Merging Separately Established Functional Equivalence Classes

The present study examined the effects of including stimuli previously trained as members of functional classes or equivalence classes on subsequent equivalence class formation, and isolated the effects of preliminary training from those of the acquired function stimuli. Fifty-six adults were assigned to 1 of 5 conditions. The control group (CONT) received no preliminary training prior to the terminal phase. Participants in the other 4 groups learned two 3-member functional classes and two 3-member equivalence classes during the preliminary phase. The terminal equivalence phase trained two 5-member classes (A → B → C → D → E) comprising abstract forms; the C stimuli in the terminal phase were (a) from the preliminary functional classes for 1 group (ACQ-F), (b) from the preliminary equivalence classes for the second experimental group (ACQ-E), (c) pictures of everyday objects for the picture control group (PIC), and (d) novel, unfamiliar stimuli for the preliminary training control group (PRE-CONT). Class formation yields were 100% in the PIC condition and 11% in the CONT condition; however, low yields in the PRE-CONT, ACQ-F, and ACQ-E conditions were unexpected, suggesting that procedural variables in preliminary training account for more of the subsequent effects on class formation than the stimulus control properties of the acquired function stimuli.

The simultaneous matching-to-sample procedures that are widely used to study stimulus equivalence in human participants have generally been unsuccessful in animals. However, functional equivalence classes have been demonstrated in pigeons and sea lions using a concurrent repeated reversal discrimination procedure. In this procedure, responding to one set of stimuli is reinforced but responding to a different set is not and the set associated with reinforcement is changed with multiple reversals during the experiment. The experiments reported here were designed to assess whether functional equivalence classes could be demonstrated in rats using similar techniques. Rats were initially trained with two sets of olfactory stimuli (six odors/set). Following many reversals, probe reversal sessions were conducted in which rats were exposed to a subset of the members of each set and, later in the session, the withheld stimuli were introduced. Responding to these delayed probe trials in accord with the reversed contingencies constituted transfer of function. There was some evidence of transfer in Experiment 1, but the effects were relatively weak and variable. Experiment 2 introduced procedural changes and found strong evidence of transfer of function consistent with the formation of functional equivalence classes. These procedures offer a promising strategy to study symbolic behavior in rodents.

In Experiments 1 and 2 a match-to-sample procedure was used to establish the prerequisite conditional discriminations for two 3-member equivalence classes (A1, B1, C1 & A2, B2, C2) where stimuli A1 and A2 were written instructions. In Experiment 1 tests for transitivity confirmed the emergence of derived relations. Following this, an off-the-baseline test was given to verify the existence of functional equivalence classes. During off-the-baseline testing stimuli B and C from both classes controlled responding appropriate to the instructions contained in A1 and A2 respectively. In Experiment 2, the omission of tests for transitivity and the reversal of instructions contained in A1 and A2 did not impede the emergence of functional equivalence classes. In Experiment 3, classes A1, B1, C1 and E1, F1, G1 were linked via D1 to produce one class (A1, B1, C1, D1, E1, F1, G1); another two classes (A2, B2, C2 & E2, F2, G2) were linked with D2 to produce A2, B2, C2, D2, E2, F2, G2. The A and G stimuli in each class contained competing sets of instructions. When transformation of function was examined in the first class, the functions at C1 and E1 were controlled by the rules at A1 and G1 respectively; D1 controlled responding in accordance with the instructions at A1. Results from all experiments are discussed in the context of procedures designed to investigate transfer/transformation of function.

