Transfinite Chomp Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof.

Slides:



Advertisements
Similar presentations
5.4 Basis And Dimension.
Advertisements

Lecture ,3.3 Sequences & Summations Proof by Induction.
Introduction to Combinatorial Game Acm Honored Class Weitao.
Section 11 Direct Products and Finitely Generated Abelian Groups One purpose of this section is to show a way to use known groups as building blocks to.
MCA 520: Graph Theory Instructor Neelima Gupta
Fall 2006Costas Busch - RPI1 Non-regular languages (Pumping Lemma)
Calculus and Elementary Analysis 1.2 Supremum and Infimum Integers, Rational Numbers and Real Numbers Completeness of Real Numbers Supremum and Infimum.
Progressively Finite Games
Finite semimodular lattices Presentation by pictures November 2012.
A rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other three sides. Find the dimensions of the pen that.
New Toads and Frogs Results By Jeff Erickson Presented by Nate Swanson.
The Mean Square Error (MSE):. Now, Examples: 1) 2)
1 Finite Model Theory Lecture 13 FO k, L k 1, ,L  1, , and Pebble Games.
The Game of Nim on Graphs: NimG By Gwendolyn Stockman With: Alan Frieze, and Juan Vera.
1 The Pumping Lemma for Context-Free Languages. 2 Take an infinite context-free language Example: Generates an infinite number of different strings.
*Department of Computing Science University of Newcastle upon Tyne **Institut für Informatik, Universität Augsburg Canonical Prefixes of Petri Net Unfoldings.
Prof. Busch - LSU1 Non-regular languages (Pumping Lemma)
Games, Hats, and Codes Mira Bernstein Wellesley College SUMS 2005.
Great Theoretical Ideas in Computer Science.
MATH – High School Common Core Vs Kansas Standards.
Great Theoretical Ideas in Computer Science.
Salvador Badillo-Rios and Verenice Mojica. Goal The goal of this research project was to provide an extended analysis of 2-D Chomp using computational.
The Binary Number System
1 Chapter 8 The Discrete Fourier Transform 2 Introduction  In Chapters 2 and 3 we discussed the representation of sequences and LTI systems in terms.
Compiled By Raj G. Tiwari
Discrete Mathematics CS 2610 March 26, 2009 Skip: structural induction generalized induction Skip section 4.5.
Systems of Linear Equation and Matrices
Chapter 13 Multiple Integrals by Zhian Liang.
{ ln x for 0 < x < 2 x2 ln 2 for 2 < x < 4 If f(x) =
Surface Area of Prisms Unit 5, Lesson 2. What is a Prism? Definition: –A three dimensional figure with 2 congruent polygon bases and rectangular sides.
Chapter 8. Section 8. 1 Section Summary Introduction Modeling with Recurrence Relations Fibonacci Numbers The Tower of Hanoi Counting Problems Algorithms.
Part II - Sums of Games Consider a game called Boxing Match which was defined in a programming contest m.short.html.
Impartial Games, Nim, Composite Games, Optimal Play in Impartial games
Chapter 2: Vector spaces
Section 1.8. Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Nonexistence Proofs Uniqueness Proofs.
Math Review Data Structures & File Management Computer Science Dept Va Tech July 2000 ©2000 McQuain WD 1 Summation Formulas Let N > 0, let A, B, and C.
UNIT 9: GEOMETRY – 6 TH GRADE LESSON 7: SURFACE AREA OF A PRISM.
Copyright © Cengage Learning. All rights reserved The Area Problem.
Sets and Whole Numbers 2.1 Sets and Operations on Sets 2.2 Sets, Counting, and the Whole Numbers 2.3 Addition and Subtraction of Whole Numbers 2.4 Multiplication.
1 1 OBJECTIVE At the end of this topic you should be able to Define sequences and series Understand finite and infinite sequence,
Week 11 - Monday.  What did we talk about last time?  Binomial theorem and Pascal's triangle  Conditional probability  Bayes’ theorem.
Induction Proof. Well-ordering A set S is well ordered if every subset has a least element. [0, 1] is not well ordered since (0,1] has no least element.
Welcome to... A Game of X’s and O’s. Another Presentation © All rights Reserved
Chap. 4 Vector Spaces 4.1 Vectors in Rn 4.2 Vector Spaces
ICS 253: Discrete Structures I Induction and Recursion King Fahd University of Petroleum & Minerals Information & Computer Science Department.
An Introduction to Game Theory Math 480: Mathematics Seminar Dr. Sylvester.
Algebraic Structure in a Family of Nim-like Arrays Lowell Abrams The George Washington University Dena Cowen-Morton Xavier University TexPoint fonts used.
Lecture Coursework 2 AGAIN. Rectangle Game Look at proof of matchsticks A rectangular board is divided into m columns by n rows. The area of the board.
Make a Model A box company makes boxes to hold popcorn. Each box is made by cutting the square corners out of a rectangular sheet of cardboard. The rectangle.
Copyright © Zeph Grunschlag, Induction Zeph Grunschlag.
Chapter 5. Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder: 1.We can reach the first rung of the ladder. 2.If we can reach.
Surface Area and Volume Mrs. Hubbard Room 152. Volume  Volume is the amount of space inside an object.  Volume is measured in cubic units, since there.
PYTHAGOREAN THEOREM TRIPLES. integers A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule: a 2 + b 2 = c 2 Example: The.
Modeling the Game of Brain Cube Leslie Muzulu, Kristal Jameson, and Kathy Radloff; St. Catherine University Game Description Brain Cube is a puzzle game.
Content Game of Nim Impartial Games Sprague-Grundy theorem
3-Dimensional Figures Surface Area.
Functions of Complex Variable and Integral Transforms
Chapter 5 Limits and Continuity.
Relations, Functions, and Matrices
Introductory Game Theory
Great Theoretical Ideas in Computer Science
Non-regular languages
Additive Combinatorics and its Applications in Theoretical CS
Polyhedron Here, we derive a representation of polyhedron and see the properties of the generators. We also see how to identify the generators. The results.
Alternating tree Automata and Parity games
Polyhedron Here, we derive a representation of polyhedron and see the properties of the generators. We also see how to identify the generators. The results.
6.4 Solving by Factoring.
Quantum Foundations Lecture 2
Presentation transcript:

