A DNC function that computes no effectively bi-immune set
|
|
- Caren Reeves
- 5 years ago
- Views:
Transcription
1 A DNC function that computes no effectively bi-immune set Achilles A. Beros Laboratoire d Informatique de Nantes Atlantique, Université de Nantes July 5, 204
2 Standard Definitions Definition f is diagonally non-computable if f (e) φ e (e). A is immune if A contains no infinite c.e. set. A is effectively immune if there is a computable (total) function, g such that W e A W e < g(e). A is bi-immune if A and A are both immune. A is effectively bi-immune (EBI) if A and A are both effectively immune. Note: If g and h witness effective immunity for A and A, respectively, then the function defined by k(x) = max{g(x), h(x)} witness effective immunity for both of them.
3 The Question and Answer In 989, Carl Jockusch proved that every DNC function computes an effectively immune set. In 202, Carl Jockusch and Andrew Lewis-Pye proved that every DNC function computes a bi-immune set. Can these results be combined? Jockusch and Lewis-Pye pose this question in their paper: Does every DNC function compute an effectively bi-immune set?
4 The Question and Answer In 989, Carl Jockusch proved that every DNC function computes an effectively immune set. In 202, Carl Jockusch and Andrew Lewis-Pye proved that every DNC function computes a bi-immune set. Can these results be combined? Jockusch and Lewis-Pye pose this question in their paper: Does every DNC function compute an effectively bi-immune set? The answer: No.
5 Results Theorem (Beros) There is a DNC function that computes no effectively bi-immune set. Corollary There is a 0 2 tree every branch of which is a DNC function that computes no effectively bi-immune set. Theorem (Kučera) Every 0 2 DNC degree computes a promptly simple set. Corollary There is a DNC function that computes no effectively bi-immune set and bounds a promptly simple set.
6 Overview Producing a DNC function is very easy: build a tree such that every level is a branching level.
7 Overview Producing a DNC function is very easy: build a tree such that every level is a branching level. For h, a branch of such a tree, to compute no EBI we must guarantee that, for every e N and for every computable function h, there is a c.e. set W a such that W a > h(a) and either W a W h e or W a W h e. This is the requirement for e and h.
8 Overview Producing a DNC function is very easy: build a tree such that every level is a branching level. For h, a branch of such a tree, to compute no EBI we must guarantee that, for every e N and for every computable function h, there is a c.e. set W a such that W a > h(a) and either W a W h e or W a W h e. This is the requirement for e and h. If the current outcome of a requirement is that there is a W a W f e, then the outcome can only be injured by a higher priority requirement as the computation is already complete.
9 Overview Producing a DNC function is very easy: build a tree such that every level is a branching level. For h, a branch of such a tree, to compute no EBI we must guarantee that, for every e N and for every computable function h, there is a c.e. set W a such that W a > h(a) and either W a W h e or W a W h e. This is the requirement for e and h. If the current outcome of a requirement is that there is a W a W f e, then the outcome can only be injured by a higher priority requirement as the computation is already complete. If the current outcome is that there is a W a W f e, then the outcome requires preservation from damage by lower priority requirements.
10 Overview Each requirement is assigned a set of roots. The roots can change a finite number of times, but they will always form a maximal antichain. The requirement can have different outcomes for each root.
11 Overview Each requirement is assigned a set of roots. The roots can change a finite number of times, but they will always form a maximal antichain. The requirement can have different outcomes for each root. The roots along a branch of the tree will adhere to the priority order and the roots of different requirements are disjoint. Root nodes for 3 requirements: Root of the 0 th requirement Root of the st requirement Root of the 2 nd requirement
12 Fix a root, σ, for the e, h requirement. For a finite set S we say σ accepts S for e if S W σ e. Conversely, σ preserves S for e if S W σ e.
13 Fix a root, σ, for the e, h requirement. For a finite set S we say σ accepts S for e if S W σ e. Conversely, σ preserves S for e if S W σ e. If W is a finite family, then a selection of W is a set that consists of one member from each member of W.
14 Fix a root, σ, for the e, h requirement. For a finite set S we say σ accepts S for e if S W σ e. Conversely, σ preserves S for e if S W σ e. If W is a finite family, then a selection of W is a set that consists of one member from each member of W. σ is -bad relative to W, m, e if σ accepts a selection of W for e. σ is (n + )-bad relative to W, m, e if σ has m immediate extensions that are n-bad relative to W, m, e.
