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Program Synthesis Meets Visual What-Comes-Next Puzzles
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Computer game technology is increasingly more complex and applied in a wide variety of domains, beyond entertainment, such as training and educational scenarios. Testing games is a difficult task requiring a lot of manual effort since the interaction space in the game is very fine grained and requires a certain level of intelligence that cannot be easily automated. This makes testing a costly activity in the overall development of games.
Natural deduction, which is a method for establishing validity of propositional type arguments, helps develop important reasoning skills and is thus a key ingredient in a course on introductory logic. We present two core components, namely solution generation and practice problem generation, for enabling computer-aided education for this important subject domain. The key enabling technology is use of an offline-computed data-structure called Universal Proof Graph (UPG) that encodes all possible applications of inference rules over all small propositions abstracted using their bitvector-based truth-table representation. This allows an efficient forward search for solution generation. More interestingly, this allows generating fresh practice problems that have given solution characteristics by performing a backward search in UPG. We obtained around 300 natural deduction problems from various textbooks. Our solution generation procedure can solve many more problems than the traditional forward-chaining based procedure, while our problem generation procedure can efficiently generate several variants with desired characteristics.
We propose computer-assisted techniques for helping with pedagogy in Algebra. In particular, given a proof problem p (of the from "Left-hand-side-term = Right-hand-side-term"), we show how to automatically generate problems that are similar to p. We believe that such a tool can be used by teachers in making examinations where they need to test students on problems similar to what they taught in class, and students in generating practice problems tailored to their specific needs. Our first insight is that we can generalize p syntactically to a query Q that implicitly represents a set of problems [Q] (which includes p). Our second insight is that we can explore the space of problems [Q] automatically, use classical results from polynomial identity testing to generate only those problems in [Q] that are correct, and then use pruning techniques to generate only unique and interesting problems. Our third insight is that with a small amount of manual tuning on the query Q, the user can interactively guide the computer to generate problems of interest to her. We present the technical details of the above mentioned steps, and also describe a tool where these steps have been implemented. We also present an empirical evaluation on a wide variety of problems from various sub-fields of algebra including polynomials, trigonometry, calculus, determinants etc. Our tool is able to generate a rich corpus of similar problems from each given problem; while some of these similar problems were already present in the textbook, several were new!
In automatic speech recognition, language models can be represented by Probabilistic Context Free Grammars (PCFGs). In this lecture we review some known algorithms which handle PCFGs; in particular an algorithm for the computation of the total probability that a PCFG generates a given sentence (Inside), an algorithm for finding the most probable parse tree (Viterbi), and an algorithm for the estimation of the probabilities of the rewriting rules of a PCFG given a corpus (Inside-Outside). Moreover, we introduce the Left-to-Right Inside algorithm, which computes the probability that successive applications of the grammar rewriting rules (beginning with the sentence start symbol s) produce a word string whose initial substring is a given one.
In this paper, we study the problem of automatically solving ruler/compass based geometry construction problems. We first introduce a logic and a programming language for describing such constructions and then phrase the automation problem as a program synthesis problem. We then describe a new program synthesis technique based on three key insights: (i) reduction of symbolic reasoning to concrete reasoning (based on a deep theoretical result that reduces verification to random testing), (ii) extending the instruction set of the programming language with higher level primitives (representing basic constructions found in textbook chapters, inspired by how humans use their experience and knowledge gained from chapters to perform complicated constructions), and (iii) pruning the forward exhaustive search using a goal-directed heuristic (simulating backward reasoning performed by humans). Our tool can successfully synthesize constructions for various geometry problems picked up from high-school textbooks and examination papers in a reasonable amount of time. This opens up an amazing set of possibilities in the context of making classroom teaching interactive.
The paper proposes a solution based on Generative Adversarial Network (GAN) for solving jigsaw puzzles. The problem assumes that an image is divided into equal square pieces, and asks to recover the image according to information provided by the pieces. Conventional jigsaw puzzle solvers often determine the relationships based on the boundaries of pieces, which ignore the important semantic information. In this paper, we propose JigsawGAN, a GAN-based auxiliary learning method for solving jigsaw puzzles with unpaired images (with no prior knowledge of the initial images). We design a multi-task pipeline that includes, (1) a classification branch to classify jigsaw permutations, and (2) a GAN branch to recover features to images in correct orders. The classification branch is constrained by the pseudo-labels generated according to the shuffled pieces. The GAN branch concentrates on the image semantic information, where the generator produces the natural images to fool the discriminator, while the discriminator distinguishes whether a given image belongs to the synthesized or the real target domain. These two branches are connected by a flow-based warp module that is applied to warp features to correct the order according to the classification results. The proposed method can solve jigsaw puzzles more efficiently by utilizing both semantic information and boundary information simultaneously. Qualitative and quantitative comparisons against several representative jigsaw puzzle solvers demonstrate the superiority of our method.
A shaded area problem in high school geometry consists of a figure annotated with facts such as lengths of line segments or angle measures, and asks to compute the area of a shaded portion of the figure. We describe a technique to generate fresh figures for these problems. Given a figure, we describe a technique to automatically synthesize shaded area problems. We demonstrate the efficacy of our synthesis techniques by synthesizing problems from fresh figures as well as figures from a corpus of problems from high-school geometry textbooks.
