DOCSLIB.ORG

Solving Mathematical Puzzles: a Challenging Competition for AI

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solving Mathematical Puzzles: a Challenging Competition for AI Articles Solving Mathematical Puzzles: A Challenging Competition for AI Federico Chesani, Paola Mello, Michela Milano I Recently, a number of noteworthy ince its origin, the holy grail of artificial intelligence results have been achieved in various has been to understand the nature of intelligence fields of artificial intelligence, and many Sand to engineer systems that exhibit such intelli- aspects of the problem-solving process gence through vision, language, emotion, motion, and have received significant attention by reasoning. In such context, AI researchers have always the scientific community. In this context, looked for challenges to push forward the limit of what the extraction of comprehensive knowl- edge suitable for problem solving and computers can do autonomously and to measure the lev- reasoning, from textual and pictorial el of “intelligence” achieved. Competitions have been problem descriptions, has been less and are currently run on conversational behavior (for investigated, but recognized as essential example, the Loebner prize1), automatic control (for for autonomous thinking in artificial example, the International Aerial Robotics Competition2 intelligence. In this work we present a or the DARPA Grand Challenge on driverless cars3), coop- challenge where methods and tools for eration and coordination in robotics (for example, the deep understanding are strongly needed RoboCup4), logic reasoning and knowledge (for example, for enabling problem solving: we propose the CADE ATP System competition for theorem to solve mathematical puzzles by means 5 of computers, starting from text and dia- provers ), and natural language (for example, the EVALI- 6 grams describing them, without any TA competition for the Italian language). Historically, human intervention. We are aware that also games have raised the interest of the AI community: the proposed challenge is hard and diffi- a number of competitions are still being held nowadays cult to solve nowadays (and in the fore- (for example, the World Computer Chess Champi- seeable future), but even studying and onship,7 Mario Championship8 [Togelius et al. 2013] and solving only single parts of the proposed its successor Platformer Competition,9 the General Game challenge would represent an important Playing competition10). For a high-level overview of the step forward for artificial intelligence. field of AI in games see the paper by Yannakakis and Togelius (2014). Some of these challenges have indeed brought many insights and advancements on various artificial intelli- gence fields. We mention in the following three of them, notable for the achieved results, as well as for their impact on the media and the consequent dissemination Copyright © 2017, Association for the Advancement of Artificial Intelligence. All rights reserved. ISSN 0738-4602 FALL 2017 83 Articles of the AI discipline to the general audience. In the last answering in an effective and efficient way. However, century, a famous challenge of AI concerned the how to integrate more complex problem-solving game of chess, considered a symbol of complexity, capabilities and reasoning techniques is still a matter problem solving, and strategic ability. As widely of research. known, the competition between humans and the The challenge we propose here goes in the direc- Deep Blue computer was definitely won by comput- tion of extracting specific knowledge from general ers in 1997 when the world chess champion Garry text and diagrams that is useful for reasoning and Kasparov was defeated. A key lesson learned from problem solving and can be stated as follows: Deep Blue’s strength is that efficient heuristic search By the middle of the 21st century, (a team of) fully can be much more effective than sophisticated rea- autonomous agent(s) shall win a mathematical puzzle soning guided search. In fact, heuristic search was so competition against primary school students, winners successful that it led Kasparov to exclaim, “I could of the most recent competitions. feel — I could smell — a new kind of intelligence Mathematical puzzles are recreational games where across the table” (Time Magazine).11 Despite a very a single human player is challenged with a problem, good result in terms of problem solving, the search- described by text and diagrams. To solve them, a intensive nature of the solution approach (in terms human player uses understanding and intuition, as of number of configurations explored, memory, and well as common sense, simple logical and mathemat- processing power) was in fact the main criticism to ical or geometry knowledge, causal relations, and the experiment. Noam Chomsky commented that many other reasoning-related capabilities. The main Deep Blue defeating Kasparov in chess was “as inter- research question then can be formulated