Merge does not trigger a n + 1 recursive function: A reply to Mendivil-Giró (2025)

Do children use language structure to discover the recursive rules of counting?

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

Basic Structures of Systems

This paper considers the question of formal complexity measurement within linguistic minimalism and argues that information structure properties should not be encoded in narrow syntax as features triggering movement, suggesting that the relevant information is established at the interfaces. Also, it argues for a minimalist model of grammar in which complexity arises out of the cyclic interaction of subderivations. This model allows one to rephrase the question of the formal complexity of a generative grammar, such that a different answer is forthcoming depending on whether we consider the grammar as a whole, or just narrow syntax. The grammar as a whole, including interface components in addition to narrow syntax, as well as recursive interaction among subderivations, is vastly more complicated than a finite-state grammar, but there is no reason for concluding that narrow syntax is not simply finite-state.

When children first learn to count, what do they understand about the structure of the count system? The present study investigated English-speaking children’s ability to generalize the rules that structure their count list to novel contexts. A total of N = 86 children (3;0 – 6;11) completed a battery of tasks aimed at measuring their understanding of the English count list: they counted as high as they could, and were asked to generate successors to English numbers (e.g., “Fifty-seven: what comes next?”). Next, they were introduced to novel decade terms, and were asked to generate successors to numbers containing those terms (e.g., “Blicky-seven: what comes next?”). Children’s ability to generate successors was predicted by their counting ability, and a sizeable subset of children were able to generate successors both for novel numbers and for English numbers outside their productive count range. These data suggest that emerging counters can use their understanding of the structure of the English count list to generate successors to unfamiliar numbers.

A central goal of cognitive neuroscience is to understand the computational properties of neural representations in visual cortex. Above and beyond the information content or visual features encoded in neural populations, we wish to understand the encoding format itself and the computations it subserves. These questions are at the heart of research into computational properties such as sparsity, dimensionality, manifold geometry, invariance, and dynamics in both biological and artificial neural networks. Here we investigate the computational properties of representations across human visual cortex using fMRI encoding models. We first fit voxelwise encoding models to predict fMRI responses to images of natural objects and scenes using both hand-engineered feature detectors (e.g. edge detectors) and pre-trained and untrained convolutional neural networks (CNNs). We then performed statistical analyses of the encoding model regressors to explore how performance is affected by several computational properties, including mixed selectivity and intrinsic dimensionality. We searched for the properties that correlated best with encoding accuracy. We find evidence for high intrinsic dimensionality as an important driver of encoding model performance across the ventral and dorsal visual streams. Furthermore, we find that several mechanisms for increasing intrinsic dimensionality lead to similar improvements in encoding model performance, including division normalization, multiplicative interaction, random nonlinear projection, and supervised learning for image classification. Thus, it appears that intrinsic dimensionality itself best explains the improvements of these varied representational mechanisms. Together, our work explores a methodology for investigating the computational properties of neural representations through an ensemble of encoding models and statistical analyses of their computational properties. We use this approach to identify high intrinsic dimensionality as a key predictor of encoding model performance, suggesting that biological vision relies on representations that are highly efficient, and may allow for the linear separability of a large number of visual properties across different tasks.

Chomsky 1993, 1995, 2000, 2001a, 2001b, following Hale and Keyser 1993, assumes that θ-roles are assigned in a certain structural configuration at logical form (LF), to be referred to as the configurational theory of θ-role licensing (CTTL). This approach eliminates the relevance of argument (θ-) structure within narrow syntax (NS), thus rendering D-structure superfluous. This study examines how argument (θ-) structure and other base properties (traditionally associated with D-structure) are captured, without D-structure, in minimalism. While Chomsky advocates the CTTL, θ-theory appears to covertly play a crucial role in NS: the properties associated with D-structure are not fully subsumed under LF. Hence, the base properties cannot be fully assumed by the conceptual-intentional interface.