The present study investigated whether simple discrimination procedures produce emergent relations that are consistent with functional and equivalence classes. In Experiment 1, 4 normally capable adults were exposed to simple successive discrimination reversal training. Responses to the S+ (A1, B1, and C1) but not S− (A2, B2, and C2) were reinforced. The participants were then exposed to repeated-reversal training, followed by tests of emergent equivalence relations. In Experiment 2, 4 additional adults were exposed to a simple successive discrimination training procedure using differential responses. Response 1 (R1) was reinforced only when emitted in the presence of A1, B1, or C1 (A1-R1, B1-R1, or C1-R1). Response 2 (R2) was reinforced only in the presence of A2, B2, or C2 (A2-R2, B2-R2, or C2-R2). A new response was then trained in the presence of 1 member of each class (A1-R3, A2-R4). A transfer-of-function test was then conducted to verify whether the new responses would be emitted in the presence of B1, B2, C1, and C2. In both experiments, emergent conditional relations were tested using a go/no-go procedure with compound stimuli. In Experiment 1, all of the participants showed functional classes, and 3 showed patterns that are consistent with the formation of equivalence classes. In Experiment 2, all 4 participants showed patterns that are consistent with functional and equivalence class formation. In these experiments, simple discrimination training generated a range of emergent stimulus relations that mirrored those that are indicative of equivalence class formation.

It has been claimed that different types of causes must be considered in biological systems, including top-down as well as same-level and bottom-up causation, thus enabling the top levels to be causally efficacious in their own right. To clarify this issue, the important distinctions between information and signs are introduced here and the concepts of information control and functional equivalence classes in those systems are rigorously defined and used to characterize when top-down causation by feedback control happens, in a way that is testable. The causally significant elements we consider are equivalence classes of lower level processes, realized in biological systems through different operations having the same outcome within the context of information control and networks.

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel’s correspondence among convex functions and Rockafellar’s extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.

Fuzzy measures or capacities are the most general representation of uncertainty functions. However, this general class has been little explored from the point of view of its information content, when degrees of uncertainty are not supposed to be numerical, and belong to a finite qualitative scale, except in the case of possibility or necessity measures. The thrust of the paper is to define an ordering relation on the set of qualitative capacities expressing the idea that one is more informative than another, in agreement with the possibilistic notion of relative specificity. To this aim, we show that the class of qualitative capacities can be partitioned into equivalence classes of functions containing the same amount of information. They only differ by the underlying epistemic attitude such as pessimism or optimism. A meaningful information ordering between capacities can be defined on the basis of the most pessimistic (resp. optimistic) representatives of their equivalence classes. It is shown that, while qualitative capacities bear strong similarities to belief functions, such an analogy can be misleading when it comes to information content.

We study counting functions on the free groups F_n and free monoids M_n for n \geq 2 , which we introduce for combinatorial approach to famous Brooks quasimorphisms on free groups. Two counting functions are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a simple graphical algorithm to determine whether two given counting functions are equivalent. In particular, this yields an algorithm to decide whether two linear combinations of Brooks quasimorphisms on F_n represent the same class in bounded cohomology.

UDC 513.88 It the article we study sublinear functionals that are defined on the space of Bochner integrable functions and possess the properties of decomposition and scalar compactness. We give an integral representation for the functionals. Such functionals are used in proving theorems on existence of continuous selections for a family of lower semicontinuous multivalued mappings with decomposable closed nonconvex values in the space of Bochner integrable functions. The article relates to the studies [1-3] and others. 1. We present basic definitions, notation, and a series of auxiliary results. Throughout what follows, X is a separable Banach space with norm H" ]l, T is a Hausdorff compact space with positive Radon measure/z and a-algebra ~ of/,-measurable subsets, and R stands for the real axis. We denote by L1 (T, X) the Banach space of equivalence classes of Bochner integrable functions u: T ---* X, with the norm I[u(.)HL1 = fJ[u(t)Hd#. T By X t and L ~ (T, X) we mean the topological dual spaces of X and L1 (T, X) with the strong (norm) topologies; (x, xl~ and (u(-), u'(.)) are the canonical bilinear forms establishing duality between X and X' and between LI(T, X) and UI(T , X) respectively. For the norm of X ~ we use the same notation as for that of X. The spaces X' and LI(T,X ) endowed with weak a(X~,X) - and a( L~ (T,X), Ll(T,X) )-topologies will be denoted by a-X' and a-L](T,X) (see [4, p. 6831). A function v : T ~ X I is called scalarly measurable if, given x E X, the scalar function (x, v(t)) is measurable. A function v : T ~ X ~ is called weakly measurable if, for every e > 0, there is a closed set Te C T, p(T \ Te) <_ e, such that the restriction of v to Te is a continuous function from T~ into a-X ~. We denote by Loo(T, X') the Banach space of equivalence classes of scalarly measurable, essentially bounded functions v : T --* X', with the norm IIv(.)Hoo = esssup Hv(t)H. Henceforth we identify lET equivalence classes in the spaces L~ (T, X) and L~ (T, X') with their representatives. It is well known (see [4, p. 807]) that U I(T,X) and Loo(T,X') are linearly isometric and that each element x'(.) e L~(T,X) is representable as (x(.),x'(.)) = f(x(t),v(t))d#, x(.) �9 LI(T,X), where T v(.) �9 Loo(T,X'). Therefore, we shall by convention regard Loo(T,X ~) as strongly dual to LI(T, X) and write (x(.),x~(.)) = f(x(t),x~(t))d#, x(.) �9 LI(T,X), considering x~(.) as an element of the space