as “What esting as a bulldozer that can win the Olympic are the requirements, functionalities, and main soft- weight-lifting competition.”12 Very recently, comput- ware components needed for autonomously solving a ers won against humans at the game Go, famous for mathematical puzzle, with no human intervention?” being one of the last games where humans were still The focus is on deep reasoning featuring (1) the better players. In March 2016, the AlphaGO (Silver et extraction of comprehensive knowledge from multi- al. 2016) prototype by Google won13 against Lee modal descriptions (texts, diagrams, sound and Sedol, one of the world’s best players. The distinct fea- speech, gesture, and others); (2) the ability of deter- ture of AlphaGO is the adoption of deep learning mining both the proper model and the correspon- techniques. ding reasoning capability; and (3) the capability of A second challenge is the Robot Soccer World Cup effectively solving the problem (possibly with a feed- (RoboCup),14 the international robotics competition back to the two previous steps). All these steps require and challenge launched in 1997. The official goal of considerable integration of many artificial intelli- the RoboCup challenge is the following: gence techniques such as natural language and dia- “By the middle of the 21st century, a team of fully gram understanding, problem solving and search, autonomous humanoid robot soccer players shall win a modeling, and machine learning. In particular, recent soccer game, complying with the official rules of FIFA, advances in machine learning open new research 15 against the winner of the most recent World Cup.” avenues for deep reasoning, as they enable general- Even if we are still far from reaching such an ambi- ization over manually collected knowledge. tious goal, the RoboCup challenge has promoted AI Recently, a number of similar challenges have been techniques in robotics, planning, self-adaptive sys- proposed. We mention here the Aristo Project (Clark tems, and machine vision, showing that even 2015), a challenge with the goal of having the com- extremely ambitious and visionary challenges can puter pass elementary school math and science bring huge benefits to a research field by approaching exams. Natural language comprehension and dia- the goal step by step. gram understanding are required as fundamental The third challenge is about Watson,16 an AI sys- steps, together with some form of inference and alge- tem developed by the IBM research team. Watson was braic or mathematical solving techniques: for this originally developed to answer questions on a quiz reason, the Aristo challenge is quite related to our show called Jeopardy. In 2011, Watson competed proposal. We will discuss Aristo and other challenges against former human winners and won. Watson can extensively in the related works and challenges sec- be considered as a very advanced question-answering tion. While the large majority of existing systems that system that applies and integrates AI techniques such provide end-to-end problem solvers stick to one spe- as natural language processing (NLP), knowledge cific solution method (for example, mathematical extraction and representation, automated reasoning, equations, logic), our challenge is broader, as the and machine learning. The system is able to access choice of the appropriate model and solution method “millions of pages” of structured and unstructured is considered a fundamental capability. content in a very efficient way, thanks to the massive We are aware that the proposed challenge is hard use of parallel processing. A huge amount of knowl- and of difficult solution nowadays, but we strongly edge is learned, processed, and used together with a believe that even studying and solving only single number of inference tasks, to perform question parts of the problem would bring important steps for- 84 AI MAGAZINE Articles ward in artificial intelligence. In addition, on the road rial can be found on the web, where a large instance to agent autonomy, it would be interesting to study set of mathematical and logic puzzles is available. which level of human intervention and interaction with the machine is needed to effectively collaborate to solve the problem. Challenge Description In a computer-aided problem-solving process, there is Mathematical Puzzles always a substantial human intervention that enables the encoding of a problem described by text and dia- Recreational mathematics is a powerful source of grams in a model and a solution algorithm. Human inspiration. Michel Criton said: intervention is essential for identifying problem com- It is a tradition that comes to us with a history of ponents (decision variables, constraints,
Load more

Recommended publications
  • Commonsense Reasoning for Natural Language Processing