15 n-bad for m = 3: A tree.
16 n-bad for m = 3: Suppose these nodes are -bad.
17 n-bad for m = 3: Three nodes are 2-bad
18 n-bad for m = 3: One node is 3-bad
19 n-bad for m = 3: The root is 4-bad
20 A diagonalization point, d, is a code such that we can control the enumeration of W d (found using the recursion theorem).
21 A diagonalization point, d, is a code such that we can control the enumeration of W d (found using the recursion theorem). Let a N be a diagonalization point. Wait for h(a).
22 A diagonalization point, d, is a code such that we can control the enumeration of W d (found using the recursion theorem). Let a N be a diagonalization point. Wait for h(a). Pick further diagonalization points b 0,..., b h(a). Wait for h(b i ).
23 A diagonalization point, d, is a code such that we can control the enumeration of W d (found using the recursion theorem). Let a N be a diagonalization point. Wait for h(a). Pick further diagonalization points b 0,..., b h(a). Wait for h(b i ). Choose disjoint subsets of W σ e of size max{h(b i ) : i h(a)}. Each W bi is set equal to one of the sets for i h(a).
24 A diagonalization point, d, is a code such that we can control the enumeration of W d (found using the recursion theorem). Let a N be a diagonalization point. Wait for h(a). Pick further diagonalization points b 0,..., b h(a). Wait for h(b i ). Choose disjoint subsets of W σ e of size max{h(b i ) : i h(a)}. Each W bi is set equal to one of the sets for i h(a). Search for m sibling nodes, τ, immediately below σ that are k-bad relative to {W b0,..., W bh(a) }, m, e for some k and some well-chosen large value of m.
25 If we cannot find enough siblings, then we can avoid one of the W bi, preserving it in W σ e.
26 If we cannot find enough siblings, then we can avoid one of the W bi, preserving it in W σ e. If we can, then a lemma guarantees we can choose a highly branching tree below σ with enough redundancy to be safe from updated computations by roots that are prefixes of σ (unless outcomes change) and every branch accepts a particular selection of {W b0,..., W bh(a) }.
27 If we cannot find enough siblings, then we can avoid one of the W bi, preserving it in W σ e. If we can, then a lemma guarantees we can choose a highly branching tree below σ with enough redundancy to be safe from updated computations by roots that are prefixes of σ (unless outcomes change) and every branch accepts a particular selection of {W b0,..., W bh(a) }. Set W a to be the selection of points accepted by every branch. W a = h(a) + and W a is contained in the set computed by e with an oracle extending any branch in the trees.
28 Large Lemma For a finite set of finite sets, W, a string σ and e, k, y N, either:. there is a y-branching tree of depth k below σ, or 2. there are m = (y ) ( Π A W A ) + that are i-bad relative to W, m, e for i k.
29 Large Lemma For a finite set of finite sets, W, a string σ and e, k, y N, either:. there is a y-branching tree of depth k below σ, or 2. there are m = (y ) ( Π A W A ) + that are i-bad relative to W, m, e for i k. This allows us to predict the maximum amount of injury that higher priority requirements that are preserving a set can do to lower priority requirements that have accepted a set.
30 Large Lemma For a finite set of finite sets, W, a string σ and e, k, y N, either:. there is a y-branching tree of depth k below σ, or 2. there are m = (y ) ( Π A W A ) + that are i-bad relative to W, m, e for i k. This allows us to predict the maximum amount of injury that higher priority requirements that are preserving a set can do to lower priority requirements that have accepted a set. Using this, the strategy does not accept unless it has enough branching to be unharmed by the maximum amount of injury all higher priority requirements can inflict (before changing outcome).
