## General structural operational semantics through categorical logic (Extended Abstract) (2008)

Citations: | 7 - 6 self |

### BibTeX

@MISC{Staton08generalstructural,

author = {Sam Staton},

title = {General structural operational semantics through categorical logic (Extended Abstract)},

year = {2008}

}

### OpenURL

### Abstract

Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rule-based definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for first-order calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the π-calculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxt-like rule format for open bisimulation in the π-calculus.

### Citations

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- 1989
(Show Context)
Citation Context ...[3; 36]). In this paper we focus on the tyft/tyxt format [20]. We argue that the questions above are not entirely answered by this research programme. Most rule formats account for languages like CCS =-=[33]-=- and various extensions thereof, but many languages are not simple enough to fit into these formats. Consider the π-calculus [34] — a relatively simple language by some standards. The syntax involves ... |

1286 | A structural approach to operational semantics
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(Show Context)
Citation Context ... π-calculus. 1. Introduction Questions about structural operational semantics. In this paper we are concerned with the semantics of programming languages in the structural style introduced by Plotkin =-=[38]-=-. One can reason in this style by induction on the language syntax, in terms of the steps that programs may make. Thus a structural operational semantics is given by a transition system specification ... |

995 | D.: A calculus of mobile processes, i
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(Show Context)
Citation Context ...s research programme. Most rule formats account for languages like CCS [33] and various extensions thereof, but many languages are not simple enough to fit into these formats. Consider the π-calculus =-=[34]-=- — a relatively simple language by some standards. The syntax involves variable binding, as in the λ-calculus, so there are concerns about α-equivalence and capture-avoiding substitution. There are va... |

784 | A.D.: A calculus for cryptographic protocols: The spi calculus - Abadi, Gordon - 1999 |

440 |
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(Show Context)
Citation Context ...ctors through Σ Ñ pΣ 1q, then the unit of the lifted monad is componentwise open. 5. Advanced examples I: name-passing calculi We now apply the developments of the previous sections to the π-calculus =-=[34; 40]-=-, a simple calculus for describing the communication of channel names along named channels. The semantics of the π-calculus involves variable binding and substitution, and so we investigate the catego... |

341 |
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(Show Context)
Citation Context ...hism is internally projective. 2.2. W-types, free algebras, and ΠW-pretoposes An important aspect of Martin-Löf’s dependent type theory is W-types, which describe well-founded induction and recursion =-=[30]-=-. W-types are central to our development, and we now summarize the important ideas in the categorical setting, following [35; 18; 5]. Signatures in categories, and their algebras. Definition 1 A signa... |

273 | Mobile values, new names, and secure communication - Abadi, Fournet - 2001 |

208 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
- 2002
(Show Context)
Citation Context ...f channel names along named channels. The semantics of the π-calculus involves variable binding and substitution, and so we investigate the categorical tyft/tyxt format in the setting of nominal sets =-=[17]-=- and nominal substitutions [12]. 5.1. Nominal sets, nominal substitutions and wideopen bisimulation We recall some definitions from the theory of nominal sets [e.g. 17]. For the remainder of this pape... |

189 |
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(Show Context)
Citation Context ...ther things, the property that bisimilarity is a congruence. Further work on more elaborate rule formats has continued since then (see surveys [3; 36]). In this paper we focus on the tyft/tyxt format =-=[20]-=-. We argue that the questions above are not entirely answered by this research programme. Most rule formats account for languages like CCS [33] and various extensions thereof, but many languages are n... |

162 |
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(Show Context)
Citation Context ...quirement of congruence for a relation on WΣ is that of Def. 4. Bisimulation from open maps. To define bisimulation in a general setting, we consider the following situation, generalizing the work of =-=[4; 26]-=-. In a category Systems, we consider a class of ‘open’ morphisms. There are various properties that might be expected of this class. We do not require anything of the morphisms for now, but the follow... |

150 | Sketches of an Elephant: a Topos Theory Compendium - Johnstone - 2002 |

134 | Toward a mathematical Operational Semantics
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Citation Context ...uestions, and to have a chance at tackling increasingly sophisticated languages, we must identify the commonality. Abstract answers: Functorial Operational Semantics. Functorial Operational Semantics =-=[44]-=- provides answers to the questions at an abstract level. Good operational semantics is seen categorically as a distributive law of a monad over a comonad, or equivalently as a lifting of a monad to a ... |

125 |
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(Show Context)
Citation Context ...vestigate operational semantics and congruence results in a even more general setting. We state general definitions and results in the setting of categories of monads. A category of monads. We follow =-=[42]-=- in defining the category Mnd of monads as follows. The objects are pairs pC, T q of a category C and a monad T ; a morphism pF, φq : pC, T q Ñ pD, Sq is a functor F : C Ñ D together with a natural tr... |

121 | Structural operational semantics
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(Show Context)
Citation Context ...pecified according to this rule format will have, among other things, the property that bisimilarity is a congruence. Further work on more elaborate rule formats has continued since then (see surveys =-=[3; 36]-=-). In this paper we focus on the tyft/tyxt format [20]. We argue that the questions above are not entirely answered by this research programme. Most rule formats account for languages like CCS [33] an... |