    Introductory Tutorial: Commonsense Reasoning for Natural Language Processing Maarten Sap 1 Vered Shwartz 1;2 Antoine Bosselut 1;2 Yejin Choi 1;2 Dan Roth 3 1 Paul G. Allen School of Computer Science & Engineering, Seattle, WA, USA 2 Allen Institute for Artificial Intelligence, Seattle, WA, USA 3 Department of Computer and Information Science, University of Pennsylvania fmsap, vereds, antoineb, yejing @cs.washington.edu, [email protected] 1 Introduction settings, such as social interactions (Sap et al., 2019b; Rashkin et al., 2018a) and physical situ- Commonsense knowledge, such as knowing that ations (Zellers et al., 2019; Talmor et al., 2019). “bumping into people annoys them” or “rain We hope that in the future, machines develop makes the road slippery”, helps humans navigate the kind of intelligence required to, for exam- everyday situations seamlessly (Apperly, 2010). ple, properly assist humans in everyday situations Yet, endowing machines with such human-like (e.g., a chatbot that anticipates the needs of an el- commonsense reasoning capabilities has remained derly person; Pollack, 2005). Current methods, an elusive goal of artificial intelligence research however, are still not powerful or robust enough to for decades (Gunning, 2018). be deployed in open-domain production settings, Commonsense knowledge and reasoning have despite the clear improvements provided by large- received renewed attention from the natural lan- scale pretrained language models. This shortcom- guage processing (NLP) community in recent ing is partially due to inadequacy in acquiring, years, yielding multiple exploratory research di- understanding and reasoning about commonsense rections into automated commonsense under- knowledge, topics which remain understudied by standing.
    [Show full text]
  • Review Articles
    review articles DOI:10.1145/2701413 AI has seen great advances of many kinds recently, but there is one critical area where progress has been extremely slow: ordinary commonsense. BY ERNEST DAVIS AND GARY MARCUS Commonsense Reasoning and Commonsense Knowledge in Artificial Intelligence WHO IS TALLER, Prince William or his baby son Prince key insights George? Can you make a salad out of a polyester shirt? ˽ To achieve human-level performance in domains such as natural language If you stick a pin into a carrot, does it make a hole processing, vision, and robotics, basic knowledge of the commonsense world— in the carrot or in the pin? These types of questions time, space, physical interactions, people, may seem silly, but many intelligent tasks, such as and so on—will be necessary. understanding texts, computer vision, planning, and ˽ Although a few forms of commonsense reasoning, such as taxonomic reasoning scientific reasoning require the same kinds of real- and temporal reasoning are well world knowledge and reasoning abilities. For instance, understood, progress has been slow. ˽ Extant techniques for implementing if you see a six-foot-tall person holding a two-foot-tall commonsense include logical analysis, handcrafting large knowledge bases, person in his arms, and you are told they are father Web mining, and crowdsourcing. Each of these is valuable, but none by itself is a and son, you do not have to ask which is which. If you full solution. need to make a salad for dinner and are out of lettuce, ˽ Intelligent machines need not replicate you do not waste time considering improvising by human cognition directly, but a better understanding of human commonsense taking a shirt of the closet and cutting might be a good place to start.
    [Show full text]
  • IPG Spring 2020 Games Titles - December 2019 Page 1
    Games Titles Spring 2020 {IPG} A Puzzle a Day A Collection of Mathematical Problems for Every Day of the School Year Vivian Lucas Summary Although primarily written for pupils ged 11-16, these puzzles do not require a high level of syllabus knowledge, making them accessible to a wide range of ages and abilities.Two levels of difficulty are provided for each entry. There are 180 puzzles, offering two similar ones on each of a wide variety of topics. They can Tarquin Group be used as a daily displayed competition for pupils to enter and a winner can be picked from the days entries. 9781899618521 Pub Date: 3/1/02 It helps to increase interest, to reinforce mathematical vocabulary and of course the puzzles are fun to do. $11.95 USD Complete with answers. Discount Code: LON Trade Paperback Contributor Bio 95 Pages Vivian Lucas is a former head of Mathematics at a school in Derby, UK. Her creative books have been used by Carton Qty: 0 tens of thousands of teachers worldwide in millions of lessons. Education / Teaching Methods & Materials EDU029010 5.5 in H | 5.5 in W Mathematical Snacks A Collection of Interesting Ideas to Fill Those Spare Moments Jon Millington Summary A collection of interesting topics and ideas to fill spare moments... 45 topics and mathematical ideas for the classroom or home, which are quick to introduce and set up but also genuinely stimulating and enriching. Although primarily designed to fill spare moments in normal lessons, they enliven clubs, quizzes and end of Tarquin Group term activities.
