7 research outputs found
Certified exact real arithmetic using co-induction in arbitrary integer base
International audienceIn this paper we describe some certified algorithms for exact real arithmetic based on co-recursion. Our work is based on previous experiences using redundant digits of base 2 but generalizes them using arbitrary integer bases. The goal is to take benefit of fast native integer computation. We extend a technique to compute converging series. We use this technique to compute the product and the inverse. We describe how we implement and certify our algorithms in the proof system Coq and evaluate the efficiency of the library inside the prover
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
Certified Exact Transcendental Real Number Computation in Coq
Reasoning about real number expressions in a proof assistant is challenging.
Several problems in theorem proving can be solved by using exact real number
computation. I have implemented a library for reasoning and computing with
complete metric spaces in the Coq proof assistant and used this library to
build a constructive real number implementation including elementary real
number functions and proofs of correctness. Using this library, I have created
a tactic that automatically proves strict inequalities over closed elementary
real number expressions by computation.Comment: This paper is to be part of the proceedings of the 21st International
Conference on Theorem Proving in Higher Order Logics (TPHOLs 2008
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
Rationality and Escalation in Infinite Extensive Games
The aim of this of this paper is to study infinite games and to prove
formally some properties in this framework. As a consequence we show that the
behavior (the madness) of people which leads to speculative crashes or
escalation can be fully rational. Indeed it proceeds from the statement that
resources are infinite. The reasoning is based on the concept of coinduction
conceived by computer scientists to model infinite computations and used by
economic agents unknowingly. When used consciously, this concept is not as
simple as induction and we could paraphrase Newton: "Modeling the madness of
people is more difficult than modeling the motion of planets".Comment: arXiv admin note: substantial text overlap with arXiv:1004.5257,
arXiv:0904.3528, and arXiv:0912.174