## Reasoning about Object-based Calculi in (Co)Inductive Type Theory and the Theory of Contexts ∗

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Citations: | 4 - 0 self |

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@MISC{Ciaffaglione_reasoningabout,

author = {Alberto Ciaffaglione and Luigi Liquori and Marino Miculan},

title = {Reasoning about Object-based Calculi in (Co)Inductive Type Theory and the Theory of Contexts ∗},

year = {}

}

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### Abstract

Abstract. We illustrate a methodology for formalizing and reasoning about Abadi and Cardelli’s object-based calculi, in (co)inductive type theory, such as the Calculus of (Co)Inductive Constructions, by taking advantage of Natural Deduction Semantics and coinduction in combination with weak Higher-Order Abstract Syntax and the Theory of Contexts. Our methodology allows to implement smoothly the calculi in the target metalanguage; moreover, it suggests novel presentations of the calculi themselves. In detail, we present a compact formalization of the syntax and semantics for the functional and the imperative variants of the ς-calculus. Our approach simplifies the proof of Subject Reduction theorems, which are proved formally in the proof assistant Coq with a relatively small overhead.

### Citations

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Citation Context ... have received little attention. We see this fact as a serious gap, because most of the foundational calculi introduced for the mathematical analysis of the object-oriented paradigm are object-based (=-=Abadi and Cardelli, 1996-=-; Fisher et al., 1994). Indeed, object-based languages simplify and generalize class-based ones: they reduce classes to more primi∗ Supported by UE project IST-CA-510996 Types and French grant CNRS AC... |

715 | A framework for defining logics
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Citation Context ...ementation in CC (Co)Ind , we have to face the problem of representing binders efficiently. To this end, one of the most suited approaches is higher-order abstract syntax (Pfenning and Elliott, 1988; =-=Harper et al., 1993-=-; Miculan, 1997). More precisely, since we work in a type theory with induction, we use weak HOAS (Miculan, 1997; Honsell et al., 2001b): binders are represented as second-order term constructors, tak... |

423 | A language with distributed scope
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Citation Context ..., Isabelle and PVS (Marché et al., 2004; Klein and Nipkow, 2003; Huisman, 2001; Van den Berg et al., 2001; Tews, 2000). On the other hand, object-based languages, such as Self (Self, 2003) and Obliq (=-=Cardelli, 1995-=-) have received little attention. We see this fact as a serious gap, because most of the foundational calculi introduced for the mathematical analysis of the object-oriented paradigm are object-based ... |

323 | Natural semantics
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Citation Context ... x is bound in b. Usual conventions about α-conversion and free variables (denoted by FV(a)) apply. Dynamic Semantics. The operational semantics of funς is expressed by a big-step reduction relation (=-=Kahn, 1987-=-; Despeyroux, 1986), relating two stores σ, σ ′ , a stack S, a term a, and a result v: σ·S ⊢AC a ❀ v·σ ′ The intended meaning is that, starting with the store σ (playing the role of a heap) and the st... |

311 |
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Citation Context ...ly, when we come to the implementation in CC (Co)Ind , we have to face the problem of representing binders efficiently. To this end, one of the most suited approaches is higher-order abstract syntax (=-=Pfenning and Elliott, 1988-=-; Harper et al., 1993; Miculan, 1997). More precisely, since we work in a type theory with induction, we use weak HOAS (Miculan, 1997; Honsell et al., 2001b): binders are represented as second-order t... |

222 | The practice of logical frameworks - Pfenning - 1996 |

169 | A lambda-calculus of objects and method specialization
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Citation Context ...ntion. We see this fact as a serious gap, because most of the foundational calculi introduced for the mathematical analysis of the object-oriented paradigm are object-based (Abadi and Cardelli, 1996; =-=Fisher et al., 1994-=-). Indeed, object-based languages simplify and generalize class-based ones: they reduce classes to more primi∗ Supported by UE project IST-CA-510996 Types and French grant CNRS ACI Modulogic. c○ 2006 ... |

150 |
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Citation Context ...ent years, there has been a lot of research about programming with, and reasoning about, datatypes in higher-order abstract syntax, and various approaches have been proposed; see e.g. (Hofmann, 1999; =-=Fiore et al., 1999-=-; Despeyroux and Leleu, 2001; Gabbay and Pitts, 2002; Honsell et al., 2001b; Momigliano and Ambler, 2003). Now, a general approach, in Logics, for increasing the expressive power of a logical system, ... |

