443 research outputs found
Foundational Extensible Corecursion
This paper presents a formalized framework for defining corecursive functions
safely in a total setting, based on corecursion up-to and relational
parametricity. The end product is a general corecursor that allows corecursive
(and even recursive) calls under well-behaved operations, including
constructors. Corecursive functions that are well behaved can be registered as
such, thereby increasing the corecursor's expressiveness. The metatheory is
formalized in the Isabelle proof assistant and forms the core of a prototype
tool. The corecursor is derived from first principles, without requiring new
axioms or extensions of the logic
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
On the Rationality of Escalation
Escalation is a typical feature of infinite games. Therefore tools conceived
for studying infinite mathematical structures, namely those deriving from
coinduction are essential. Here we use coinduction, or backward coinduction (to
show its connection with the same concept for finite games) to study carefully
and formally the infinite games especially those called dollar auctions, which
are considered as the paradigm of escalation. Unlike what is commonly admitted,
we show that, provided one assumes that the other agent will always stop,
bidding is rational, because it results in a subgame perfect equilibrium. We
show that this is not the only rational strategy profile (the only subgame
perfect equilibrium). Indeed if an agent stops and will stop at every step, we
claim that he is rational as well, if one admits that his opponent will never
stop, because this corresponds to a subgame perfect equilibrium. Amazingly, in
the infinite dollar auction game, the behavior in which both agents stop at
each step is not a Nash equilibrium, hence is not a subgame perfect
equilibrium, hence is not rational.Comment: 19 p. This paper is a duplicate of arXiv:1004.525
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
First-Order Guarded Coinduction in Coq
We introduce two coinduction principles and two proof translations which, under certain conditions, map coinductive proofs that use our principles to guarded Coq proofs. The first principle provides an "operational" description of a proof by coinduction, which is easy to reason with informally. The second principle extends the first one to allow for direct proofs by coinduction of statements with existential quantifiers and multiple coinductive predicates in the conclusion. The principles automatically enforce the correct use of the coinductive hypothesis. We implemented the principles and the proof translations in a Coq plugin
(Mechanical) Reasoning on Infinite Extensive Games
In order to better understand reasoning involved in analyzing infinite games
in extensive form, we performed experiments in the proof assistant Coq that are
reported here.Comment: 11
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