    [Show full text]
  • Downloading Programs
    Futures of Artificial Intelligence through Technology Readiness Levels Fernando Mart´ınez-Plumed1,2, Emilia Gomez´ 1, Jose´ Hernandez-Orallo´ 2 Abstract Artificial Intelligence (AI) offers the potential to transform our lives in radical ways. However, the main unanswered questions about this foreseen transformation are its depth, breadth and timelines. To answer them, not only do we lack the tools to determine what achievements will be attained in the near future, but we even ignore what various technologies in present-day AI are capable of. Many so-called breakthroughs in AI are associated with highly-cited research papers or good performance in some particular benchmarks. However, research breakthroughs do not directly translate into a technology that is ready to use in real-world environments. In this paper, we present a novel exemplar- based methodology to categorise and assess several AI technologies, by mapping them onto Technology Readiness Levels (TRL) (representing their depth in maturity and availability). We first interpret the nine TRLs in the context of AI, and identify several categories in AI to which they can be assigned. We then introduce a generality dimension, which represents increasing layers of breadth of the technology. These two dimensions lead to the new readiness-vs-generality charts, which show that higher TRLs are achievable for low-generality technologies, focusing on narrow or specific abilities, while high TRLs are still out of reach for more general capabilities. We include numerous examples of AI technologies in a variety of fields, and show their readiness-vs-generality charts, serving as exemplars. Finally, we show how the timelines of several AI technology exemplars at different generality layers can help forecast some short-term and mid-term trends for AI.
    [Show full text]
  • Commonsense Knowledge in Wikidata
    Commonsense Knowledge in Wikidata Filip Ilievski1, Pedro Szekely1, and Daniel Schwabe2 1 Information Sciences Institute, University of Southern California 2 Dept. of Informatics, Pontificia Universidade Cat´olicaRio de Janeiro filievski,[email protected], [email protected] Abstract. Wikidata and Wikipedia have been proven useful for reason- ing in natural language applications, like question answering or entity linking. Yet, no existing work has studied the potential of Wikidata for commonsense reasoning. This paper investigates whether Wikidata con- tains commonsense knowledge which is complementary to existing com- monsense sources. Starting from a definition of common sense, we devise three guiding principles, and apply them to generate a commonsense subgraph of Wikidata (Wikidata-CS). Within our approach, we map the relations of Wikidata to ConceptNet, which we also leverage to integrate Wikidata-CS into an existing consolidated commonsense graph. Our ex- periments reveal that: 1) albeit Wikidata-CS represents a small portion of Wikidata, it is an indicator that Wikidata contains relevant com- monsense knowledge, which can be mapped to 15 ConceptNet relations; 2) the overlap between Wikidata-CS and other commonsense sources is low, motivating the value of knowledge integration; 3) Wikidata-CS has been evolving over time at a slightly slower rate compared to the overall Wikidata, indicating a possible lack of focus on commonsense knowl- edge. Based on these findings, we propose three recommended actions to improve the coverage and quality of Wikidata-CS further. Keywords: Commonsense Knowledge · Wikidata · Knowledge Graphs 1 Introduction Common sense is \the basic ability to perceive, understand, and judge things that are shared by nearly all people and can be reasonably expected of nearly all people without need for debate" [10].