31 Thank You
The Turing Definability of the Relation of Computably Enumerable In. S. Barry Cooper
The Turing Definability of the Relation of Computably Enumerable In S. Barry Cooper Computability Theory Seminar University of Leeds Winter, 1999 2000 1. The big picture Turing definability/invariance
More informationAn effective perfect-set theorem
An effective perfect-set theorem David Belanger, joint with Keng Meng (Selwyn) Ng CTFM 2016 at Waseda University, Tokyo Institute for Mathematical Sciences National University of Singapore The perfect
More informationLecture l(x) 1. (1) x X
Lecture 14 Agenda for the lecture Kraft s inequality Shannon codes The relation H(X) L u (X) = L p (X) H(X) + 1 14.1 Kraft s inequality While the definition of prefix-free codes is intuitively clear, we
More informationGödel algebras free over finite distributive lattices
TANCL, Oxford, August 4-9, 2007 1 Gödel algebras free over finite distributive lattices Stefano Aguzzoli Brunella Gerla Vincenzo Marra D.S.I. D.I.COM. D.I.C.O. University of Milano University of Insubria
More informationJune 11, Dynamic Programming( Weighted Interval Scheduling)
Dynamic Programming( Weighted Interval Scheduling) June 11, 2014 Problem Statement: 1 We have a resource and many people request to use the resource for periods of time (an interval of time) 2 Each interval
More informationThe Traveling Salesman Problem. Time Complexity under Nondeterminism. A Nondeterministic Algorithm for tsp (d)
The Traveling Salesman Problem We are given n cities 1, 2,..., n and integer distances d ij between any two cities i and j. Assume d ij = d ji for convenience. The traveling salesman problem (tsp) asks
More information3 The Model Existence Theorem
3 The Model Existence Theorem Although we don t have compactness or a useful Completeness Theorem, Henkinstyle arguments can still be used in some contexts to build models. In this section we describe
More informationFORCING AND THE HALPERN-LÄUCHLI THEOREM. 1. Introduction This document is a continuation of [1]. It is intended to be part of a larger paper.
FORCING AND THE HALPERN-LÄUCHLI THEOREM NATASHA DOBRINEN AND DAN HATHAWAY Abstract. We will show the various effects that forcing has on the Halpern-Läuchli Theorem. We will show that the the theorem at
More informationarxiv: v2 [math.lo] 13 Feb 2014
A LOWER BOUND FOR GENERALIZED DOMINATING NUMBERS arxiv:1401.7948v2 [math.lo] 13 Feb 2014 DAN HATHAWAY Abstract. We show that when κ and λ are infinite cardinals satisfying λ κ = λ, the cofinality of the
More informationLecture 5: Tuesday, January 27, Peterson s Algorithm satisfies the No Starvation property (Theorem 1)
Com S 611 Spring Semester 2015 Advanced Topics on Distributed and Concurrent Algorithms Lecture 5: Tuesday, January 27, 2015 Instructor: Soma Chaudhuri Scribe: Nik Kinkel 1 Introduction This lecture covers
More informationBest response cycles in perfect information games
P. Jean-Jacques Herings, Arkadi Predtetchinski Best response cycles in perfect information games RM/15/017 Best response cycles in perfect information games P. Jean Jacques Herings and Arkadi Predtetchinski
More informationarxiv: v1 [math.co] 31 Mar 2009
A BIJECTION BETWEEN WELL-LABELLED POSITIVE PATHS AND MATCHINGS OLIVIER BERNARDI, BERTRAND DUPLANTIER, AND PHILIPPE NADEAU arxiv:0903.539v [math.co] 3 Mar 009 Abstract. A well-labelled positive path of
More informationIntroduction to Greedy Algorithms: Huffman Codes
Introduction to Greedy Algorithms: Huffman Codes Yufei Tao ITEE University of Queensland In computer science, one interesting method to design algorithms is to go greedy, namely, keep doing the thing that
More informationAlgorithmic Game Theory and Applications. Lecture 11: Games of Perfect Information
Algorithmic Game Theory and Applications Lecture 11: Games of Perfect Information Kousha Etessami finite games of perfect information Recall, a perfect information (PI) game has only 1 node per information
More informationTABLEAU-BASED DECISION PROCEDURES FOR HYBRID LOGIC
TABLEAU-BASED DECISION PROCEDURES FOR HYBRID LOGIC THOMAS BOLANDER AND TORBEN BRAÜNER Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is well-known
More informationbeing saturated Lemma 0.2 Suppose V = L[E]. Every Woodin cardinal is Woodin with.