116 | Bisimulation from open maps
- Joyal, Nielsen, et al.
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(Show Context)
Citation Context ...quirement of congruence for a relation on WΣ is that of Def. 4. Bisimulation from open maps. To define bisimulation in a general setting, we consider the following situation, generalizing the work of =-=[4; 26]-=-. In a category Systems, we consider a class of ‘open’ morphisms. There are various properties that might be expected of this class. We do not require anything of the morphisms for now, but the follow... |

97 |
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Citation Context ...s has an asymmetric monoidal structure: the tensor is given by pA, A 1 , #Aq pB, B 1 , #Bq A Z B, A 1 Z B 1 , #A Z #B Y tpa, bq | a P A 1 , b P Bu § where p q § indicates symmetric closure. Following =-=[10]-=-, this monoidal structure induces a biclosed monoidal structure on Set Env . We write for the monoidal product, and rXsp q for the right adjoint of X p q. The functor rN sp q introduces new binding na... |

63 | Higher-level synchronising devices in Meije-SCCS
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(Show Context)
Citation Context ...resent paper we bring the two approaches together, providing rule formats in a general categorical setting. Concrete answers: rule formats. In a first attempt to answer the questions above, de Simone =-=[11]-=- introduced a rule format: a list of syntactic conditions about rules, stipulating which variables and operators may appear in the various parts of the rule. Any language specified according to this r... |

46 |
A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so
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(Show Context)
Citation Context ...pen. Here, we have assumed that Systems is locally presentable and that the monads are finitary because the result can then be proved using an inductive characterization of the sum pT1 T2q, following =-=[27]-=-. Relevance of tyft/tyxt. The conservativity result for tyft/tyxt (Thm. 8) can be understood in the context of Thm. 14, through Thm. 13 and the following result. Proposition 15 If all the rules in a c... |

41 |
A completeness theorem for open maps
- Joyal, Moerdijk
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(Show Context)
Citation Context ...ystems, we consider a class of ‘open’ morphisms. There are various properties that might be expected of this class. We do not require anything of the morphisms for now, but the following axioms (from =-=[25; 24]-=-) will be referred to in this development. • A1. Isomorphisms are open, and open maps are closed under composition. • A3 (Descent). In a pullback square f 1 Y 1 X 1 g � �� Y � �� Y if g is epi and f 1... |

39 | A theory of bisimulation for the pi-calculus
- Sangiorgi
- 1996
(Show Context)
Citation Context ...lculi by working in more elaborate categories. In Sec. 5 we study the categorical format in a category of nominal substitutions, achieving congruence results for a simplification of open bisimilarity =-=[39]-=-. In Sec. 6, we work with (genuine) open bisimilarity by studying the format in a category of presheaves. The resulting format is essentially that of [45]. Acknowledgements. Many thanks to Marcelo Fio... |

38 |
Algebraic Set Theory
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(Show Context)
Citation Context ...ystems, we consider a class of ‘open’ morphisms. There are various properties that might be expected of this class. We do not require anything of the morphisms for now, but the following axioms (from =-=[25; 24]-=-) will be referred to in this development. • A1. Isomorphisms are open, and open maps are closed under composition. • A3 (Descent). In a pullback square f 1 Y 1 X 1 g � �� Y � �� Y if g is epi and f 1... |

37 | Wellfounded trees in categories
- Moerdijk, Palmgren
- 2000
(Show Context)
Citation Context ...e, we find that we are able to formulate results about the tyft/tyxt format in any ΠW-pretopos. Such categories arise as models of first-order logic, dependent type theory, and well-founded induction =-=[35; 18; 5]-=-. From a logical perspective, then, we develop a meta-theory for tyft/tyxt specifications in a constructive logic. Finally, once the concrete results are understood in general, categorical terms, the ... |

31 | A conservative look at operational semantics with variable binding - Fokkink, Verhoef - 1998 |

30 | Combining effects: sum and tensor - Hyland, Plotkin, et al. - 2006 |

26 | Wellfounded trees and dependent polynomial functors
- Gambino, Hyland
- 2004
(Show Context)
Citation Context ...e, we find that we are able to formulate results about the tyft/tyxt format in any ΠW-pretopos. Such categories arise as models of first-order logic, dependent type theory, and well-founded induction =-=[35; 18; 5]-=-. From a logical perspective, then, we develop a meta-theory for tyft/tyxt specifications in a constructive logic. Finally, once the concrete results are understood in general, categorical terms, the ... |

24 | Propositions as [types - Awodey, Bauer |

21 | Abstract syntax and variable binding (extended abstract
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...e category of nominal substitutions and homomorphisms between them. The category of nominal substitutions is a Grothendieck topos. Indeed, it is a sheaf subcategory of the category Set F , studied by =-=[13]-=- (see [41, Sec. 7.3.1]). The forgetful functor NSub Ñ Nom preserves products, and we can lift the additional structure of Nom along it. The nominal set of names has an evident nominal substitution str... |