100 | Semantical analysis of higher-order abstract syntax
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Citation Context ...k) HOAS. In recent years, there has been a lot of research about programming with, and reasoning about, datatypes in higher-order abstract syntax, and various approaches have been proposed; see e.g. (=-=Hofmann, 1999-=-; Fiore et al., 1999; Despeyroux and Leleu, 2001; Gabbay and Pitts, 2002; Honsell et al., 2001b; Momigliano and Ambler, 2003). Now, a general approach, in Logics, for increasing the expressive power o... |

93 | Reasoning with higher-order abstract syntax in a logical framework
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Citation Context ... HOAS encodings in linear logical frameworks. Another possibility is to use Felty’s elegant two-level approach for encoding sub-structural logics within CC (Co)Ind (Felty, 2002), in the tradition of (=-=McDowell and Miller, 1997-=-). In this approach the metalanguage is used for representing the sequents of the logics, and all the peculiar structural rules one possibly needs. Therefore, besides the known judgements (typing, eva... |

86 | A multiple-conclusion meta-logic
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Citation Context ...possible the management of stacks and typing structures to the meta-level proof context. Unluckily, the structural features of Natural Deduction prevent us to internalise also the store. As shown by (=-=Miller, 1994-=-; Chirimar, 1995), stores can be neatly internalised in linear logical frameworks, such as Forum or LLF (Cervesato and Pfenning, 2002). However, these systems do not provide a native support for coind... |

80 |
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Citation Context ...ged in the assumptions. Using this rule, we can build types for results just by visiting the store and following the pointers it contains. The idea of using coinductive rules for typing goes back to (=-=Milner and Tofte, 1991-=-), but actually we have been inspired by modern type theories, such as CC (Co)Ind , where coinduction is natively provided. 19 paper.tex; 19/06/2006; 17:08; p.19s20 wt([li:Bi] i∈I ) ∀i ∈ I : s(ιi) ≡ λ... |

71 | The Krakatoa tool for certification of Java/JavaCard programs annotated - Marché, Paulin-Mohring, et al. |

46 |
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Citation Context ...in b. Usual conventions about α-conversion and free variables (denoted by FV(a)) apply. Dynamic Semantics. The operational semantics of funς is expressed by a big-step reduction relation (Kahn, 1987; =-=Despeyroux, 1986-=-), relating two stores σ, σ ′ , a stack S, a term a, and a result v: σ·S ⊢AC a ❀ v·σ ′ The intended meaning is that, starting with the store σ (playing the role of a heap) and the stack S, the term a ... |

41 | Proof Theoretic Approach to Specification Languages
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Citation Context ... management of stacks and typing structures to the meta-level proof context. However, the structural features of Natural Deduction prevent us to internalize also the store. As shown by (Miller, 1994; =-=Chirimar, 1995-=-), stores can be neatly internalized in linear logical frameworks, such as Forum or LLF (Cervesato and Pfenning, 2002). How39 paper.tex; 7/05/2006; 11:30; p.39s40 ever, these systems do not provide a ... |

16 | Recursion over objects of functional type - Despeyroux, Leleu |

14 |
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(Show Context)
Citation Context ...Ind , we have to face the problem of representing binders efficiently. To this end, one of the most suited approaches is higher-order abstract syntax (Pfenning and Elliott, 1988; Harper et al., 1993; =-=Miculan, 1997-=-). More precisely, since we work in a type theory with induction, we use weak HOAS (Miculan, 1997; Honsell et al., 2001b): binders are represented as second-order term constructors, taking as argument... |

13 | Multi-level meta-reasoning with higher order abstract syntax - Momigliano, Ambler - 2003 |

12 | Developing (meta)theory of lambda-calculus in the theory of contexts - Miculan - 2001 |

11 | Certified Reasoning on Real Numbers and Objects in Coinductive Type Theory - Ciaffaglione - 1993 |

11 | Implementing a program logic of objects in a higher-order logic theorem prover - Hofmann, Tang - 2000 |

10 | A certified compiler for an imperative language
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Citation Context ...s the certification of tools, such as interpreters, compilers and type-checkers. Some results in this direction, using Coq and Isabelle, are the certification of compilers for an imperative language (=-=Bertot, 1998-=-) and Java (Strecker, 2002). However, none of these works adopts higher-order abstract syntax for dealing with binders: we believe that the use of Natural Deduction Semantics and HOAS can simplify the... |

10 |
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Citation Context ...ion stacks, stores, store types, etc. A straightforward representation of these structures as lists would lead to complicated judgements and proofs. Then, followpaper.tex; 19/06/2006; 17:08; p.2sing (=-=Burstall and Honsell, 1990-=-; Miculan, 1994), we use hypothetic-general judgements à la Martin-Löf for internalising those structures which obey to a stack-discipline. Hence, stacks and typing environments “disappear” from the f... |