On NS ω1 being saturated Ralf Schindler 1 Institut für Mathematische Logik und Grundlagenforschung, Universität Münster Einsteinstr. 62, 48149 Münster, Germany Definition 0.1 Let δ be a cardinal. We say
More informationSy D. Friedman. August 28, 2001
0 # and Inner Models Sy D. Friedman August 28, 2001 In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0 #. We show, assuming that 0 # exists, that such
More informationGeneralising the weak compactness of ω
Generalising the weak compactness of ω Andrew Brooke-Taylor Generalised Baire Spaces Masterclass Royal Netherlands Academy of Arts and Sciences 22 August 2018 Andrew Brooke-Taylor Generalising the weak
More informationOutline. Objective. Previous Results Our Results Discussion Current Research. 1 Motivation. 2 Model. 3 Results
On Threshold Esteban 1 Adam 2 Ravi 3 David 4 Sergei 1 1 Stanford University 2 Harvard University 3 Yahoo! Research 4 Carleton College The 8th ACM Conference on Electronic Commerce EC 07 Outline 1 2 3 Some
More information0.1 Equivalence between Natural Deduction and Axiomatic Systems
0.1 Equivalence between Natural Deduction and Axiomatic Systems Theorem 0.1.1. Γ ND P iff Γ AS P ( ) it is enough to prove that all axioms are theorems in ND, as MP corresponds to ( e). ( ) by induction
More informationStructural Induction
Structural Induction Jason Filippou CMSC250 @ UMCP 07-05-2016 Jason Filippou (CMSC250 @ UMCP) Structural Induction 07-05-2016 1 / 26 Outline 1 Recursively defined structures 2 Proofs Binary Trees Jason
More informationContinuous images of closed sets in generalized Baire spaces ESI Workshop: Forcing and Large Cardinals
Continuous images of closed sets in generalized Baire spaces ESI Workshop: Forcing and Large Cardinals Philipp Moritz Lücke (joint work with Philipp Schlicht) Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität
More informationCourse Information and Introduction
August 20, 2015 Course Information 1 Instructor : Email : arash.rafiey@indstate.edu Office : Root Hall A-127 Office Hours : Tuesdays 12:00 pm to 1:00 pm in my office (A-127) 2 Course Webpage : http://cs.indstate.edu/
More informationOn the Optimality of a Family of Binary Trees Techical Report TR
On the Optimality of a Family of Binary Trees Techical Report TR-011101-1 Dana Vrajitoru and William Knight Indiana University South Bend Department of Computer and Information Sciences Abstract In this
More informationUndecidability and 1-types in Intervals of the Computably Enumerable Degrees
Undecidability and 1-types in Intervals of the Computably Enumerable Degrees Klaus Ambos-Spies Mathematisches Institut, Universität Heidelberg, D-69120 Heidelberg, Germany Denis R. Hirschfeldt Department
More informationmaps 1 to 5. Similarly, we compute (1 2)(4 7 8)(2 1)( ) = (1 5 8)(2 4 7).
Math 430 Dr. Songhao Li Spring 2016 HOMEWORK 3 SOLUTIONS Due 2/15/16 Part II Section 9 Exercises 4. Find the orbits of σ : Z Z defined by σ(n) = n + 1. Solution: We show that the only orbit is Z. Let i,
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 3 Tuesday, February 2, 2016 1 Inductive proofs, continued Last lecture we considered inductively defined sets, and
More informationGeneralization by Collapse
Generalization by Collapse Monroe Eskew University of California, Irvine meskew@math.uci.edu March 31, 2012 Monroe Eskew (UCI) Generalization by Collapse March 31, 2012 1 / 19 Introduction Our goal is
More informationPRIORITY QUEUES. binary heaps d-ary heaps binomial heaps Fibonacci heaps. Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley
PRIORITY QUEUES binary heaps d-ary heaps binomial heaps Fibonacci heaps Lecture slides by Kevin Wayne Copyright 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos Last updated
More informationSemantics and Verification of Software
Semantics and Verification of Software Thomas Noll Software Modeling and Verification Group RWTH Aachen University http://moves.rwth-aachen.de/teaching/ws-1718/sv-sw/ Recap: CCPOs and Continuous Functions
More informationExpectations. Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or
Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or µ X, is E(X ) = µ X = x D x p(x) Definition Let X be a discrete
More informationStrong normalisation and the typed lambda calculus
CHAPTER 9 Strong normalisation and the typed lambda calculus In the previous chapter we looked at some reduction rules for intuitionistic natural deduction proofs and we have seen that by applying these
