154 research outputs found
A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators
First, we reconstruct Wim Veldman's result that Open Induction on Cantor
space can be derived from Double-negation Shift and Markov's Principle. In
doing this, we notice that one has to use a countable choice axiom in the proof
and that Markov's Principle is replaceable by slightly strengthening the
Double-negation Shift schema. We show that this strengthened version of
Double-negation Shift can nonetheless be derived in a constructive intermediate
logic based on delimited control operators, extended with axioms for
higher-type Heyting Arithmetic. We formalize the argument and thus obtain a
proof term that directly derives Open Induction on Cantor space by the shift
and reset delimited control operators of Danvy and Filinski
Resumptions, Weak Bisimilarity and Big-Step Semantics for While with Interactive I/O: An Exercise in Mixed Induction-Coinduction
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 trace-based semantics for While, we define several big-step
semantics for While with interactive I/O, based on resumptions and
termination-sensitive weak bisimilarity. These require nesting inductive
definitions in coinductive definitions, which is interesting both
mathematically and from the point-of-view 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
delay-free 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 delay-free 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.Comment: In Proceedings SOS 2010, arXiv:1008.190
Resumption-based big-step and small-step interpreters for While with interactive I/O
In this tutorial, we program big-step and small-step total interpreters for the While language extended with input and output primitives. While is a simple imperative language consisting of skip, assignment, sequence, conditional and loop. We first develop trace-based interpreters for While. Traces are potentially infinite nonempty sequences of states. The interpreters assign traces to While programs: for us, traces are denotations of While programs. The trace is finite if the program is terminating and infinite if the program is non-terminating. However, we cannot decide (i.e., write a program to determine), for any given program, whether its trace is finite or infinite, which amounts to deciding the halting problem. We then extend While with interactive input/output primitives. Accordingly, we extend the interpreters by generalizing traces to resumptions. The tutorial is based on our previous work with T. Uustalu on reasoning about interactive programs in the setting of constructive type theory
Classical call-by-need sequent calculi : The unity of semantic artifacts
International audienceWe systematically derive a classical call-by-need sequent calculus, which does not require an unbounded search for the standard redex, by using the unity of semantic artifacts proposed by Danvy et al. The calculus serves as an intermediate step toward the generation of an environment-based abstract machine. The resulting abstract machine is context-free, so that each step is parametric in all but one component. The context-free machine elegantly leads to an environment-based CPS transformation. This transformation is observationally different from a natural classical extension of the transformation of Okasaki et al., due to duplication of un-evaluated bindings
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