780,405 research outputs found
Axiomatizing Flat Iteration
Flat iteration is a variation on the original binary version of the Kleene
star operation P*Q, obtained by restricting the first argument to be a sum of
atomic actions. It generalizes prefix iteration, in which the first argument is
a single action. Complete finite equational axiomatizations are given for five
notions of bisimulation congruence over basic CCS with flat iteration, viz.
strong congruence, branching congruence, eta-congruence, delay congruence and
weak congruence. Such axiomatizations were already known for prefix iteration
and are known not to exist for general iteration. The use of flat iteration has
two main advantages over prefix iteration: 1.The current axiomatizations
generalize to full CCS, whereas the prefix iteration approach does not allow an
elimination theorem for an asynchronous parallel composition operator. 2.The
greater expressiveness of flat iteration allows for much shorter completeness
proofs.
In the setting of prefix iteration, the most convenient way to obtain the
completeness theorems for eta-, delay, and weak congruence was by reduction to
the completeness theorem for branching congruence. In the case of weak
congruence this turned out to be much simpler than the only direct proof found.
In the setting of flat iteration on the other hand, the completeness theorems
for delay and weak (but not eta-) congruence can equally well be obtained by
reduction to the one for strong congruence, without using branching congruence
as an intermediate step. Moreover, the completeness results for prefix
iteration can be retrieved from those for flat iteration, thus obtaining a
second indirect approach for proving completeness for delay and weak congruence
in the setting of prefix iteration.Comment: 15 pages. LaTeX 2.09. Filename: flat.tex.gz. On A4 paper print with:
dvips -t a4 -O -2.15cm,-2.22cm -x 1225 flat. For US letter with: dvips -t
letter -O -0.73in,-1.27in -x 1225 flat. More info at
http://theory.stanford.edu/~rvg/abstracts.html#3
Tangent Graeffe Iteration
Graeffe iteration was the choice algorithm for solving univariate polynomials
in the XIX-th and early XX-th century. In this paper, a new variation of
Graeffe iteration is given, suitable to IEEE floating-point arithmetics of
modern digital computers. We prove that under a certain generic assumption the
proposed algorithm converges. We also estimate the error after N iterations and
the running cost. The main ideas from which this algorithm is built are:
classical Graeffe iteration and Newton Diagrams, changes of scale
(renormalization), and replacement of a difference technique by a
differentiation one. The algorithm was implemented successfully and a number of
numerical experiments are displayed
Intuition, iteration, induction
In Mathematical Thought and Its Objects, Charles Parsons argues that our
knowledge of the iterability of functions on the natural numbers and of the
validity of complete induction is not intuitive knowledge; Brouwer disagrees on
both counts. I will compare Parsons' argument with Brouwer's and defend the
latter. I will not argue that Parsons is wrong once his own conception of
intuition is granted, as I do not think that that is the case. But I will try
to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's
claims can be justified in more detail than he has done; (2) There are certain
elements in Parsons' discussion that, when developed further, would lead to
Brouwer's notion thus analysed, or at least something relevantly similar to it.
(This version contains a postscript of May, 2015.)Comment: Elaboration of a presentation on December 5, 2013 at `Intuition and
Reason: International Conference on the Work of Charles Parsons', Van Leer
Jerusalem Institute, Jerusale
- …