More informationAlgorithms PRIORITY QUEUES. binary heaps d-ary heaps binomial heaps Fibonacci heaps. binary heaps d-ary heaps binomial heaps Fibonacci heaps
Priority queue data type Lecture slides by Kevin Wayne Copyright 05 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos PRIORITY QUEUES binary heaps d-ary heaps binomial heaps Fibonacci
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 5 [1] Consider an infinitely repeated game with a finite number of actions for each player and a common discount factor δ. Prove that if δ is close enough to zero then
More informationCupping and noncupping in the enumeration degrees of Σ 0 2 sets
Cupping and noncupping in the enumeration degrees of Σ 0 2 sets S. Barry Cooper School of Mathematics, University of Leeds LS2 9JT, England Andrea Sorbi Department of Mathematics, University of Siena 53100
More informationSequential allocation of indivisible goods
1 / 27 Sequential allocation of indivisible goods Thomas Kalinowski Institut für Mathematik, Universität Rostock Newcastle Tuesday, January 22, 2013 joint work with... 2 / 27 Nina Narodytska Toby Walsh
More informationCharacterizing large cardinals in terms of layered partial orders
Characterizing large cardinals in terms of layered partial orders Philipp Moritz Lücke Joint work with Sean D. Cox (VCU Richmond) Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn
More informationThe Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.
The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.854J / 18.415J Advanced Algorithms Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advanced
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2015 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationChain conditions, layered partial orders and weak compactness
Chain conditions, layered partial orders and weak compactness Philipp Moritz Lücke Joint work with Sean D. Cox (VCU Richmond) Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn http://www.math.uni-bonn.de/people/pluecke/
More informationSchnorr trivial sets and truth-table reducibility
Schnorr trivial sets and truth-table reducibility Johanna N. Y. Franklin and Frank Stephan Abstract In this paper, we give several characterizations of Schnorr trivial sets, including a new lowness notion
More information1 Solutions to Tute09
s to Tute0 Questions 4. - 4. are straight forward. Q. 4.4 Show that in a binary tree of N nodes, there are N + NULL pointers. Every node has outgoing pointers. Therefore there are N pointers. Each node,
More informationMATH 121 GAME THEORY REVIEW
MATH 121 GAME THEORY REVIEW ERIN PEARSE Contents 1. Definitions 2 1.1. Non-cooperative Games 2 1.2. Cooperative 2-person Games 4 1.3. Cooperative n-person Games (in coalitional form) 6 2. Theorems and
More informationCopyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the
Copyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the open text license amendment to version 2 of the GNU General
More informationADDING A LOT OF COHEN REALS BY ADDING A FEW II. 1. Introduction
ADDING A LOT OF COHEN REALS BY ADDING A FEW II MOTI GITIK AND MOHAMMAD GOLSHANI Abstract. We study pairs (V, V 1 ), V V 1, of models of ZF C such that adding κ many Cohen reals over V 1 adds λ many Cohen
More informationNotes on Natural Logic
Notes on Natural Logic Notes for PHIL370 Eric Pacuit November 16, 2012 1 Preliminaries: Trees A tree is a structure T = (T, E), where T is a nonempty set whose elements are called nodes and E is a relation
More informationCopyright 1973, by the author(s). All rights reserved.
Copyright 1973, by the author(s). All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are
More informationLecture 10: The knapsack problem
Optimization Methods in Finance (EPFL, Fall 2010) Lecture 10: The knapsack problem 24.11.2010 Lecturer: Prof. Friedrich Eisenbrand Scribe: Anu Harjula The knapsack problem The Knapsack problem is a problem
More informationV. Fields and Galois Theory
Math 201C - Alebra Erin Pearse V.2. The Fundamental Theorem. V. Fields and Galois Theory 4. What is the Galois roup of F = Q( 2, 3, 5) over Q? Since F is enerated over Q by {1, 2, 3, 5}, we need to determine
More information10.1 Elimination of strictly dominated strategies
Chapter 10 Elimination by Mixed Strategies The notions of dominance apply in particular to mixed extensions of finite strategic games. But we can also consider dominance of a pure strategy by a mixed strategy.