Two experiments investigated possible interactions between existing equivalence relations among stimuli and the acquisition of simple discriminations involving the same stimuli. During the first phase of Experiment 1, two adult humans learned two sets of six conditional relations among arbitrary visual stimuli designed to produce two independent sets of three, three-member equivalence classes (Set 1 and Set 2). Subjects whose performance met accuracy criteria on tests for equivalence relations proceeded to a second phase in which subjects were taught, via reinforcement, to make the same response to a cluster of three stimuli (e.g., response 1 in the presence of A1, B1, and C1). The clusters of three stimuli were initially consistent with documented equivalence relations for both Set 1 and Set 2 stimuli. In the second half of this phase, the contingencies were realigned such that the simple discriminations being established were consistent with documented equivalence relations for Set 1 stimuli but inconsistent for Set 2 stimuli. Results showed that the simple discriminations were acquired faster for Set 1 stimuli than for Set 2 stimuli. Experiment 2 introduced a third set of stimuli as a control set and systematically replicated Experiment 1 with four subjects. This study found that three of four subjects acquired simple discriminations faster when the cluster of stimuli were drawn from within equivalence classes rather than across equivalence classes. The development of equivalence relations with Set 3 stimuli constrained the utility of the control condition in this experiment. The results are best interpreted as suggesting that existing equivalence relations among stimuli can interact with the development of analytic units such as simple discriminations to either facilitate or retard their development.

A modified version of the original Derived Relational Intrusions Following Training paradigm (DRIFT; Guinther & Dougher, Journal of the Experimental Analysis of Behavior, 93, 329–347; 2010) was used in an attempt to engineer contextually controlled semantic false memories. Participants completed a contextually controlled interrelated conditional discrimination training procedure (match to sample) without tests of symmetry or transitivity. Study list words were nominally assigned to participate in a functional equivalence class with 1 set of non-study words in 1 context but with a different set of non-study words in a different context. The study list words were then presented in 1 of the 2 contexts, between subjects, for free-recall memory testing. Subsequent transfer of the study list words’ remembering function to non-study words (i.e., false recall) was found to be partially dependent upon the context in which the free recall test was administered, thus demonstrating partial contextual control of semantic false memory.

