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Numbo: a study in cognition and recognition
... RPM has been long considered an interesting benchmark for abstract reasoning systems, along with Bongard problems [2], Hofstadter's analogies [3], Numbo [14], or Sudoku, to name a few. The advent of deep learning only intensified this interest, with an outpour of studies proposing various architectures and learning approaches. ...
One of the challenges in learning to perform abstract reasoning is that problems are often posed as monolithic tasks, with no intermediate subgoals. In Raven Progressive Matrices (RPM), the task is to choose one of the available answers given a context, where both contexts and answers are composite images featuring multiple objects in various spatial arrangements. As this high-level goal is the only guidance available, learning is challenging and most contemporary solvers tend to be opaque. In this study, we propose a deep learning architecture based on the transformer blueprint which, rather than directly making the above choice, predicts the visual properties of individual objects and their arrangements. The multidimensional predictions obtained in this way are then directly juxtaposed to choose the answer. We consider a few ways in which the model parses the visual input into tokens and several regimes of masking parts of the input in self-supervised training. In experimental assessment, the models not only outperform state-of-the-art methods but also provide interesting insights and partial explanations about the inference. The design of the method also makes it immune to biases that are known to exist in some RPM benchmarks.
... Despite the fact that games based on four-operations are very common, the limited number of scientific studies have been conducted in this regard. Defays [2], [3] used artificial intelligence search methods to solve these games. Hutton [4] developed a simple but ineffective functional program for solving four-operations problems. ...
Combination problems are one of the most important issues of
probability theory. The four-operations combination problem underlies
the basis of some competition programs broadcasted in many national
channels. In these competition programs, the competitors are expected
to reach the target number by using six numbers and four basic
arithmetic operators. The numbers are used at most once, the operators
can be used any desired number to reach the target number. In this
problem, all four-operations combinations include the operation blocks
consisting of two numbers and an operator. Therefore, the fouroperations combination problem is solved by developing a "Type-2 Tree
Structure" which is a new approach to accurately model the operation
blocks. The performance of the proposed method for the four-operations
combination problem is examined by a simulation study. Also, the
statistics from experimental results are given in this study.
... Other relevant models include those devised by Lenat and Brown (1984) and by Fajtlowicz (1988Fajtlowicz ( , 1989 to model mathematicians' conjecturing and those that Siegler and his colleagues (Siegler & Jenkins, 1989) are developing with regard to children's strategy choice in arithmetic. Defays' (1988Defays' ( , 1990 computer program Numbo is probably the most relevant to the present study. It plays a game involving the construction of a given number (the "target") from a given set of five other numbers (the "bricks"), using the arithmetical operations of addition, multiplication, and subtraction. ...
Selected groups of 44 academic ''pure'' mathematicians, 44 accountants, 44 psychology students, and 44 English students were given Levine's (1982) computational estimation task, which involved mentally estimating the products and quotients of 20 multiplication and division problems and describing their strategies. The mathematicians were the most accurate estimators, and the English students the least accurate, with psychology students and accountants obtaining similar scores intermediate between the other groups. All groups demonstrated an impressively versatile use of appropriate strategies. The mathematicians and accountants used significantly larger numbers of appropriate strategies than the other groups and strongly resembled one another in this respect, despite the significantly greater accuracy of the mathematicians. All the non-mathematician groups used significantly larger numbers of inappropriate strategies than did the mathematicians. We discuss (1) the implications for cognitive psychology of the great variability of strategy use in an apparently simple task; and (2) the relationship between people's mathematical knowledge and experience and their estimation accuracy and strategy variability.
The November 1997 issue of Index magazine featured a rather unusual piece by avantgardist John Jesurun, which apparently had surprised even the editors – and this despite having commissioned the contribution themselves. Building on the already troubled conversion from James M. Cain’s 1941 novel Mildred Pierce to the 1945 film produced by Jerry Wald via multiple screenwriters and many more rewrites, this essay approaches the theme of betrayal so conspicuous in both works less from the narrative angle than from a processual angle inspired by the principle of incommensurability. To this end, it juxtaposes the ‘classical’ adaptation from the Hollywood studio era with Jesurun’s experimental reimagining of the betrayal theme as a homology-based remodelling.
A cousin concept of dramaturgy, the notion of ‘mediaturgy’ developed by performance scholar Bonnie Marranca reroutes connotations
from a text-based linear progression of sorts to a media-induced sense of simultaneity as organizing principle. To ‘mediaturge’/director John Jesurun, the latter has led in our contemporary networked societies
to ‘troubled tensions’ between traditional conceptions of ‘meaning’ and an increasing awareness of the processes that bring
them about. Mediaturgies, accordingly, are artistic creations concerned with staging the brittle balance between formal complexity
and processual logic. However, beyond this self-reflexivity the concept equally evokes similar tensions with the notion of
adaptation. While retracing the adaptation history of John Jesurun’s 1990-production Everything That Rises Must Converge, it will accordingly be argued that juxtaposing the notions of ‘mediaturgy’, ‘adaptation’, and ‘convergence’ offers an integrative
framework that allows for reflection across formal distinctions by focusing on analogies as unifying agents.
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