More informationDecidability and Recursive Languages
Decidability and Recursive Languages Let L (Σ { }) be a language, i.e., a set of strings of symbols with a finite length. For example, {0, 01, 10, 210, 1010,...}. Let M be a TM such that for any string
More informationQuadrant marked mesh patterns in 123-avoiding permutations
Quadrant marked mesh patterns in 23-avoiding permutations Dun Qiu Department of Mathematics University of California, San Diego La Jolla, CA 92093-02. USA duqiu@math.ucsd.edu Jeffrey Remmel Department
More informationMath-Stat-491-Fall2014-Notes-V
Math-Stat-491-Fall2014-Notes-V Hariharan Narayanan December 7, 2014 Martingales 1 Introduction Martingales were originally introduced into probability theory as a model for fair betting games. Essentially
More informationUnary PCF is Decidable
Unary PCF is Decidable Ralph Loader Merton College, Oxford November 1995, revised October 1996 and September 1997. Abstract We show that unary PCF, a very small fragment of Plotkin s PCF [?], has a decidable
More informationFinding Equilibria in Games of No Chance
Finding Equilibria in Games of No Chance Kristoffer Arnsfelt Hansen, Peter Bro Miltersen, and Troels Bjerre Sørensen Department of Computer Science, University of Aarhus, Denmark {arnsfelt,bromille,trold}@daimi.au.dk
More informationINFLATION OF FINITE LATTICES ALONG ALL-OR-NOTHING SETS TRISTAN HOLMES J. B. NATION
INFLATION OF FINITE LATTICES ALONG ALL-OR-NOTHING SETS TRISTAN HOLMES J. B. NATION Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA Phone:(808)956-4655 Abstract. We introduce a
More informationLecture 7: Bayesian approach to MAB - Gittins index
Advanced Topics in Machine Learning and Algorithmic Game Theory Lecture 7: Bayesian approach to MAB - Gittins index Lecturer: Yishay Mansour Scribe: Mariano Schain 7.1 Introduction In the Bayesian approach
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationTR : Knowledge-Based Rational Decisions
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2009 TR-2009011: Knowledge-Based Rational Decisions Sergei Artemov Follow this and additional works
More informationPARTITIONS OF 2 ω AND COMPLETELY ULTRAMETRIZABLE SPACES
PARTITIONS OF 2 ω AND COMPLETELY ULTRAMETRIZABLE SPACES WILLIAM R. BRIAN AND ARNOLD W. MILLER Abstract. We prove that, for every n, the topological space ω ω n (where ω n has the discrete topology) can
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 3 Tuesday, January 30, 2018 1 Inductive sets Induction is an important concept in the theory of programming language.
More informationCS360 Homework 14 Solution
CS360 Homework 14 Solution Markov Decision Processes 1) Invent a simple Markov decision process (MDP) with the following properties: a) it has a goal state, b) its immediate action costs are all positive,
More informationRisk-neutral Binomial Option Valuation
Risk-neutral Binomial Option Valuation Main idea is that the option price now equals the expected value of the option price in the future, discounted back to the present at the risk free rate. Assumes
More informationPhilipp Moritz Lücke
Σ 1 -partition properties Philipp Moritz Lücke Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn http://www.math.uni-bonn.de/people/pluecke/ Logic & Set Theory Seminar Bristol, 14.02.2017
More informationCSE 417 Algorithms. Huffman Codes: An Optimal Data Compression Method
CSE 417 Algorithms Huffman Codes: An Optimal Data Compression Method 1 Compression Example 100k file, 6 letter alphabet: a 45% b 13% c 12% d 16% e 9% f 5% File Size: ASCII, 8 bits/char: 800kbits 2 3 >
More informationComputing Unsatisfiable k-sat Instances with Few Occurrences per Variable
Computing Unsatisfiable k-sat Instances with Few Occurrences per Variable Shlomo Hoory and Stefan Szeider Abstract (k, s)-sat is the propositional satisfiability problem restricted to instances where each
More informationIntroduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Splay Trees Date: 9/27/16
600.463 Introduction to lgoritms / lgoritms I Lecturer: Micael initz Topic: Splay Trees ate: 9/27/16 8.1 Introduction Today we re going to talk even more about binary searc trees. -trees, red-black trees,
More informationMATH 116: Material Covered in Class and Quiz/Exam Information