10 |
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Citation Context ...ay, α-conversion of abstractions is automatically ensured by the parametricity of the set of variables, still retaining the benefits of inductive definitions and without the presence of exotic terms (=-=Despeyroux et al., 1995-=-). The main drawback of (weak) HOAS is that it is difficult to reason about the encodings. For instance, for proving Subject Reduction we have to prove several properties concerning variable renaming,... |

10 | Two-level meta-reasoning in Coq - Felty |

9 | A case study of co-induction in Isabelle
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Citation Context ...ly closures, like a compiler for a register machine would do. The importance of having a native support for coinduction is confirmed also by Frost’s implementation of Milner-Tofte’s work in Isabelle (=-=Frost, 1995-=-). Frost reports that in the implementation in Isabelle/HOL using an impredicative, higher-order encoding of greatest fixed points, “4/5 of the work was about the management of fixed points”. Moreover... |

6 |
Lectures on [Co]Induction and [Co]Algebras
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Citation Context ...s monotone (and hence the definition is sound) because the application of the coinductive hypothesis is always guarded. For further details about coinductive proof systems and guarded induction, see (=-=Crole, 1998-=-; Giménez, 1995). paper.tex; 19/06/2006; 17:08; p.20sEXAMPLE 14. Let us recall the store with a loop of Example 1, obtained by the evaluation of a term with imperative update: ∅ ⊢ND eval(∅, [l = ς(x)x... |

6 | Mechanising Hankin and Barendregt using the Gordon-Melham axioms - Norrish - 2003 |

5 |
Codifying guarded recursion definitions with recursive schemes
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Citation Context ...patibly with its content; however, these structures are not easy-to-use in a proof assistant. Luckily, nowadays type theories provide coinduction for dealing with circular, non well-founded entities (=-=Giménez, 1995-=-). Inspired by this feature, we elaborate an original coinductive system for typing values, without using store types, and instead recovering the types from the content of store locations. Using our s... |

4 | 2002): A New Approach to Abstract Syntax with Variable Binding. Formal Aspects of Computing 13(3–5 - Gabbay, Pitts |

4 | The expressive power of structural operational semantics with explicit assumptions
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Citation Context ...pes, etc. A straightforward representation of these structures as lists would lead to complicated judgements and proofs. Then, followpaper.tex; 19/06/2006; 17:08; p.2sing (Burstall and Honsell, 1990; =-=Miculan, 1994-=-), we use hypothetic-general judgements à la Martin-Löf for internalising those structures which obey to a stack-discipline. Hence, stacks and typing environments “disappear” from the formal judgement... |

3 | 2000, ‘A formalization of a concurrent object calculus up to alpha-conversion - Gillard - 2000 |

2 | 2003a, ‘Imperative object-calculi in (Co)inductive type theories - Ciaffaglione, Liquori, et al. - 2003 |

2 | 2003c, ‘The Web Appendix of this paper - Ciaffaglione, Liquori, et al. - 2005 |

2 |
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Citation Context ...y redesign of such systems. To sum up, we present the first systematic formalization of (Abadi and Cardelli’s) object-based calculi, in proof assistants based on type theories (the closest works are (=-=Laurent, 1997-=-; Gillard, 2000), which deal with functional semantics only, and (Hofmann and Tang, 2000), that does not formalize the operational semantics directly). We believe that the work described in this paper... |

2 | 2001b, ‘On the formalization of the modal µ-calculus - Miculan |

2 | 2002, ‘Reasoning about Names - Scagnetto - 2002 |

1 | Nipkow (eds.): 1994 - Barendregt, T |

1 | Scagnetto: 2006, ‘Consistency of the Theory of Contexts - Bucalo, Hofmann, et al. - 1990 |

1 | 2003b, ‘Reasoning on an Imperative objectcalculus in Higher-Order Abstract Syntax - Ciaffaglione, Liquori, et al. |

1 | Scagnetto: 2003, ‘Plug and play the theory of contexts in higher-order abstract syntax - Ciaffaglione, I |

1 | 2001, ‘A Higher-Order Embedding of a Logic of Objects - Hofmann, Tang |

1 | Momigliano (eds.): 2003, ‘Eighth ACM SIGPLAN Workshop on Mechanized reasoning about languages with variable binding - Honsell, Miculan, et al. - 1994 |

1 | 2001, ‘Reasoning about Java programs in higer order logic with PVS and Isabelle - Huisman |

1 | Nipkow (eds - Barendregt, T |