Transfinite Chomp Scott Huddleston and Jerry Shurman Presented by Ehren Winterhof

Chomp Invented by David Gale, 1974 Invented by David Gale, 1974 Non-partisan combinatorial Non-partisan combinatorial Played on ℕ Played on ℕ d for d in ℤ + A move consists of choosing a lattice point in the position and removing it along with all points outward Transfinite chomp uses ordinals for notation

Ordinals Ordinals Ω Ordinals Ω extend Natural Numbers ℕ to include infinite numbers Totally ordered (mex, sup) ⋆ (not commutative) ⊎, ⋆ (not commutative) Smallest infinite number is ω (little omega) In ascending order: ⋆ ⋆⋆⋆ In ascending order: ω, ω ⊎1, ω ⊎2, …, ω ⋆ 2, ω ⋆ 2 ⊎1, …, ω ⋆ 3, …, ω 2, ω 2 ⊎1, …, ω 2 ⋆ 2, ω 3, …, ω ω …

Chomp Notation Each ordinal a is the set of all ordinals less than a. ie. 5 = { } Each ordinal a is the set of all ordinals less than a. ie. 5 = { } A rectangular game is written as a x b A rectangular game is written as a x b 5 x 3 = { } x { } 5 x 3 = { } x { } A bite from a two dimensional game is A bite from a two dimensional game is ⌐⌐⌐ ⌐(a b) = ⌐a x ⌐b = { y | y ≥ a } x { z | z ≥ b } Notation extends to any number of dimensions

Chomp Size Every Chomp position X has ordinal size, size(X) Every Chomp position X has ordinal size, size(X) Decompose position into finite, overlapping sum of boxes S Decompose position into finite, overlapping sum of boxes S Each component box has each side length ω e, for non-negative integer e Each component box has each side length ω e, for non-negative integer e Discard any box contained within another to form Discard any box contained within another to form S’ If Y is reachable from X, size(Y) < size(X) If Y is reachable from X, size(Y) < size(X) Chomp terminates after finitely many moves Chomp terminates after finitely many moves