MATH 116: Material Covered in Class and Quiz/Exam Information August 23 rd. Syllabus. Divisibility and linear combinations. Example 1: Proof of Theorem 2.4 parts (a), (c), and (g). Example 2: Exercise
More information4 Martingales in Discrete-Time
4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1
More informationIEOR E4004: Introduction to OR: Deterministic Models
IEOR E4004: Introduction to OR: Deterministic Models 1 Dynamic Programming Following is a summary of the problems we discussed in class. (We do not include the discussion on the container problem or the
More informationCOMPARING NOTIONS OF RANDOMNESS
COMPARING NOTIONS OF RANDOMNESS BART KASTERMANS AND STEFFEN LEMPP Abstract. It is an open problem in the area of effective (algorithmic) randomness whether Kolmogorov-Loveland randomness coincides with
More informationarxiv:math/ v1 [math.lo] 9 Dec 2006
arxiv:math/0612246v1 [math.lo] 9 Dec 2006 THE NONSTATIONARY IDEAL ON P κ (λ) FOR λ SINGULAR Pierre MATET and Saharon SHELAH Abstract Let κ be a regular uncountable cardinal and λ > κ a singular strong
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationOn Finite Strategy Sets for Finitely Repeated Zero-Sum Games
On Finite Strategy Sets for Finitely Repeated Zero-Sum Games Thomas C. O Connell Department of Mathematics and Computer Science Skidmore College 815 North Broadway Saratoga Springs, NY 12866 E-mail: oconnellt@acm.org
More informationCS792 Notes Henkin Models, Soundness and Completeness
CS792 Notes Henkin Models, Soundness and Completeness Arranged by Alexandra Stefan March 24, 2005 These notes are a summary of chapters 4.5.1-4.5.5 from [1]. 1 Review indexed family of sets: A s, where
More informationTR : Knowledge-Based Rational Decisions and Nash Paths
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2009 TR-2009015: Knowledge-Based Rational Decisions and Nash Paths Sergei Artemov Follow this and
More informationAdmissibility in Quantitative Graph Games
Admissibility in Quantitative Graph Games Guillermo A. Pérez joint work with R. Brenguier, J.-F. Raskin, & O. Sankur (slides by O. Sankur) CFV 22/04/16 Seminar @ ULB Reactive Synthesis: real-world example
More informationHEIKE MILDENBERGER AND SAHARON SHELAH
A VERSION OF κ-miller FORCING HEIKE MILDENBERGER AND SAHARON SHELAH Abstract. Let κ be an uncountable cardinal such that 2 ω, 2 2
More informationQuality Sensitive Price Competition in. Secondary Market Spectrum Oligopoly- Multiple Locations
Quality Sensitive Price Competition in 1 Secondary Market Spectrum Oligopoly- Multiple Locations Arnob Ghosh and Saswati Sarkar arxiv:1404.6766v3 [cs.gt] 11 Oct 2015 Abstract We investigate a spectrum
More informationGenerating all nite modular lattices of a given size
Generating all nite modular lattices of a given size Peter Jipsen and Nathan Lawless Dedicated to Brian Davey on the occasion of his 65th birthday Abstract. Modular lattices, introduced by R. Dedekind,
More informationPart 3: Trust-region methods for unconstrained optimization. Nick Gould (RAL)
Part 3: Trust-region methods for unconstrained optimization Nick Gould (RAL) minimize x IR n f(x) MSc course on nonlinear optimization UNCONSTRAINED MINIMIZATION minimize x IR n f(x) where the objective
More informationExtender based forcings, fresh sets and Aronszajn trees
Extender based forcings, fresh sets and Aronszajn trees Moti Gitik August 31, 2011 Abstract Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model
More informationSupporting Information
Supporting Information Novikoff et al. 0.073/pnas.0986309 SI Text The Recap Method. In The Recap Method in the paper, we described a schedule in terms of a depth-first traversal of a full binary tree,
More informationCOMBINATORICS OF REDUCTIONS BETWEEN EQUIVALENCE RELATIONS
COMBINATORICS OF REDUCTIONS BETWEEN EQUIVALENCE RELATIONS DAN HATHAWAY AND SCOTT SCHNEIDER Abstract. We discuss combinatorial conditions for the existence of various types of reductions between equivalence
More informationAn orderly algorithm to enumerate finite (semi)modular lattices
An orderly algorithm to enumerate finite (semi)modular lattices BLAST 23 Chapman University October 6, 23 Outline The original algorithm: Generating all finite lattices Generating modular and semimodular
More informationSAT and DPLL. Introduction. Preliminaries. Normal forms DPLL. Complexity. Espen H. Lian. DPLL Implementation. Bibliography.
