Are There Enough Injective Sets?

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.     

Individuation and reference in memory: Proper names and definite descriptions

A theory is presented which assumes that individuals are represented by unique nodes in memory. To test the theory, simple facts were predicated of an individual person. Some facts referred to him by proper name, and other facts by his profession. In a before condition, subjects learned that the profession and name referred to the same individual before learning the facts, while in an after condition, they learned the identity after learning the facts. Subsequent to learning the facts and identities, subjects verified sentences based on what they had learned. Verification latencies indicated that in the before condition, one memory node was created to represent the individual, but two nodes were set up in the after condition. Assymmetries between proper names and professions indicate that the two types of referring expressions are treated differently in long-term memory.     

Testing protects against proactive interference in face–name learning

Learning face–name pairings at a social function becomes increasingly more difficult the more individuals one meets. This phenomenon is attributable to proactive interference—the negative influence of prior learning on subsequent learning. Recent evidence suggests that taking a memory test can alleviate proactive interference in verbal list learning paradigms. We apply this technique to face–name pair learning. Participants studied four lists of 12 face–name pairings and either attempted to name the 12 faces just studied after every list or did not. Recall attempts after every list improved learning of the fourth list by over 100%. Moreover, no reduction in learning of face–name pairings occurred from list 1 to list 4 for participants who attempted to name studied faces between lists. These results suggest that testing oneself on the names of a group of new acquaintances before moving on to the next group is an effective mnemonic technique for social functions.     

On the logic of testing the independence assumption in the process-dissociation procedure

Theprocess-dissociation procedure is designed to provide quantitative estimates of the influence of explicit and implicit memory in a variety of tasks. The procedure relies on the assumption that these two forms of memory produce independent influences on performance. Prior investigators have attempted to test this assumption by determining whether the parameter representing the influence of implicit memory (denotedA) is constant across experimental conditions. I argue that the constancy of A cannot provide an appropriate test of the independence assumption, because (1) the prediction of constancy can be generated without the assumption of independence, obviating the need to posit independence; and (2) the constancy ofA does not necessarily imply independence, even if one assumes that a dependency hypothesis, supplemented by ancillary assumptions (Curran & Hintzman, 1995), predicts differences inA. I close by emphasizing that we can test the independence assumption by using standard procedures that compare the fit of a model that assumes independence with the fit of a model that assumes dependence.     

Recognition memory with little or no remembering: Implications for a detection model

Remembering and knowing are two states of awareness that reflect autonoetic and noetic consciousness. Recent extensions of signal detection theory have attempted to fitremember andknow responses, which measure these states of awareness, to a continuum of trace strength or familiarity. The model assumes there are two response criteria, a remembering criterion, which is more strict, and a recognition criterion, which is more lenient and leads to any positive recognition response. The most important prediction of this model is that bias-free estimates of memory should be the same whether derived from overall hit and false alarm rates or fromremember hit and false alarm rates. We describe evidence that disconfirms this prediction and discuss other findings that the model cannot accommodate.     

Models for ordinal hierarchical classes analysis

This paper proposes an ordinal generalization of the hierarchical classes model originally proposed by De Boeck and Rosenberg (1998). Any hierarchical classes model implies a decomposition of a two-way two-mode binary arrayM into two component matrices, called bundle matrices, which represent the association relation and the set-theoretical relations among the elements of both modes inM. Whereas the original model restricts the bundle matrices to be binary, the ordinal hierarchical classes model assumes that the bundles are ordinal variables with a prespecified number of values. This generalization results in a classification model with classes ordered along ordinal dimensions. The ordinal hierarchical classes model is shown to subsume Coombs and Kao's (1955) model for nonmetric factor analysis. An algorithm is described to fit the model to a given data set and is subsequently evaluated in an extensive simulation study. An application of the model to student housing data is discussed.     