21 | A proof search specification of the π-calculus - Tiu, Miller - 2004 |

20 | A congruence rule format for name-passing process calculi
- Fiore, Staton
- 2009
(Show Context)
Citation Context ...nnels. The semantics of the π-calculus involves variable binding and substitution, and so we investigate the categorical tyft/tyxt format in the setting of nominal sets [17] and nominal substitutions =-=[12]-=-. 5.1. Nominal sets, nominal substitutions and wideopen bisimulation We recall some definitions from the theory of nominal sets [e.g. 17]. For the remainder of this paper we fix an infinite set N of n... |

19 | Compositional SOS and beyond: a coalgebraic view of open systems - Corradini, Heckel, et al. |

19 | Variable binding operators in transition system specifications - Middelburg |

16 | The tyft/tyxt format reduces to tree rules
- Fokkink
- 1994
(Show Context)
Citation Context ...TSS. 1. The multiplication of the lifted monad is componentwise open. 2. If the TSS is pure and pP Ñ Rq is internally projective then the functorial action of the lifted monad preserves open maps. In =-=[14; 20]-=- it is argued that the purity and well-foundedness conditions can be eliminated from the congruence theorem, because the initial labelled transition system induced by any tyft/tyxt TSS can also be ind... |

16 | The π-calculus in FM - Gabbay |

16 | A congruence format for name-passing calculi, Electron
- Ziegler, Miller, et al.
- 2006
(Show Context)
Citation Context ...lts for a simplification of open bisimilarity [39]. In Sec. 6, we work with (genuine) open bisimilarity by studying the format in a category of presheaves. The resulting format is essentially that of =-=[45]-=-. Acknowledgements. Many thanks to Marcelo Fiore for his guidance. It was helpful to be able to discuss Prop. 6 with Thorsten Altenkirch, Nicola Gambino and Thomas Streicher, and to discuss [45] with ... |

14 | Formalising the pi-calculus using nominal logic - Bengtson, Parrow - 2008 |

13 | A unifying model of variables and names, in: V
- Miculan, Yemane
- 2005
(Show Context)
Citation Context ...d example II: Open bisimulation For a more sophisticated example, we consider categorical tyft/tyxt TSSs in a presheaf category. This presheaf category has much in common with the category studied in =-=[19; 31]-=-. In this category, the categorical tyft/tyxt format is essentially the format proposed by [45]. Indeed, fol10lowing [8] we arrive at a treatment of the open bisimilarity of [39]. Pragmatic motivatio... |

10 | Relationally staged computation in calculi of mobile processes
- Ghani, Yemane, et al.
(Show Context)
Citation Context ...d example II: Open bisimulation For a more sophisticated example, we consider categorical tyft/tyxt TSSs in a presheaf category. This presheaf category has much in common with the category studied in =-=[19; 31]-=-. In this category, the categorical tyft/tyxt format is essentially the format proposed by [45]. Indeed, fol10lowing [8] we arrive at a treatment of the open bisimilarity of [39]. Pragmatic motivatio... |

9 | Modular correspondence between dependent type theories and categorical universes including pretopoi and topoi - Maietti |

8 | Name-passing process calculi: operational models and structural operational semantics - Staton - 2007 |

5 | Bialgebraic methods in structural operational semantics
- Klin
- 2007
(Show Context)
Citation Context ...ory of coalgebras. The initial work [44] illustrated the general framework with congruence results for CCSlike languages, but subsequent work demonstrated the ideas in various different settings (see =-=[28]-=- for a survey). A potential criticism of much of this work is that it is too abstract. One must really squint hard to view a distributive law as a collection of rules, and to view the naturality condi... |

4 | I.: pi-calculus in (co)inductive-type theory - Honsell, Miculan, et al. - 2001 |

2 | A categorical version of the Brouwer-HeytingKolmogorov interpretation - Palmgren |

1 |
Indexed containers. Available online
- Altenkirch, Ghani, et al.
- 2006
(Show Context)
Citation Context ...e, we find that we are able to formulate results about the tyft/tyxt format in any ΠW-pretopos. Such categories arise as models of first-order logic, dependent type theory, and well-founded induction =-=[35; 18; 5]-=-. From a logical perspective, then, we develop a meta-theory for tyft/tyxt specifications in a constructive logic. Finally, once the concrete results are understood in general, categorical terms, the ... |

1 |
Open bisimulation, revisited. Theoret
- Briais, Nestmann
(Show Context)
Citation Context ...y. This presheaf category has much in common with the category studied in [19; 31]. In this category, the categorical tyft/tyxt format is essentially the format proposed by [45]. Indeed, fol10lowing =-=[8]-=- we arrive at a treatment of the open bisimilarity of [39]. Pragmatic motivation. Wide-open bisimilarity can be criticised for being too discriminating. Suppose that we add a name-equality check to th... |

1 |
SOS formats and meta-theory
- Mousavi, Reniers, et al.
(Show Context)
Citation Context ...pecified according to this rule format will have, among other things, the property that bisimilarity is a congruence. Further work on more elaborate rule formats has continued since then (see surveys =-=[3; 36]-=-). In this paper we focus on the tyft/tyxt format [20]. We argue that the questions above are not entirely answered by this research programme. Most rule formats account for languages like CCS [33] an... |