The Computational Complexity of Some Games and Puzzles with Theoretical Applications
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Parity Games: Descriptive Complexity and Algorithms for New Solvers
Imperial College London Department of Computing Parity Games: Descriptive Complexity and Algorithms for New Solvers Huan-Pu Kuo 2013 Supervised by Prof Michael Huth, and Dr Nir Piterman Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Computing of Imperial College London and the Diploma of Imperial College London 1 Declaration I herewith certify that all material in this dissertation which is not my own work has been duly acknowledged. Selected results from this dissertation have been disseminated in scientific publications detailed in Chapter 1.4. Huan-Pu Kuo 2 Copyright Declaration The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 Abstract Parity games are 2-person, 0-sum, graph-based, and determined games that form an important foundational concept in formal methods (see e.g., [Zie98]), and their exact computational complexity has been an open prob- lem for over twenty years now. In this thesis, we study algorithms that solve parity games in that they determine which nodes are won by which player, and where such decisions are supported with winning strategies. We modify and so improve a known algorithm but also propose new algorithmic approaches to solving parity games and to understanding their descriptive complexity. -
Words Should Be Fun: Scrabble As a Tool for Language Preservation in Tuvan and Other Local Languages1
Vol. 4 (2010), pp. 213-230 http://nflrc.hawaii.edu/ldc/ http://hdl.handle.net/10125/4480 Words should be fun: Scrabble as a tool for language preservation in Tuvan and other local languages1 Vitaly Voinov The University of Texas at Arlington One small but practical way of empowering speakers of an endangered language to maintain their language’s vitality amidst a climate of rapid globalization is to introduce a mother-tongue version of the popular word game Scrabble into their society. This paper examines how versions of Scrabble have been developed and used for this purpose in various endangered or non-prestige languages, with a focus on the Tuvan language of south Siberia, for which the author designed a Tuvan version of the game. Playing Scrab- ble in their mother tongue offers several benefits to speakers of an endangered language: it presents a communal approach to group literacy, promotes the use of a standardized orthography, creates new opportunities for intergenerational transmission of the language, expands its domains of usage, and may heighten the language’s external and internal prestige. Besides demonstrating the benefits of Scrabble, the paper also offers practical suggestions concerning both linguistic factors (e.g., choice of letters to be included, cal- culation of letter frequencies, dictionary availability) and non-linguistic factors (board de- sign, manufacturing, legal issues, etc.) relevant to producing Scrabble in other languages for the purpose of revitalization. 1. INTRODUCTION.2 The past several decades have seen globalization penetrating even the most remote corners of the world, bringing with them popular American exports such as Coca-Cola and Hollywood movies. -
8 CHAPTER 2 THEORETICAL FRAMEWORK in Order to Discuss
CHAPTER 2 THEORETICAL FRAMEWORK In order to discuss and analyse the research, this chapter provides several essential theories needed about Scrabble, Zyzzyva, and Morphology. 2.1 SCRABBLE This section will introduce more about Scrabble to give readers deeper view and insight. Furthermore, this section will be discussing the history of Scrabble, explaining the game details, and finally elaborating the rules of playing the game. 2.1.1 The History of Scrabble Alfred Mosher Butts, an out-of-work architect from Poughkeepsie, New York, invented Scrabble during the Great Depression. Butts divided games into three categories: number games (dice and bingo), move games (chess and checkers), and word games (anagrams). He enjoyed anagrams and crosswords a lot that he created a game with both feature and named it LEXIKO, the game was later called CRISS CROSS WORDS (National Scrabble Association, 2012). 8 9 Figure 2.1 Alfred Mosher Butts Originally, the game was played by forming words using letter tiles and placing them on a crossword-concept board, the length of the word determined the score. After Burr studied the letter occurrence on the front page of the New York Times, the point of each letter tile was valued based on the frequency of the letters appeared in that newspaper (Halpern & Wai, 2007). 10 Figure 2.2 Criss Cross Words,an early version of Scrabble game Butts’ invention has been rejected by several established game manufactures until he met James Brunot, a game-loving entrepreneur who liked the concept and the idea of the game. They both finally made some refinements on the rules and the name SCRABBLE was created. -