Three ideal observer models for rule learning in simple languages

Children learning the inflections of their native language show the ability to generalize beyond the perceptual particulars of the examples they are exposed to. The phenomenon of “rule learning”—quick learning of abstract regularities from exposure to a limited set of stimuli—has become an important model system for understanding generalization in infancy. Experiments with adults and children have revealed differences in performance across domains and types of rules. To understand the representational and inferential assumptions necessary to capture this broad set of results, we introduce three ideal observer models for rule learning. Each model builds on the next, allowing us to test the consequences of individual assumptions. Model 1 learns a single rule, Model 2 learns a single rule from noisy input, and Model 3 learns multiple rules from noisy input. These models capture a wide range of experimental results—including several that have been used to argue for domain-specificity or limits on the kinds of generalizations learners can make—suggesting that these ideal observers may be a useful baseline for future work on rule learning.     

A Bayesian Theory of Sequential Causal Learning and Abstract Transfer

Two key research issues in the field of causal learning are how people acquire causal knowledge when observing data that are presented sequentially, and the level of abstraction at which learning takes place. Does sequential causal learning solely involve the acquisition of specific cause‐effect links, or do learners also acquire knowledge about abstract causal constraints? Recent empirical studies have revealed that experience with one set of causal cues can dramatically alter subsequent learning and performance with entirely different cues, suggesting that learning involves abstract transfer, and such transfer effects involve sequential presentation of distinct sets of causal cues. It has been demonstrated that pre‐training (or even post‐training) can modulate classic causal learning phenomena such as forward and backward blocking. To account for these effects, we propose a Bayesian theory of sequential causal learning. The theory assumes that humans are able to consider and use several alternative causal generative models, each instantiating a different causal integration rule. Model selection is used to decide which integration rule to use in a given learning environment in order to infer causal knowledge from sequential data. Detailed computer simulations demonstrate that humans rely on the abstract characteristics of outcome variables (e.g., binary vs. continuous) to select a causal integration rule, which in turn alters causal learning in a variety of blocking and overshadowing paradigms. When the nature of the outcome variable is ambiguous, humans select the model that yields the best fit with the recent environment, and then apply it to subsequent learning tasks. Based on sequential patterns of cue‐outcome co‐occurrence, the theory can account for a range of phenomena in sequential causal learning, including various blocking effects, primacy effects in some experimental conditions, and apparently abstract transfer of causal knowledge.     

A simple computational algorithm of model-based choice preference

A broadly used computational framework posits that two learning systems operate in parallel during the learning of choice preferences—namely, the model-free and model-based reinforcement-learning systems. In this study, we examined another possibility, through which model-free learning is the basic system and model-based information is its modulator. Accordingly, we proposed several modified versions of a temporal-difference learning model to explain the choice-learning process. Using the two-stage decision task developed by Daw, Gershman, Seymour, Dayan, and Dolan (2011), we compared their original computational model, which assumes a parallel learning process, and our proposed models, which assume a sequential learning process. Choice data from 23 participants showed a better fit with the proposed models. More specifically, the proposed eligibility adjustment model, which assumes that the environmental model can weight the degree of the eligibility trace, can explain choices better under both model-free and model-based controls and has a simpler computational algorithm than the original model. In addition, the forgetting learning model and its variation, which assume changes in the values of unchosen actions, substantially improved the fits to the data. Overall, we show that a hybrid computational model best fits the data. The parameters used in this model succeed in capturing individual tendencies with respect to both model use in learning and exploration behavior. This computational model provides novel insights into learning with interacting model-free and model-based components.     

The influence of hierarchy on probability judgment

Consider the task of predicting which soccer team will win the next World Cup. The bookmakers may judge Brazil to be the team most likely to win, but also judge it most likely that a European rather than a Latin American team will win. This is an example of a non-aligned hierarchy structure: the most probable event at the subordinate level (Brazil wins) appears to be inconsistent with the most probable event at the superordinate level (a European team wins). In this paper we exploit such structures to investigate how people make predictions based on uncertain hierarchical knowledge. We distinguish between aligned and non-aligned environments, and conjecture that people assume alignment. Participants were exposed to a non-aligned training set in which the most probable superordinate category predicted one outcome, whereas the most probable subordinate category predicted a different outcome. In the test phase participants allowed their initial probability judgments about category membership to shift their final ratings of the probability of the outcome, even though all judgments were made on the basis of the same statistical data. In effect people were primed to focus on the most likely path in an inference tree, and neglect alternative paths. These results highlight the importance of the level at which statistical data are represented, and suggest that when faced with hierarchical inference problems people adopt a simplifying heuristic that assumes alignment.     

