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33
Some domain theory and denotational semantics in Coq
, 2009
"... Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance recursive domain equations, and the existence of invariant relations on those solutions. We then define operatio ..."
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Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance recursive domain equations, and the existence of invariant relations on those solutions. We then define operational and denotational semantics for both a simplytyped CBV language with recursion and an untyped CBV language, and establish soundness and adequacy results in each case. 1
A functional specification of effects
, 2009
"... This dissertation is about effects and type theory. Functional programming languages such as Haskell demonstrate that monads can safely encapsulated side effects. If you want to reason about effectful code, however, it is not enough to only encapsulate effects: you must explain their meaning. In thi ..."
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This dissertation is about effects and type theory. Functional programming languages such as Haskell demonstrate that monads can safely encapsulated side effects. If you want to reason about effectful code, however, it is not enough to only encapsulate effects: you must explain their meaning. In this dissertation I provide pure specifications of effects in type theory. After initial investigations using Haskell, I show how such specifications can be made total in a language with dependent types. Hoare Type Theory takes an alternative approach to incorporating effects in a dependently typed language. Instead of giving functional specifications, they follow Haskell’s lead and postulate the existence of primitive functions. This dissertation shows how some of these primitives may be implemented in a programming language with dependent types. I believe that functional specifications may provide the foundations on top of which richer logics may be constructed. The results presented in this dissertation may be used to write and verify
Resumptions, Weak Bisimilarity and BigStep Semantics for While with Interactive I/O: An Exercise in Mixed InductionCoinduction
"... We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive tracebased semantics for While [17], we define several bigstep semantics for While with interactive I/O, based on resumptions an ..."
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We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive tracebased semantics for While [17], we define several bigstep semantics for While with interactive I/O, based on resumptions and terminationsensitive weak bisimilarity. These require nesting inductive definitions in coinductive definitions, which is interesting both mathematically and from the pointofview of implementation in a proof assistant. After first defining a basic semantics of statements in terms of resumptions with explicit internal actions (delays), we introduce a semantics in terms of delayfree resumptions that essentially removes finite sequences of delays on the fly from those resumptions that are responsive. Finally, we also look at a semantics in terms of delayfree resumptions supplemented with a silent divergence option. This semantics hinges on decisions between convergence and divergence and is only equivalent to the basic one classically. We have fully formalized our development in Coq. 1
Beating the Productivity Checker Using Embedded Languages
"... Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures th ..."
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Cited by 6 (3 self)
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Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problemspecific language as a data type, writing the program in the problemspecific language, and writing a guarded interpreter for this language. 1
Tracebased coinductive operational semantics for While; Bigstep and smallstep, relational and functional styles
 In Theorem Proving in Higher Order Logics, 22nd International Conference, TPHOLs 2009, volume 5674 of LNCS
, 2009
"... Abstract. We present four coinductive operational semantics for the While language accounting for both terminating and nonterminating program runs: bigstep and smallstep relational semantics and bigstep and smallstep functional semantics. The semantics employ traces (possibly infinite sequences ..."
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Abstract. We present four coinductive operational semantics for the While language accounting for both terminating and nonterminating program runs: bigstep and smallstep relational semantics and bigstep and smallstep functional semantics. The semantics employ traces (possibly infinite sequences of states) to record the states that program runs go through. The relational semantics relate statementstate pairs to traces, whereas the functional semantics return traces for statementstate pairs. All four semantics are equivalent. We formalize the semantics and their equivalence proofs in the constructive setting of Coq. 1
A constructive denotational semantics for Kahn networks in Coq
, 2007
"... Semantics of programming languages and interactive environments for the development of proofs and programs are two important aspects of Gilles Kahn’s scientific contributions. In his paper “The semantics of a simple language for parallel programming ” [11], he proposed an interpretation of (determin ..."
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Semantics of programming languages and interactive environments for the development of proofs and programs are two important aspects of Gilles Kahn’s scientific contributions. In his paper “The semantics of a simple language for parallel programming ” [11], he proposed an interpretation of (deterministic) parallel programs (now called Kahn networks) as stream transformers based on the theory of complete partial orders (cpos). A restriction of this language to synchronous programs is the basis of the dataflow Lustre language which is used for the development of critical embedded systems [14, 10]. We present a formalization of this seminal paper in the Coq proof assistant [4, 15]. For that purpose, we developed a general library for cpos. Our cpos are defined with an explicit function computing the least upper bound (lub) of an increasing sequence of elements. This is different from what G. Kahn developed for the standard Coq library where only the existence of lubs (for arbitrary directed sets) is required, giving no way to explicitly compute a fixpoint. We define a cpo structure for the type of possibly infinite streams. It is then possible to define formally what is a Kahn network and what is its semantics, achieving the goal of having a concept closed by composition and recursion. The library is illustrated by the example taken from the original paper as well as the Sieve of Eratosthenes, an example of a dynamic network. 1
Recursive definitions of monadic functions
 In Proc. of PAR 2010
, 2010
"... Using standard domaintheoretic fixedpoints, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the stateexception monad of Isabelle/HOL’s imperative programming extension, which results in a convenient ..."
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Using standard domaintheoretic fixedpoints, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the stateexception monad of Isabelle/HOL’s imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define. For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically. 1
Mixing Induction and Coinduction
, 2009
"... Purely inductive definitions give rise to treeshaped values where all branches have finite depth, and purely coinductive definitions give rise to values where all branches are potentially infinite. If this is too restrictive, then an alternative is to use mixed induction and coinduction. This techn ..."
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Purely inductive definitions give rise to treeshaped values where all branches have finite depth, and purely coinductive definitions give rise to values where all branches are potentially infinite. If this is too restrictive, then an alternative is to use mixed induction and coinduction. This technique appears to be fairly unknown. The aim of this paper is to make the technique more widely known, and to present several new applications of it, including a parser combinator library which guarantees termination of parsing, and a method for combining coinductively defined inference systems with rules like transitivity. The developments presented in the paper have been formalised and checked in Agda, a dependently typed programming language and proof assistant.
Formalizing Domains, Ultrametric Spaces and Semantics of Programming Languages
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... We describe a Coq formalization of constructive ωcpos, ultrametric spaces and ultrametricenriched categories, up to and including the inverselimit construction of solutions to mixedvariance recursive equations in both categories enriched over ωcppos and categories enriched over ultrametric spac ..."
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Cited by 3 (1 self)
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We describe a Coq formalization of constructive ωcpos, ultrametric spaces and ultrametricenriched categories, up to and including the inverselimit construction of solutions to mixedvariance recursive equations in both categories enriched over ωcppos and categories enriched over ultrametric spaces. We show how these mathematical structures may be used in formalizing semantics for three representative programming languages. Specifically, we give operational and denotational semantics for both a simplytyped CBV language with recursion and an untyped CBV language, establishing soundness and adequacy results in each case, and then use a Kripke logical relation over a recursivelydefined metric space of worlds to give an interpretation of types over a stepcounting operational semantics for a language with recursive types and general references.