Dykstra's projection algorithm: Difference between revisions

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In computer science, specifically in the field of [[formal verification]], '''well-structured transition systems (WSTSs)''' are a general class of infinite state systems for which many verification problems are [[decidable language|decidable]], owing to the existence of a kind of [[well-quasi-ordering|order]] between the states of the system which is compatible with the transitions of the system. WSTS decidability results can be applied to [[Petri nets]], lossy channel systems, and more.
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== Formal definition ==
Recall that a [[well-quasi-ordering]] <math>\leq</math> on a set <math>X</math> is a [[quasi-ordering]] (i.e., a [[reflexive relation|reflexive]], [[transitive relation|transitive]] [[binary relation]]) such that any infinite sequence of elements <math>x_0, x_1, x_2, \ldots</math>, from <math>X</math> contains an increasing pair <math>x_i \leq x_j</math> with <math>i < j</math>. The set <math>X</math> is said to be '''well-quasi-ordered''', or shortly '''wqo'''.
 
For our purposes, a ''[[transition system]]'' is a structure <math>\mathcal S = \langle S, \rightarrow, \cdots \rangle</math>, where <math>S</math> is any set (its elements are called ''states''), and <math>\rightarrow \subseteq S \times S</math> (its elements are called ''transitions''). In general a transition system may have additional structure like initial states, labels on transitions, accepting states, etc, but they do not concern us here.
 
A '''well-structured transition system''' consists of a transition system <math>\langle S, \to, \leq \rangle</math>, such that
* <math>\leq \subseteq S \times S</math> is a well-quasi-ordering on the set of states.
* <math>\leq</math> is upward compatible with <math>\to</math>: that is, for all transitions <math>s_1 \to s_2</math> (by this we mean <math>(s_1, s_2) \in \to</math>) and for all <math>t_1</math> such that <math>s_1 \leq t_1</math>, there exists <math>t_2</math> such that <math>t_1 \xrightarrow{*} t_2</math> (that is, <math>t_2</math> can be reached from <math>t_1</math> by a sequence of zero or more transitions) and <math>s_2 \leq t_2</math>.
 
== References ==
* A. Finkel and Ph. Schnoebelen, [http://www.lsv.ens-cachan.fr/Publis/PAPERS/PDF/FinSch-TCS99.pdf Well-Structured Transition Systems Everywhere!], Theoretical Computer Science 256(1–2), pages 63–92, 2001.
 
{{comp-sci-theory-stub}}
 
[[Category:Wellfoundedness]]
[[Category:Automata (computation)]]
[[Category:Automata theory]]

Revision as of 22:13, 24 February 2014

A lot of folks think home improvement is not something that they can do. Stories of home improvement failures may make people avoid trying. Home improvement does not have to be a giant pain in the neck. This article contains some helpful tips to get you started.

Anything simple, like a lampshade, can look and be boring. Brighten up a lamp shade by using stencils and acrylic paint. This will add a personal touch, making the room reflect your personality. This can take a boring room and make it more exciting.

If you'd like to re-purpose a luggage rack, try transforming it into a side table! Simply secure a sheet of glass to the top of the rack using permanent adhesive. Add some books, a pretty lamp, and your side table is complete.

Use two-liter bottles that are empty to store things in your kitchen. They are quite useful in that you can see through them and will hold up well when protecting sugar and flour or other dry goods. You can keep them in a freezer, refrigerator, or on a If you beloved this article and you simply would like to be given more info regarding home improvements diy generously visit our page. shelf. When you need something all you need to do is take the lid off and pour.

Chair caning tends to sag over time, which is very natural. It is relatively simple to restore form to your caning. Take a sponge and wet the bottom part of your chair's sagging caning. Be sure that the water you're using is warm. Allow the caning to dry overnight. Continue doing this until your caning is back to normal.

Don't forget to improve your outdoor living space. Your backyard can make a wonderful extension to your indoor living area. This is a great place for family to gather or to relax on your own.

If possible, make home improvements immediately after noticing an issue. You may be able to live with the damage that happens. You don't want to give into this impulse. A small issue can rapidly grow into a multifaceted and expensive problem. This may occur more rapidly than you expect.

When thinking about replacing the cabinets in your home, look to wood cabinets. Wood cabinets will last for years, withstanding much wear and tear. You should invest in cherry, maple or oak woods for best results. One additional benefit of these woods is that they take stains well. This makes it relatively easy to alter the color scheme of your cabinets.

Plan your next home improvement project. Avoiding the planning until the time you begin your project can cause indecisiveness and many lost dollars due to poor, spur-of-the-minute decisions. If you plan in advance, you will have a much easier time completing the project.

If your ceiling is starting to leak, odds are you need to replace your roof. If you don't hurry, you may end up with a disaster on your hands. Search your area for a high-quality roof specialist, and have them inspect your roof. Do some background research on the roof specialist you want to hire to make sure they are properly qualified.

Always remember, approach each new project with a fresh mind, work on a step-by-step basis, and don't be afraid to break out of your comfort zone by trying something new. While the thoughts of home improvement projects may seem terrifying to you, it is possible to successfully complete many tasks with just a little knowledge. If you take the advice in this piece to heart, you will be poised for true home improvement success.

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