Learning Board Game Rules from an Instruction Manual Chad Mills A
Learning Board Game Rules from an Instruction Manual Chad Mills A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science University of Washington 2013 Committee: Gina-Anne Levow Fei Xia Program Authorized to Offer Degree: Linguistics – Computational Linguistics ©Copyright 2013 Chad Mills University of Washington Abstract Learning Board Game Rules from an Instruction Manual Chad Mills Chair of the Supervisory Committee: Professor Gina-Anne Levow Department of Linguistics Board game rulebooks offer a convenient scenario for extracting a systematic logical structure from a passage of text since the mechanisms by which board game pieces interact must be fully specified in the rulebook and outside world knowledge is irrelevant to gameplay. A representation was proposed for representing a game’s rules with a tree structure of logically-connected rules, and this problem was shown to be one of a generalized class of problems in mapping text to a hierarchical, logical structure. Then a keyword-based entity- and relation-extraction system was proposed for mapping rulebook text into the corresponding logical representation, which achieved an f-measure of 11% with a high recall but very low precision, due in part to many statements in the rulebook offering strategic advice or elaboration and causing spurious rules to be proposed based on keyword matches. The keyword-based approach was compared to a machine learning approach, and the former dominated with nearly twenty times better precision at the same level of recall. This was due to the large number of rule classes to extract and the relatively small data set given this is a new problem area and all data had to be manually annotated. -
THE ART of Puzzle Game Design
THE ART OF Puzzle Design Scott Kim & Alexey Pajitnov with Bob Bates, Gary Rosenzweig, Michael Wyman March 8, 2000 Game Developers Conference These are presentation slides from an all-day tutorial given at the 2000 Game Developers Conference in San Jose, California (www.gdconf.com). After the conference, the slides will be available at www.scottkim.com. Puzzles Part of many games. Adventure, education, action, web But how do you create them? Puzzles are an important part of many computer games. Cartridge-based action puzzle gamse, CD-ROM puzzle anthologies, adventure game, and educational game all need good puzzles. Good News / Bad News Mental challenge Marketable? Nonviolent Dramatic? Easy to program Hard to invent? Growing market Small market? The good news is that puzzles appeal widely to both males and females of all ages. Although the market is small, it is rapidly expanding, as computers become a mass market commodity and the internet shifts computer games toward familiar, quick, easy-to-learn games. Outline MORNING AFTERNOON What is a puzzle? Guest Speakers Examples Exercise Case studies Question & Design process Answer We’ll start by discussing genres of puzzle games. We’ll study some classic puzzle games, and current projects. We’ll cover the eight steps of the puzzle design process. We’ll hear from guest speakers. Finally we’ll do hands-on projects, with time for question and answer. What is a Puzzle? Five ways of defining puzzle games First, let’s map out the basic genres of puzzle games. Scott Kim 1. Definition of “Puzzle” A puzzle is fun and has a right answer. -
Can You Change MAN to APE? Computers Can
Can a Computer Solve a Word Puzzle? - or - Can You Change MAN to APE? Computers can add; they can store numbers; they can solve equations. But word puzzles ask for abilities we don't usually associate with computers. Can we apply computational thinking to these unusual problems? To do so, we have to understand how we solve these puzzles, and identify those parts of our thought processes that can be \explained" to a computer. Computational Thinking In this discussion, we will look at a simple word puzzle. If we think about how we solve such puzzles, we can identify some mental pro- cesses (human thinking) that a computer would have to mimic: • memory: we need to know a lot of words; • imagination: we need to imagine possible changes to a word; • evaluation: given several possible changes, we need to choose the one most likely to take us to our goal; • backtracking: when a choice doesn't work out, we need to backtrack and search for an alternate choice; If we want a computer to solve these puzzles, we have to understand how we do them first, and then try to translate our thinking into computational thinking. DOUBLETS: Invented by Lewis Carroll DOUBLETS: Invented by Lewis Carroll Lewis Carroll, who wrote the children's book \Alice in Wonderland", was very fond of word games and puzzles. He asked a riddle that no one has solved: Why is a raven like a writing desk?. He wrote poems like Jabberwocky full of nonsense words, a few of which were absorbed into English: burbled and gallumphing. -
Complexity in Simulation Gaming
