Infinitary Logic First published Sun Jan 23, 2000; substantive revision Wed Feb 22, 2012 Traditionally, expressions in formal systems have been regarded as signifying finite inscriptions which are—at least in principle—capable of actually being written out in primitive notation. However, the fact that (first-order) formulas may be identified with natural numbers (via “Gödel numbering”) and hence with finite sets makes it no longer necessary to regard formulas as inscriptions, and suggests the possibility of fashioning “languages” some of whose formulas would be naturally identified as infinite sets. 1. Given a pair κ, λ of infinite cardinals such that λ ≤ κ, we define a class of infinitary languages in each of which we may form conjunctions and disjunctions of sets of formulas of cardinality < κ, and quantifications over sequences of variables of length < λ. Let L — the (finitary) base language — be an arbitrary but fixed first-order language with any number of extralogical symbols. ∧i∈I φ φ0 ∧ φ1 ∧ … 2.
How to Win at Rock-Paper-Scissors Rock-paper-scissors* isn’t obviously interesting to look at mathematically. The Nash-equilibrium strategy is very simple: choose equally and randomly from the three choices, and (in the long run) your opponent will not beat you (nor will you beat your opponent). Nevertheless, it’s still possible for a computer strategy to beat a human player over a long run of games. My nine-year-old daughter showed me one solution with a Scratch program that she wrote that won every time by looking at your choice before making its decision! But I will walk you through a simple solution that wins without cheating. While the equal random choice is unbeatable, we can rely on the fact that humans are not very good at being random. I thought about writing the algorithm as a topic within our Computer-Based Math™ statistics course. First we need to be able to play the game. Random player The code is mostly user interface, display, and game rules. where 1 represents rock, 2 paper, and 3 scissors.
Second-order and Higher-order Logic First published Thu Dec 20, 2007; substantive revision Wed Mar 4, 2009 Second-order logic is an extension of first-order logic where, in addition to quantifiers such as “for every object (in the universe of discourse),” one has quantifiers such as “for every property of objects (in the universe of discourse).” This augmentation of the language increases its expressive strength, without adding new non-logical symbols, such as new predicate symbols. For classical extensional logic (as in this entry), properties can be identified with sets, so that second-order logic provides us with the quantifier “for every set of objects.” There are two approaches to the semantics of second-order logic. 1. In symbolic logic, the formula (Px → Px) will be true, no matter what object in the universe of discourse is assigned to the variable x. In first-order languages, there are some things we can say, and some that we cannot. ∃x Px → ∃x(Px & ∀y(Py → (y = x v x < y))) ∀X[X0 & ∀y(Xy → XSy) → ∀y Xy] 2.
Building a Balanced Tree From a List in Linear Time | Harder, Better, Faster, Stronger The usual way of forming a search tree from a list is to scan the list and insert each of its element, one by one, into the tree, leading to a(n expected) run-time of However, if the list is sorted (in ascending order, say) and the tree is not one of the self-balancing varieties, insertion is , because the “tree” created by the successive insertions of sorted key is in fact a degenerate tree, a list. So, what if the list is already sorted and don’t really want to have a self-balancing tree? Let us make the simplifying assumption that we can have two types of nodes in the tree: leaves, that contains the actual data, and internal nodes (or just nodes for the remainder of this post) that holds only a key. The first strategy that comes to mind, using this assumption, is to use a method reminiscent of how Huffman Codes are constructed. (Notice the metaphor: leaves are green, internal nodes brown.) Then we do one pass of merges: Then another: …and finally: What do you notice? (as I show here). , so
Model Theory 1. Basic notions of model theory Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. For example I might say He is killing all of them, and offer the interpretation that ‘he’ is Alfonso Arblaster of 35 The Crescent, Beetleford, and that ‘them’ are the pigeons in his loft. The structure I in the previous paragraph involves one fixed object and one fixed class. Note that the objects and classes in a structure carry labels that steer them to the right expressions in the sentence. If the same class is used to interpret all quantifiers, the class is called the domain or universe of the structure. One of those thingummy diseases is killing all the birds. 2. Take for example the following set of first-order sentences:
Workflow Patterns | Patterns | Control Downloads of the original and revised control-flow patterns papers: N. Russell, A.H.M. ter Hofstede, W.M.P. van der Aalst, and N. Mulyar. Workflow Control-Flow Patterns: A Revised View. W.M.P van der Aalst, A.H.M. ter Hofstede, B. Introduction The Workflow Patterns Initiative was established with the aim of delineating the fundamental requirements that arise during business process modelling on a recurring basis and describe them in an imperative way. Revisiting the Original Patterns Here we present a revised description of the original twenty control-flow patterns previously presented in [vdAtHKB03]. New Control Flow Patterns Review of the patterns associated with the control-flow perspective over the past few years has led to the recognition that there are a number of distinct modelling constructs that can be identified during process modelling that are not adequately captured by the original set of twenty patterns. Basic Control Flow Patterns 1. 6. Multiple Instance Patterns 12. 16. 19.
Fuzzy logic Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. Compared to traditional binary sets (where variables may take on true or false values) fuzzy logic variables may have a truth value that ranges in degree between 0 and 1. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific functions. The term "fuzzy logic" was introduced with the 1965 proposal of fuzzy set theory by Lotfi A. Overview Classical logic only permits propositions having a value of truth or falsity. Both degrees of truth and probabilities range between 0 and 1 and hence may seem similar at first. Applying truth values Fuzzy logic temperature In this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale.
Opinie > Meningen : Algoritmes Nemen Alles Over vrijdag 15 november 2013 | 29 reacties Althans als je een 'believer of algorithms' bent. Wat weet u van algoritmes? Mijn eerste ervaring met algoritmes is van drie jaar terug. U denkt, nou wow, knap. En ik u vertel dat je hiermee in milliseconden inzicht hebt in bijna de volledige werkelijkheid wat betreft werking, en vooral niet-werking, van interne-controlemaatregelen? Nog leuker wordt het als je met een aantal handelingen kan voorspellen hoe deze processen gaan opereren, door subprocesstappen weg te laten of toe te voegen. In mijn en ik hoop ook uw wereld van processen, transacties, risico's en interne controle gaat process mining iets heel 'groots' worden. Zover mijn wereld. Een robotstofzuiger aansturen en 'weten' wanneer uw vloer wel of niet schoon genoeg is. Bij dit laatste gaat het regelmatig fout. Zelf vind ik de inzet van algoritmes in de wereld van Venture Capitalists een bijzonder interessant aandachtsgebied. Opgelost! Pieter de Kok Reacties (29) | Reageer @Arnout van Kempen
Interception of Two Moving Objects in 2D Space Introduction This information will allow you to determine the direction that an object ("Chaser") must move so that it is able to intercept a target object ("Runner"). It assumes that you know the positions of both objects in 2D space. It assumes that you know the speed at which your Chaser can move. It assumes that you know the current speed and direction of the Runner. Both the Chaser and the Runner are abstracted to points on a plane. Background In most video games, it suffices to approximate real-world physical phenomena. In this article, I'll describe a process for finding an interception point assuming that inertia and acceleration are being approximated or ignored, and that the game can re-compute this interception point regularly (every game frame if needed). There is an article here that gives a very thorough methodology for doing a similar thing. This approach makes heavy use of vectors, described in great detail in many other articles. Some 2D Vector Math The Dot Product 2D vs 3D