    [Show full text]
  • Natural Language Understanding with Commonsense Reasoning
    E.T.S. DE INGENIEROS INFORMÁTICOS UNIVERSIDAD POLITÉCNICA DE MADRID MASTER TESIS MSc IN ARTIFICIAL INTELLIGENCE (MUIA) NATURAL LANGUAGE UNDERSTANDING WITH COMMONSENSE REASONING: APPLICATION TO THE WINOGRAD SCHEMA CHALLENGE AUTHOR: ALFONSO LÓPEZ TORRES SUPERVISOR: MARTÍN MOLINA GONZÁLEZ JUNE, 2016 This is for my children Carla and Alonso, and my wife Véronique Thanks for their unconditional support and patient (also for the coming adventures…) v Acknowledgments: I’d like to thank the advices and help received from Martín. I was very lucky being your student. vi RESUMEN En 1950, Alan Turing propuso un test para evaluar el grado de inteligencia humana que podría presentar una máquina. La idea principal era realmente sencilla: llevar a cabo una charla abierta entre un evaluador y la máquina. Si dicho evaluador era incapaz de discernir si el examinado era una persona o una máquina, podría afirmarse que el test había sido superado. Desde entonces, a lo largo de los últimos 60 años se han presentado numerosas propuestas a través de los cuales se han puesto al descubierto ciertas debilidades del test. Quizás la más importante es el hecho de centrarse en la inteligencia humana, dejando a un lado otros tipos de inteligencia. El test obliga en gran medida a definir en la máquina un comportamiento antropomórfico y de imitación con el único fin de pasar el test. Con el fin de superar estos y otros puntos débiles, Hector Levesque propuso en 2011 un nuevo reto, “The Winograd Schema Challenge”. Un sencillo test basado en Pregunta y Respuesta sobre una frase que describe una situación cotidiana.
    [Show full text]
  • Evaluating Commonsense Reasoning in Neural Machine Translation
    The Box is in the Pen: Evaluating Commonsense Reasoning in Neural Machine Translation Jie Hey∗, Tao Wangz∗, Deyi Xiongy, and Qun Liux y College of Intelligence and Computing, Tianjin University, Tianjin, China z School of Computer Science and Technology, Soochow University, Suzhou, China x Huawei Noah’s Ark Lab, Hong Kong, China [email protected], [email protected] [email protected], [email protected] Abstract evidence for the need of commonsense knowl- edge in machine translation is “The box is in the Does neural machine translation yield transla- pen”, where machine translation is expected to per- tions that are congenial with common sense? form reasoning on the relative sizes of “box” and In this paper, we present a test suite to evalu- “pen”. Bar-Hillel also doubts that a machine, even ate the commonsense reasoning capability of equipped with extra-linguistic knowledge, would neural machine translation. The test suite consists of three test sets, covering lexical be able to reason with such knowledge sponta- and contextless/contextual syntactic ambiguity neously as human translators do (Bar-Hillel, 1960a; that requires commonsense knowledge to re- Macklovitch, 1995). solve. We manually create 1,200 triples, each Modern natural language processing (NLP) has of which contain a source sentence and two made tremendous progress, not only in building contrastive translations, involving 7 different abundant resources to develop linguistic insights, common sense types. Language models pre- but also in plenty of methodological practices. On trained on large-scale corpora, such as BERT, GPT-2, achieve a commonsense reasoning ac- the one hand, machine translation has been sub- curacy of lower than 72% on target transla- stantially advanced with large-scale parallel data tions of this test suite.