Complexity in Simulation Gaming Marcin Wardaszko Kozminski University [email protected] ABSTRACT The paper offers another look at the complexity in simulation game design and implementation. Although, the topic is not new or undiscovered the growing volatility of socio-economic environments and changes to the way we design simulation games nowadays call for better research and design methods. The aim of this article is to look into the current state of understanding complexity in simulation gaming and put it in the context of learning with and through complexity. Nature and understanding of complexity is both field specific and interdisciplinary at the same time. Analyzing understanding and role of complexity in different fields associated with simulation game design and implementation. Thoughtful theoretical analysis has been applied in order to deconstruct the complexity theory and reconstruct it further as higher order models. This paper offers an interdisciplinary look at the role and place of complexity from two perspectives. The first perspective is knowledge building and dissemination about complexity in simulation gaming. Second, perspective is the role the complexity plays in building and implementation of the simulation gaming as a design process. In the last section, the author offers a new look at the complexity of the simulation game itself and perceived complexity from the player perspective. INTRODUCTION Complexity in simulation gaming is not a new or undiscussed subject, but there is still a lot to discuss when it comes to the role and place of complexity in simulation gaming. In recent years, there has been a big number of publications targeting this problem from different perspectives and backgrounds, thus contributing to the matter in many valuable ways. -
Prueba Específica De Certificación De Nivel Avanzado C1 De Inglés. Junio 2019
ESCUELAS OFICIALES DE IDIOMAS DEL PRINCIPADO DE ASTURIAS PRUEBA ESPECÍFICA DE CERTIFICACIÓN DE NIVEL AVANZADO C1 DE INGLÉS. JUNIO 2019 Comisión de Evaluación de la EOI de COMPRENSIÓN DE TEXTOS ESCRITOS Puntuación total /20 puntos Calificación /10 puntos Apellidos: Nombre: DNI/NIE: LEA LAS SIGUIENTES INSTRUCCIONES A continuación va a realizar una prueba que contiene tres ejercicios de comprensión de textos escritos. Los ejercicios tienen la siguiente estructura: se presentan unos textos y se especifican unas tareas que deberá realizar en relación a dichos textos. Las tareas o preguntas serán del siguiente tipo: Opción múltiple: preguntas o frases incompletas, seguidas de una serie de respuestas posibles o de frases que las completan. En este caso deberá elegir la respuesta correcta rodeando con un círculo la letra de su opción en la HOJA DE RESPUESTAS. Sólo una de las opciones es correcta. Ejemplo: 1 A B C Si se confunde, tache la respuesta equivocada y rodee la opción que crea verdadera. 1 A B C Pregunta de relacionar. Se presentan una serie de proposiciones que deberá relacionar con su respuesta correspondiente de entre las proporcionadas. En este caso deberá elegir la respuesta correcta y escribir la letra de su opción en la HOJA DE RESPUESTAS. Ejemplo: 1 A B C D E Si se confunde, tache la respuesta equivocada y rodee la opción que crea verdadera. 1 A B C D E Pregunta de Verdadero / Falso. Se presentan una serie de preguntas y se deberá decidir si la información facilitada es verdadera o falsa. Ejemplo: 1 True False Si se confunde, tache la respuesta equivocada y rodee la opción que crea verdadera. -
Zero Knowledge and Circuit Minimization
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 68 (2014) Zero Knowledge and Circuit Minimization Eric Allender1 and Bireswar Das2 1 Department of Computer Science, Rutgers University, USA [email protected] 2 IIT Gandhinagar, India [email protected] Abstract. We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RPMCSP. This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these problems share, as candidate NP-intermediate problems. 1 Introduction For as long as there has been a theory of NP-completeness, there have been attempts to understand the computational complexity of the following two problems: – Graph Isomorphism (GI): Given two graphs G and H, determine if there is permutation τ of the vertices of G such that τ(G) = H. – The Minimum Circuit Size Problem (MCSP): Given a number i and a Boolean function f on n variables, represented by its truth table of size 2n, determine if f has a circuit of size i. (There are different versions of this problem depending on precisely what measure of “size” one uses (such as counting the number of gates or the number of wires) and on the types of gates that are allowed, etc. For the purposes of this paper, any reasonable choice can be used.) Cook [Coo71] explicitly considered the graph isomorphism problem and mentioned that he “had not been able” to show that GI is NP-complete. -
Is It Easier to Prove Theorems That Are Guaranteed to Be True?