\n O processo de envelhecimento pode gerar perdas cognitivas que, se agravadas, podem acarretar o diagnóstico de demência, sendo a Demência de Alzheimer (DA) a mais comum. A DA é uma doença neurodegenerativas que afeta principalmente a memória e a linguagem. Procedimentos de ensino baseados no paradigma de equivalência de estímulos e no ensino por exclusão, apesar de testados com sucesso em diferentes populações com problemas de desenvolvimento, têm gerado poucas pesquisas com idosos com perdas cognitivas. Esta pesquisa apresenta três estudos, com objetivos distintos, mas complementares. O Estudo 1 teve como objetivo avaliar se idosos institucionalizados sem DA formariam classes de estímulos equivalente a partir de um procedimento de ensino de relações condicionais entre palavras ditadas, fotos, nomes impressos de profissões e de graus de parentesco; adicionalmente, pretendia-se verificar que essas relações se mantinham ao longo de um mês, em três testes de manutenção. Cinco idosos institucionalizados passaram pelo ensino de relações arbitrárias, em uma estrutura de treino linear com ensino de novas relações por exclusão. Testes de equivalência foram conduzidos em seguida à Fase de Ensino, e após cinco, 15 e 30 dias. Todos os idosos demonstraram desempenho por exclusão na Fase de Ensino e aprenderam todas as relações com o número mínimo de blocos de ensino programados (exceto uma participante). Todos os idosos apresentaram formação de classes de equivalência. Quatro dos idosos apresentaram manutenção dessas classes em todos os testes de manutenção. O procedimento, portanto, mostrou-se efetivo para esses idosos e gerou formação de equivalência, apesar da sua condição de institucionalização. O Estudo 2 teve como objetivo avaliar se idosos com DA, também institucionalizados, demonstrariam desempenho por exclusão e aprenderiam relações condicionais entre quatro nomes ditados e quatro fotos, em um procedimento de ensino por exclusão. Seis idosas com diagnóstico de DA passaram por tarefas de emparelhamento ao modelo, em que as relações entre modelos ditados e fotos foram ensinadas por exclusão. Apesar de todas as idosas terem respondido por exclusão na primeira apresentação de um estímulo modelo novo e seu comparação correspondente em pelo menos duas das três tentativas em que tal desempenho era possível, apenas duas mantiveram essa relação ao longo das demais tentativas e aprenderam as relações ensinadas. Possíveis motivos para esse desempenho são discutidos. O Estudo 3 avaliou se idosos institucionalizados, com DA e comprometimento cognitivo, formariam classes de equivalência a partir de um ensino de relações arbitrárias com estrutura de treino um-para-muitos, procedimento de dica atrasada e ensino por exclusão. Quatro idosas passaram pelo ensino de relações condicionais entre estímulos visuais (nomes escritos, fotos e nomes de profissões). Todas aprenderam as relações ensinadas com o número mínimo de blocos programados, porém apresentaram dificuldade em formar classes de equivalência e manter tais relações nos testes de manutenção após cinco e oito dias. Os procedimentos de ensino empregados foram eficientes em gerar a aprendizagem das relações, mas tal aprendizagem, nessas idosas, não sustentou a formação e manutenção de classes de equivalência.\n

Equivalence, linearization, and decomposition of holomorphic operator functions

Stimulus Class Formation in Humans and Animals

Multiple problems in engineering and science have fundamental limits that can be determined by computing network coding capacity regions. Network coding capacity regions, in turn, are implicitly and exactly characterized in terms of the entropy region. For four or more random variables, the boundary of the entropy region is unknown. At four random variables, it is known that reaching the unknown part of the entropy region necessitates nonlinear code constructions. Those nonlinear constructions can, in turn, be parameterized by the supports of the their induced probability distributions. Since different supports can reach identical collections of entropy vectors, and thus the same part of the entropy region, supports are grouped into equivalence classes based on the part of the entropy region they map to. Equivalence classes can easily be formalized through group actions by encoding supports as sets of set partitions. For supports with a fixed number of random variables and a fixed number of atoms, a single representative from every equivalence class can be exhaustively listed utilizing the Leiterspiel algorithm from the theory of group actions. A structural analysis of supports, leveraging sub-modularity of a particular class of functions, is presented that enables a large number of support equivalence classes to be proven incapable of reaching the unknown part of the entropy region. The tools underlying this structural analysis can also be utilized to prove when one support generates entropy vectors which push out farther in a given direction in entropy space than another support; a relationship identified as dominance of the former support over the latter. Furthermore, proofs of violation and dominance can be generated automatically utilizing linear programs constructed from the supports. The power of the structural theory of dominance is demonstrated by providing a highly concise structural characterization of those supports that maximally violate the Ingleton inequality.