    [Show full text]
  • For the Low Achiever Student Inmathematics
    DOCUMZNT RESUME ED 025 437 By-Zimmerman, Joseph Central Iowa Low Achiever MathematicsProject ESP. Central Iowa Low-Achiever MathematicsProject, Des Moines. Elementary and Secondary Education. Spons Agency-Office of Education(DHEW), Washington, D.C. Bureau of Pub Date (Nov 68] Grant- OEG- 3965 Note- 64p EDRS Price MF-$0.50 HC-$3.30 *Instructional Materials, Descriptors-Curriculum, Curriculum Development,*Elementary School Mathematics, *Low Achievers, *Mathematics,*Problem Solving. SecondarySchool Mathematics Identifiers-Central Iowa Low AchieverMathematics Project The materials in this EnrichmentStudent Project (ESP) aredesigned especially for the low achiever student inmathematics. The booklet is aselfontained unit consisting of fourelements--a mathematicalpuzzle, a set of instructions,response sheets, and a suitable containerfor keeping the unittogether. ESP is amotivational the student's interest andpromoting hisinvolvement in a idea aimed at attracting been collected portion of mathematicsthat can be enjoyed. Thematerials which have for this ESP, complete with solutionof problems for the teacher'sconvenience involve prepared under ESEATitle peg, dissection,cube, and topologypuzzles. This work was III contract. (RP) Central Iowa LowAchiever Mathematics Project U.S. DEPARTMENT Of HEALTH, EDUCATION & WELFARE OFFICE Of EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE ED025437 PERSON OR ORGANIZATION ORIGINATING II.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. The work presented or rennrted herein was performed our, suant to a Grant, OE No. 3969, from the U.S. Office of Educe. tion, Deprirtment of Hei Ith. Ed,initi-n, and Welfare.However, the opinions exnrer,sed hrTiin do not necesssrily reflect the positionorpolicy of the U.S.
    [Show full text]
  • An Approach to Evaluate AI Commonsense Reasoning Systems∗
    An Approach to Evaluate AI Commonsense Reasoning Systems∗ Stellan Ohlsson and Robert H. Sloan Gyorgy¨ Turan´ Daniel Uber and Aaron Urasky University of Illinois at Chicago University of Illinois at Chicago, University of Illinois at Chicago University of Szeged Abstract ery schoolchild and the methods for making obvious infer- ences from this knowledge are called common sense” (Davis We propose and give a preliminary test of a new metric 1990). Without a generally accepted performance standard, for the quality of the commonsense knowledge and rea- it is impossible to evaluate claims and document progress. soning of large AI databases: Using the same measure- ment as is used for a four-year-old, namely, an IQ test We propose that tests of intelligence developed by psy- for young children. We report on results obtained us- chometricians can serve as one type of assessment of com- ing test questions we wrote in the spirit of the questions mon sense reasoning systems. Psychologists face the same of the Wechsler Preschool and Primary Scale of Intel- problem as artificial intelligence researchers: How to mea- ligence, Third Edition (WPPSI-III) on the ConceptNet sure something as broad and varied as common sense? Their system, which were, on the whole, quite strong. solution is to generate a multitude of diverse but simple tasks, called test items, and then collect empirical data on the association between performance on the proposed test items 1 Introduction and some criterion variable. This statistical solution is en- Computer systems have long outperformed humans on nu- capsulated in what are generally known as intelligence tests.
    [Show full text]
  • Commonsense Reasoning About Task Instructions
    Commonsense Reasoning about Task Instructions Rakesh Gupta Ken Hennacy Honda Research Institute USA, Inc. University of Maryland 800 California Street, Suite 300 Institute for Advanced Computer Studies Mountain View, CA 94041 College Park, MD 20742 [email protected] [email protected] Abstract to handle the various difficulties associated with change and scalability (Minker 1991). We have developed an approach for commonsense reason- ing using knowledge collected from volunteers over the web. As an alternative to formal methods that guarantee solu- This knowledge is collected in natural language, and includes tions, implementational approaches place more emphasis on information such as task instructions, locations of objects in adaptation and knowledge acquisition. Studies involving hu- homes, causes and effects, and uses of objects in the home. man interactions and natural language, for example, have This knowledge stored in tables in a relational database is shown promise in developing agents that can learn and rea- filtered using statistical methods and rule-based inference. son about human-centric information (W. L. Johnson 2004). Missing details within natural language task instructions are According to Rasmussen et al (1983), human behavior in fa- reasoned to determine the steps to be executed in the task. miliar environments doesn’t require extensive decision mak- These missing details are handled by meta-rules which work ing. It is controlled by a set of rules or steps which have across the knowledge categories and interact with the appro- priate tables to extract the right information. Our reasoning proven successful previously. The sequence of steps may approach is illustrated for common household tasks.