Is it Easier to Prove Theorems that are Guaranteed to be True? Rafael Pass∗ Muthuramakrishnan Venkitasubramaniamy Cornell Tech University of Rochester [email protected] [email protected] April 15, 2020 Abstract Consider the following two fundamental open problems in complexity theory: • Does a hard-on-average language in NP imply the existence of one-way functions? • Does a hard-on-average language in NP imply a hard-on-average problem in TFNP (i.e., the class of total NP search problem)? Our main result is that the answer to (at least) one of these questions is yes. Both one-way functions and problems in TFNP can be interpreted as promise-true distri- butional NP search problems|namely, distributional search problems where the sampler only samples true statements. As a direct corollary of the above result, we thus get that the existence of a hard-on-average distributional NP search problem implies a hard-on-average promise-true distributional NP search problem. In other words, It is no easier to find witnesses (a.k.a. proofs) for efficiently-sampled statements (theorems) that are guaranteed to be true. This result follows from a more general study of interactive puzzles|a generalization of average-case hardness in NP|and in particular, a novel round-collapse theorem for computationally- sound protocols, analogous to Babai-Moran's celebrated round-collapse theorem for information- theoretically sound protocols. As another consequence of this treatment, we show that the existence of O(1)-round public-coin non-trivial arguments (i.e., argument systems that are not proofs) imply the existence of a hard-on-average problem in NP=poly. -
The Complexity of Nash Equilibria by Constantinos Daskalakis Diploma
The Complexity of Nash Equilibria by Constantinos Daskalakis Diploma (National Technical University of Athens) 2004 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Christos H. Papadimitriou, Chair Professor Alistair J. Sinclair Professor Satish Rao Professor Ilan Adler Fall 2008 The dissertation of Constantinos Daskalakis is approved: Chair Date Date Date Date University of California, Berkeley Fall 2008 The Complexity of Nash Equilibria Copyright 2008 by Constantinos Daskalakis Abstract The Complexity of Nash Equilibria by Constantinos Daskalakis Doctor of Philosophy in Computer Science University of California, Berkeley Professor Christos H. Papadimitriou, Chair The Internet owes much of its complexity to the large number of entities that run it and use it. These entities have different and potentially conflicting interests, so their interactions are strategic in nature. Therefore, to understand these interactions, concepts from Economics and, most importantly, Game Theory are necessary. An important such concept is the notion of Nash equilibrium, which provides us with a rigorous way of predicting the behavior of strategic agents in situations of conflict. But the credibility of the Nash equilibrium as a framework for behavior-prediction depends on whether such equilibria are efficiently computable. After all, why should we expect a group of rational agents to behave in a fashion that requires exponential time to be computed? Motivated by this question, we study the computational complexity of the Nash equilibrium. We show that computing a Nash equilibrium is an intractable problem. -
Game Complexity Vs Strategic Depth
Game Complexity vs Strategic Depth Matthew Stephenson, Diego Perez-Liebana, Mark Nelson, Ahmed Khalifa, and Alexander Zook The notion of complexity and strategic depth within games has been a long- debated topic with many unanswered questions. How exactly do you measure the complexity of a game? How do you quantify its strategic depth objectively? This seminar answered neither of these questions but instead presents the opinion that these properties are, for the most part, subjective to the human or agent that is playing them. What is complex or deep for one player may be simple or shallow for another. Despite this, determining generally applicable measures for estimating the complexity and depth of a given game (either independently or comparatively), relative to the abilities of a given player or player type, can provide several benefits for game designers and researchers. There are multiple possible ways of measuring the complexity or depth of a game, each of which is likely to give a different outcome. Lantz et al. propose that strategic depth is an objective, measurable property of a game, and that games with a large amount of strategic depth continually produce challenging problems even after many hours of play [1]. Snakes and ladders can be described as having no strategic depth, due to the fact that each player's choices (or lack thereof) have no impact on the game's outcome. Other similar (albeit subjective) evaluations are also possible for some games when comparing relative depth, such as comparing Tic-Tac-Toe against StarCraft. However, these comparative decisions are not always obvious and are often biased by personal preference.