Hierarchically organized behavior and its neural foundations: a reinforcement learning perspective

Research on human and animal behavior has long emphasized its hierarchical structure—the divisibility of ongoing behavior into discrete tasks, which are comprised of subtask sequences, which in turn are built of simple actions. The hierarchical structure of behavior has also been of enduring interest within neuroscience, where it has been widely considered to reflect prefrontal cortical functions. In this paper, we reexamine behavioral hierarchy and its neural substrates from the point of view of recent developments in computational reinforcement learning. Specifically, we consider a set of approaches known collectively as hierarchical reinforcement learning, which extend the reinforcement learning paradigm by allowing the learning agent to aggregate actions into reusable subroutines or skills. A close look at the components of hierarchical reinforcement learning suggests how they might map onto neural structures, in particular regions within the dorsolateral and orbital prefrontal cortex. It also suggests specific ways in which hierarchical reinforcement learning might provide a complement to existing psychological models of hierarchically structured behavior. A particularly important question that hierarchical reinforcement learning brings to the fore is that of how learning identifies new action routines that are likely to provide useful building blocks in solving a wide range of future problems. Here and at many other points, hierarchical reinforcement learning offers an appealing framework for investigating the computational and neural underpinnings of hierarchically structured behavior.     

The shortest possible length of the longest implicational axiom

A four-valued matrix is presented which validates all theorems of the implicational fragment, IF, of the classical sentential calculus in which at most two distinct sentence letters occur. The Wajsberg/Diamond-McKinsley Theorem for IF follows as a corollary: every complete set of axioms (with substitution and detachment as rules) must include at least one containing occurrences of three or more distinct sentence letters.Additionally, the matrix validates all IF theses built from nine or fewer occurrences of connectives and letters. So the classic result of Jaskovski for the full sentential calculus —that every complete axiom set must contain either two axioms of length at least nine or else one of length at least eleven—can be improved in the implicational case: every complete axiom set for IF must contain at least one axiom eleven or more characters long.Both results are best possible, and both apply as well to most subsystems of IF, e.g., the implicational fragments of the standard relevance logics, modal logics, the relatives of implicational intutionism, and logics in the ukasiewicz family.Earlier proofs of these results, utilizing a five-valued matrix built from the product matrix of C2 with itself via the method of [8], were obtained in 1988 while the author was a Visiting Research Fellow at the Automated Reasoning Project, Research School of Social Sciences, Australian National University, and were presented in [9]. The author owes thanks to the RSSS for creating the Project, and to the members of the Project generally for the stimulating atmosphere they created in turn, but especially to Robert K. Meyer for making the visit possible, and for many discussions over the years.     

Retrieval of propositional information from long-term memory

Three experiments are reported in which subjects learn propositions like A hippie is in the park. The experiments manipulate the number of such propositions involving a particular person (e.g., hippie) or a particular location (e.g., park). After learning the material, subjects are asked to judge whether particular probe propositions are from the study set. Times to make these judgments about probe propositions increase with the number of study propositions involving the person or location used in the probe proposition. A model is presented which assumes a subject simultaneously accesses memory from all concepts in a probe proposition and serially searches through all study propositions involving each concept. Search of memory terminates as soon as one search process from a concept finds the probe proposition or exhausts the study propositions attached to that concept.     

Optimal Hierarchical Learning Path Design With Reinforcement Learning

E-learning systems are capable of providing more adaptive and efficient learning experiences for learners than traditional classroom settings. A key component of such systems is the learning policy. The learning policy is an algorithm that designs the learning paths or rather it selects learning materials for learners based on information such as the learners’ current progresses and skills, learning material contents. In this article, the authors address the problem of finding the optimal learning policy. To this end, a model for learners’ hierarchical skills in the E-learning system is first developed. Based on the hierarchical skill model and the classical cognitive diagnosis model, a framework to model various mastery levels related to hierarchical skills is further developed. The optimal learning path in consideration of the hierarchical structure of skills is found by applying a model-free reinforcement learning method, which does not require any assumption about learners’ learning transition processes. The effectiveness of the proposed framework is demonstrated via simulation studies.     