Size Example Size (X) = Size (S’) = ω *3 + 1

Grundy Values G(X) = mex{G(Y) : Y is reachable from X } G(X) = mex{G(Y) : Y is reachable from X } Poison Cookie has Grundy value 1 Poison Cookie has Grundy value 1 P-Positions have Grundy value 1 because they are reversible P-Positions have Grundy value 1 because they are reversible P-positions typically have value 0, but unrestricted misere Chomp is “tame” P-positions typically have value 0, but unrestricted misere Chomp is “tame”

Extension Two Chomp Positions A and B of dimension d and d-1, (with 1 < d < ) Two Chomp Positions A and B of dimension d and d-1, (with 1 < d < ω) Ordinal h Ordinal h E(A, B, h) = A + (B x Ω) - ⌐(0,…,0,h) E(A, B, h) = A + (B x Ω) - ⌐(0,…,0,h) A plus an infinite “column” of B, truncated to height h in the last direction A plus an infinite “column” of B, truncated to height h in the last direction “Extension of A by B to height h “Extension of A by B to height h

Fundamental Theorem For any A and B, there is a unique ordinal h such that E(A, B, h) is a P position For any A and B, there is a unique ordinal h such that E(A, B, h) is a P position Uniqueness is easy given existence Uniqueness is easy given existence Existence requires complicated double- induction Existence requires complicated double- induction h is tricky to calculate, but if you choose B to be the d-1 dimension poison square, h is bounded by size(A – (B x Ω)) h is tricky to calculate, but if you choose B to be the d-1 dimension poison square, h is bounded by size(A – (B x Ω))

Consequences Assuming we can find h, such that E(A, B, h) is a P-position, we can: Assuming we can find h, such that E(A, B, h) is a P-position, we can: Find the Grundy Value of a position Find the Grundy Value of a position Construct positions of arbitrary Grundy value Construct positions of arbitrary Grundy value For finite A and ordinal h, G(A + (1 d-1 x h)) has the same highest term as h. For finite A and ordinal h, G(A + (1 d-1 x h)) has the same highest term as h. (General Beanstalk Lemma)

P-Ordered Positions A Chomp Position is P-Ordered if its P subpositions are totally ordered by inclusion A Chomp Position is P-Ordered if its P subpositions are totally ordered by inclusion 2 x 2 x ω { (1 x (i+1) + (2 x i) : 0 ≤ I < ω } { (1 x a) + (a x 1) : 0 < a } If P is a P ordered Chomp Position, then G(X x P) = G(X)

Side-Top Positions A 2 dimensional Position of the form A 2 dimensional Position of the form U = 2 x U = 2 x ω + (2,0) + (0, ω) Def. ☐S = (2 x s) + (2, 0) Side-Top Theorem- In a S.T. Position, if H(S,2) is finite, U is an N-Position. If it is infinite: U is P position iff G

Two-Wide Chomp Two Columns h, k of ordinal height Two Columns h, k of ordinal height h = ω i * u + a k = ω j * v + b If h and k differ by a factor of ω, by an extension of the beanstalk lemma, the Grundy value is infinite Limiting examination to i=j and u=v we get the following

Finite Two Wide Grundy Values If columns are of finite heights u, v If i = j = 1, and u = v

More Two Wide Grundies When 2 < i = j < ω, and u = v When i = j v When ω ≤ i = j

Question In the sum of these three 2-wide Chomp positions, what is the winning move that reduces the game size the most? In the sum of these three 2-wide Chomp positions, what is the winning move that reduces the game size the most? A. ( A. ( ω * 2 + 3) x 2 B. (ω 4 * ), (ω 4 * ) C. (ω 3 * ), (ω 3 * )

Other Topics Covered but omitted here Side – Top Theorem Side – Top Theorem N and P analysis of 3 wide chomp N and P analysis of 3 wide chomp ω ω x 3 is a P Position Open Questions