SAT and Espen H. Lian Ifi, UiO Implementation May 4, 2010 Espen H. Lian (Ifi, UiO) SAT and May 4, 2010 1 / 59 Espen H. Lian (Ifi, UiO) SAT and May 4, 2010 2 / 59 Introduction Introduction SAT is the problem
More informationMultirate Multicast Service Provisioning I: An Algorithm for Optimal Price Splitting Along Multicast Trees
Mathematical Methods of Operations Research manuscript No. (will be inserted by the editor) Multirate Multicast Service Provisioning I: An Algorithm for Optimal Price Splitting Along Multicast Trees Tudor
More informationCounting Basics. Venn diagrams
Counting Basics Sets Ways of specifying sets Union and intersection Universal set and complements Empty set and disjoint sets Venn diagrams Counting Inclusion-exclusion Multiplication principle Addition
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 7 Sampling Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 2014 Pearson Education, Inc. Chap 7-1 Learning Objectives
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games Michael Levet June 23, 2016 1 Introduction Game Theory is a mathematical field that studies how rational agents make decisions in both competitive and cooperative situations.
More informationComputing Unsatisfiable k-sat Instances with Few Occurrences per Variable
Computing Unsatisfiable k-sat Instances with Few Occurrences per Variable Shlomo Hoory and Stefan Szeider Department of Computer Science, University of Toronto, shlomoh,szeider@cs.toronto.edu Abstract.
More informationCOMPUTER SCIENCE 20, SPRING 2014 Homework Problems Recursive Definitions, Structural Induction, States and Invariants
COMPUTER SCIENCE 20, SPRING 2014 Homework Problems Recursive Definitions, Structural Induction, States and Invariants Due Wednesday March 12, 2014. CS 20 students should bring a hard copy to class. CSCI
More informationConditional Rewriting
Conditional Rewriting Bernhard Gramlich ISR 2009, Brasilia, Brazil, June 22-26, 2009 Bernhard Gramlich Conditional Rewriting ISR 2009, July 22-26, 2009 1 Outline Introduction Basics in Conditional Rewriting
More informationDesign and Analysis of Algorithms 演算法設計與分析. Lecture 9 November 19, 2014 洪國寶
Design and Analysis of Algorithms 演算法設計與分析 Lecture 9 November 19, 2014 洪國寶 1 Outline Advanced data structures Binary heaps(review) Binomial heaps Fibonacci heaps Data structures for disjoint sets 2 Mergeable
More informationBlackwell Optimality in Markov Decision Processes with Partial Observation
Blackwell Optimality in Markov Decision Processes with Partial Observation Dinah Rosenberg and Eilon Solan and Nicolas Vieille April 6, 2000 Abstract We prove the existence of Blackwell ε-optimal strategies
More informationCARDINALITIES OF RESIDUE FIELDS OF NOETHERIAN INTEGRAL DOMAINS
CARDINALITIES OF RESIDUE FIELDS OF NOETHERIAN INTEGRAL DOMAINS KEITH A. KEARNES AND GREG OMAN Abstract. We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality
More informationSHORT EXTENDER FORCING
SHORT EXTENDER FORCING MOTI GITIK AND SPENCER UNGER 1. Introduction These notes are based on a lecture given by Moti Gitik at the Appalachian Set Theory workshop on April 3, 2010. Spencer Unger was the
More informationGAME THEORY. Department of Economics, MIT, Follow Muhamet s slides. We need the following result for future reference.
14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Existence and Continuity of Nash Equilibria Follow Muhamet s slides. We need the following result for future reference. Theorem 1. Suppose
More informationOutline. 1 Introduction. 2 Algorithms. 3 Examples. Algorithm 1 General coordinate minimization framework. 1: Choose x 0 R n and set k 0.
Outline Coordinate Minimization Daniel P. Robinson Department of Applied Mathematics and Statistics Johns Hopkins University November 27, 208 Introduction 2 Algorithms Cyclic order with exact minimization
More information