    [Show full text]
  • Using Conceptnet to Teach Common Sense to an Automated Theorem Prover∗
    Using ConceptNet to Teach Common Sense to an Automated Theorem Prover∗ Claudia Schon Institute for Web Science and Technologies, University of Koblenz-Landau, 56070 Koblenz, Germany [email protected] Sophie Siebert Frieder Stolzenburg Harz University of Applied Sciences, Automation and Computer Sciences Department, 38855 Wernigerode, Germany fssiebert,[email protected] The CoRg system is a system to solve commonsense reasoning problems. The core of the CoRg sys- tem is the automated theorem prover Hyper that is fed with large amounts of background knowledge. This background knowledge plays a crucial role in solving commonsense reasoning problems. In this paper we present different ways to use knowledge graphs as background knowledge and discuss challenges that arise. 1 Introduction In recent years, numerous benchmarks for commonsense reasoning have been presented which cover different areas: the Choice of Plausible Alternatives Challenge (COPA) [17] requires causal reasoning in everyday situations, the Winograd Schema Challenge [8] addresses difficult cases of pronoun disam- biguation, the TriangleCOPA Challenge [9] focuses on human relationships and emotions, and the Story Cloze Test with the ROCStories Corpora [11] focuses on the ability to determine a plausible ending for a given short story, to name just a few. In our system, we focus on the COPA challenge where each problem consists of a problem description (the premise), a question, and two answer candidates (called alternatives). See Fig. 1 for an example. Most approaches tackling these problems are based on machine learning or exploit statistical properties of the natural language input (see e.g. [14, 16]) and are therefore unable to provide explanations for the decisions made.
    [Show full text]
  • MAA Rocky Mountain Section Meeting
    e Mathematical Association of America Rocky Mountain Section Meeting April Ôâ and ÔÞ, òýÔý Colorado State University Fort Collins, CO Rocky Mountain Section Meeting Annual Book Sale April 16 & 17, 2010 All listed titles available for shipment, free shipping & handling. A full catalog and order form for shipped orders is included on the middle pages of this booklet for your convenience. Over 100 titles available (in limited quantities) for immediate purchase at our display. All catalog prices have been discounted 10% below membership prices, and 10% of book proceeds will be returned to the section – enlarge your book collection while benefiting the section today! To place an order, visit us in the Cherokee Park Ballroom. Discount prices available to all meeting participants; prices good only on orders placed at the meeting. All order forms must be returned to a display staff member for processing!! Payment by check, Visa or MasterCard accepted (sorry - no cash sales). Please make checks payable to: MAA Rocky Mountain Section. MATHEMATICAL ASSOCIATION OF AMERICA ò Schedule Friday, April Ôâ :ýý-Ôò:ýý Section NExT workshop (Virginia Dale) À:çý-ÔÔ:çý Workshop: Proposal writing for the NSF DUE (Lory òÔÞ) Stephanie Fitchett, NSF and University of Northern Colorado ÔÔ:¥ -Ôò:¥ Luncheon for Dept. Chairs and MAA Liaisons (Lory òçý) ÔÔ:ýý-¥:çý Registration (Lory òòÞ) Ô:ýý-Ô:Ôý Opening Remarks and Welcome (North Ballroom) Ô:Ôý-Ô: Burton W. Jones Teaching Award Lecture (North Ballroom) Richard Grassl, University of Northern Colorado Ô:ýý- :çý Publisher
    [Show full text]