisli categories. Unpublished manuscript
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12 | A testing scenario for probabilistic automata
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Citation Context ...gebras. F op C op P ⊤ A M S op together with MP δ =⇒ PF op In fact, it is noticed in [45] that such a situation not only hosts a modal logic but also a more general notion of testing (in the sense of =-=[57, 53]-=-, also called testing scenarios). Therefore we shall call the above situation a testing situation. In the last technical section of the paper we investigate coalgebraic trace semantics for the special... |

11 | Generic Forward and Backward Simulations
- Hasuo
- 2006
(Show Context)
Citation Context ...ms; coinduction yielding process semantics; and morphisms of coalgebras as behavior-preserving maps. In this paper we study the first two in a Kleisli category. What about morphisms of coalgebras? In =-=[14]-=- this question is answered by identifying lax/oplax morphisms of coalgebras in a Kleisli category as forward/backward simulations. Use of traces and simulations is a common technique in formal verific... |

11 | B.: Context-free languages via coalgebraic trace semantics
- Hasuo, Jacobs
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(Show Context)
Citation Context ...bject Classification: F.3.1, F.3.2, G.3. Key words and phrases: coalgebra, category theory, trace semantics, monad, Kleisli category, process semantics, non-determinism, probability. Earlier versions =-=[16, 17]-=- of this paper are presented at the 1st International Conference on Algebra and Coalgebra in Computer Science (CALCO 2005), Swansea, UK, September 2005, and at the 8th International Workshop on Coalge... |

11 | The microcosm principle and concurrency in coalgebras, 2007. preprint, available from http://www.cs.ru.nl/ ichiro/papers
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Citation Context ...tional semantics (SOS) rule determines its dynamics, that is, its coalgebraic structure. This is where “algebra meets coalgebra” and the interaction is studied e.g. in [55, 4, 31]. In our recent work =-=[18]-=- we claim the importance of the microcosm principle in this context and provide a “general compositionality theorem”: under suitable assumptions, the final coalgebra semantics is compatible with the a... |

10 | From bialgebraic semantics to congruence formats
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Citation Context ....P � a.Q) and a structural operational semantics (SOS) rule determines its dynamics, that is, its coalgebraic structure. This is where “algebra meets coalgebra” and the interaction is studied e.g. in =-=[55, 4, 31]-=-. In our recent work [18] we claim the importance of the microcosm principle in this context and provide a “general compositionality theorem”: under suitable assumptions, the final coalgebra semantics... |