Economically organized hierarchies in WordNet and the Oxford English Dictionary

Good definitions consist of words that are more basic than the defined word. There are, however, many ways of satisfying this desideratum. For example, at one extreme, there could be a small set of atomic words that are used to define all other words; i.e., there would be just two hierarchical levels. Alternatively, there could be many hierarchical levels, where a small set of atomic words is used to define a larger set of words, and these are, in turn, used to define the next hierarchically higher set of words, and so on to the top-level of very specific, complex words. Importantly, some possible organizations are more economical than others in the amount of space required to record all the definitions. Here I ask, How economical are dictionaries? I present a simple model for an optimal set of definitions, predicting on the order of seven hierarchical levels. I test the model via measurements from WordNet and the Oxford English Dictionary, and find that the organization of each possesses the signature features expected for an economical dictionary.     

一个含有原子的自然模型∑（A）

1989年A.Blass和A.Scedrov构造了含有原子的模型V（A）（A是所有原子的集合,参见文献[1]）并证明了V（A）是ZFA（ZFA=ZF＋A,公理A断言：存在所有原子的集合）的模型。由于集合论的公理系统GB是ZF的一个保守扩充,因此,集合论的公理系统GBA（GBA=GB＋A,其中GB是集合论的含有集合和类的哥德尔－贝奈斯公理系统）也是ZFA的一个保守扩充。本文的目的是在集合论的含有原子和集合的公理系统ZFA的自然模型V（A）的基础上,为集合论的含有原子、集合和类的公理系统GBA建立模型。因此,我们首先介绍了A.Blass和A.Scedrov的含有原子的模型V（A）;第二,给出并证明V（A）具有的一些基本性质;第三,扩充了集合论的公理系统ZFA的形式语言LZFA并定义含有原子和集合的类C;第四,构造含有原子、集合和类的模型∑（A）,称它为自然模型,最后,证明了∑（A）是GBA的模型。     

Dimensionality of the Latent Structure and Item Selection Via Latent Class Multidimensional IRT Models

With reference to a questionnaire aimed at assessing the performance of Italian nursing homes on the basis of the health conditions of their patients, we investigate two relevant issues: dimensionality of the latent structure and discriminating power of the items composing the questionnaire. The approach is based on a multidimensional item response theory model, which assumes a two-parameter logistic parameterization for the response probabilities. This model represents the health status of a patient by latent variables having a discrete distribution and, therefore, it may be seen as a constrained version of the latent class model. On the basis of the adopted model, we implement a hierarchical clustering algorithm aimed at assessing the actual number of dimensions measured by the questionnaire. These dimensions correspond to disjoint groups of items. Once the number of dimensions is selected, we also study the discriminating power of every item, so that it is possible to select the subset of these items which is able to provide an amount of information close to that of the full set. We illustrate the proposed approach on the basis of the data collected on 1,051 elderly people hosted in a sample of Italian nursing homes.     

Learning to learn during visual discrimination in group housed dwarf goats (Capra hircus)

Using an automated learning device, we investigated "learning to learn" by dwarf goats (Capra hircus) in what was for them a familiar environment and normal social settings. Nine problems, each consisting of four discriminable black symbols, each with one S-super+ and three different S-super(-), were presented on a computer screen. Mean daily learning success improved over the course of the first four problems, and the improvement was maintained throughout the remaining five problems. The number of trials to reach the learning criterion decreased significantly beginning with problem four. Such results may be interpreted as evidence that the goats were developing a learning set. In the present case, the learning set appeared to have two components. One involved gaining familiarity and apparent understanding of the learning device and the basic requirements of the discrimination task. The second component involved learning potential error factors to be ignored, as well as learning commonalities that carried over from one problem to the next. Among the error factors, evidence of apparent preferences for specific symbols was seen, which had a predictable effect on performances.