A long-term goal of game design research is to achieve end-to-end automation of much of the design process, one aspect of which is creating effective level progressions. A key difficulty is getting the player to practice with interesting combinations of learned skills while maintaining their engagement. Although recent work in task generation and sequencing has reduced this effort, we still lack end-to-end automation of the entire content design process. We approach this goal by incorporating ideas from intelligent tutoring systems and proposing progression strategies that seek to achieve mastery of not only base concepts but arbitrary combinations of these concepts. The input to our system is a model of what the player needs to do to complete each level, expressed as either an imperative procedure for producing solutions or a representation of features common to all solutions. The output is a progression of levels that can be adjusted by changing high-level parameters. We apply our framework to a popular math puzzle game and present results from 2,377 players showing that our automatic level progression is comparable to expert-crafted progression after a few design iterations based on a key engagement metric.
This paper presents a semi-automated methodology for generating geometric proof problems of the kind found in a high-school curriculum. We formalize the notion of a geometry proof problem and describe an algorithm for generating such problems over a user-provided figure. Our experimental results indicate that our problem generation algorithm can effectively generate proof problems in elementary geometry. On a corpus of 110 figures taken from popular geometry textbooks, our system generated an average of about 443 problems per figure in an average time of 4.7 seconds per figure. Copyright © 2014, Association for the Advancement of Artificial Intelligence.
This paper presents intial work on a system that bridges from robust, broad-coverage natural language processing to precise semantics and automated reasoning, focusing on solving logic puzzles drawn from sources such as the Law School Admission Test (LSAT) and the analytic section of the Graduate Record Exam (GRE). We highlight key challenges, and discuss the representations and performance of the prototype system.
Sketching is a synthesis methodology that aims to bridge the gap between a programmer’s high-level insights about a problem and the computer’s ability to manage low-level details. In sketching, the programmer uses a partial program, a sketch, to describe the desired implementation strategy, and leaves the low-level details of the implementation to an automated synthesis procedure. In order to generate an implementation from the programmer provided sketch, the synthesizer uses counterexample-guided inductive synthesis (CEGIS). Inductive synthesis refers to the process of generating candidate implementations from concrete examples of correct or incorrect behavior. CEGIS combines a SAT-based inductive synthesizer with an automated validation procedure, a bounded model-checker, that checks whether the candidate implementation produced by inductive synthesis is indeed correct and to produce new counterexamples. The result is a synthesis procedure that is able to handle complex problems from a variety of domains including ciphers, scientific programs, and even concurrent data-structures.
Spot-the-difference is a type of puzzles, where users try to find the different parts of two perceptually similar but actually different images. We propose a semi-automatic system to produce various spot-the-difference puzzle images tagged with their difficulties from a single input image. First, we extract regions to modify from the input by our modified maximal similarity-based region merging algorithm with little user intervention and then apply a variety of image editing techniques for each region to create a modified image. We evaluate the difficulty of a pair of the input and the modified images by considering the saliency and the perceptual difference of the modified region. We provide an empirical model to estimate the time to solve a pair of the images with respect to its difficulty. We show our experimental results and quantitative user research results to evaluate the effectiveness of the proposed method.
We describe a framework that combines deductive, numeric, and inductive reasoning to solve geometric problems. Applications include the generation of geometric models and animations, as well as problem solving in the context of intelligent tutoring systems.
Our novel methodology uses (i) deductive reasoning to generate a partial program from logical constraints, (ii) numerical methods to evaluate the partial program, thus creating geometric models which are solutions to the original problem, and (iii) inductive synthesis to read off new constraints that are then applied to one more round of deductive reasoning leading to the desired deterministic program. By the combination of methods we were able to solve problems that each of the methods was not able to solve by itself.
The number of nondeterministic choices in a partial program provides a measure of how close a problem is to being solved and can thus be used in the educational context for grading and providing hints.
We have successfully evaluated our methodology on 18 Scholastic Aptitude Test geometry problems, and 11 ruler/compass-based geometry construction problems. Our tool solved these problems using an average of a few seconds per problem.
In this paper, we are proposing a method which is different from many practical computer programs have been developed to exhibit useful types of learning. For problems such as speech recognition, different algorithms based on machine learning outperform all other approaches that have been attempted to date. In the field known as data mining, machine learning algorithms are being used commonly to discover valuable knowledge from large commercial databases containing equipment maintenance records, loan applications, financial transactions, medical records etc. Thus, it seems inevitable that machine learning will play an integral role in computer science and computer technology. In this paper, modeling and designing of a general learning system is proposed that presents new machine learning procedures used to arrive at "knowledgeable" static evaluators for checker board positions. The static evaluators are compared with each other, and with the linear polynomial using two different numerical indices reflecting the extent to which they agree with the choices of checker experts in the course of tabulated book games. The new static evaluators are found to perform about equally well, despite the relative simplicity of the second; and they perform noticeably better than the linear polynomial.
I propose to consider the question, “Can machines think?”♣ This should begin with definitions of the meaning of the terms “machine” and “think”. The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous. If the meaning of the words “machine” and “think” are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, “Can machines think?” is to be sought in a statistical survey such as a Gallup poll.
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