Unsolvable Problems in the Theory of Computable Numbers

No syntax saltation in language evolution

A long tradition takes word order differences within one and the same language and across languages to be a consequence of movement. More recently Chomsky has made the suggestion that, given the architecture of UG, movement in narrow syntax should have interpretive effects at both interfaces (LF and PF). Any putative movement that has an effect on the PF side but no effect on the LF side should thus be expunged from narrow syntax, with the consequence that any word order difference devoid of LF effects should be confined to the PF interface. This requires an algorithm at PF that converts hierarchical structures to linear order. To assume as an algorithm the apparent fact that the physics of speech leaves just two options (a head either precedes or follows its complements) seems far from being sufficient in that it falls short of accounting for a number of linear order generalizations. Here it is argued that some meaningless movement must be assumed if we want to capture these generalizations and provide a theory of externalization based on the Linear Correspondence Axiom, an algorithm compatible with these generalizations.

Hierarchical structures are commonly observed in nature and possess unique properties. The fabrication of hierarchical structures with well-controlled sizes in different length scales, however, is still a great challenge. To further understand the morphologies and properties of the hierarchical structures, here we present a novel strategy to prepare hierarchical polymer structures by combining the modified breath figure method and the template method. Poly(methyl methacrylate) (PMMA) honeycomb films with regular micropores are first prepared using the modified breath figure method by dipping PMMA films into mixtures of chloroform and methanol. The polymer chains on the honeycomb films are then annealed and wetted into the nanopores of anodic aluminum oxide templates via capillary forces, resulting in the formation of hierarchical polymer structures. The morphologies of the polymer structures, which can be controlled by the molecular weights of the polymers and the concentrations of the polymer solutions, are characterized by scanning electron microscopy. The surface wettabilities of the polymer structures are also examined by water contact angle measurements, and the hierarchical structures are observed to be more hydrophobic than the flat films and honeycomb films. This work not only provides a feasible approach to fabricate hierarchical polymer structures with controlled sizes but also gives a better understanding of the relationship between surface morphologies and properties.

Offshore structures should have a high degree of structural safety even under extreme enviromental loadings. The authors have studied a composite steel-concrete structure of sandwich system for the strength members of huge offshore structures, where concrete is placed between steel plates.In the previous papers, the authors carried out both experimental and theoretical investigations into the strength of the composite structure. It was clarified that the composite structure has various excelent properties: The ultimate load-bearing capacity of the composite structure is very high and it can absorb a great deal of energy until failure under any type of loading conditions.In the present paper, the authors carried out both experimental and theoretical investigations into the dynamic toughness of the sandwich composite structures. Experiments were carried out using the two dimensional models of composite structure under high speed loading. A nonlinear analysis was developed to predict the toughness of sandwich beam under dynamic load, where the material non linearities of both concrete and steel with strain rate were taken into account. The nonlinear analysis accurately represented the behavior and toughness of the sandwich beam structure.

In this work, one three-dimensional periodic composite structure consisting of alumina frame and inscribed nickel spheres is proposed for designing the micron composite porous structure. The finite-difference time domain method is applied for predicting the radiation spectra of proposed composite structure and distribution of absorbed radiative power. The spectral characteristics of composite structure are compared with the hollow ceramic frame structure and stacked metal spheres for analysis of radiative properties of discussed micron composite porous structure, then the effects by changing the number of layers and inner structure are summarized. Strong spectral selectivity is observed, while radiative properties of composite structure are harmonic results of hollow ceramic frame structure and stacked metal spheres. The absorptivity changes with increasing layers hardly, while the reflectivity and transmissivity are affected by number of layers obviously until the number of layers reaches three. When the number of layers exceeds three, the effects of increasing layers are ignorable. Besides, effects on the distribution of absorbed power inside composite porous structure are produced by upper two layers mainly. Furthermore, the radiation spectra are changed by placing hollow ceramic frame layers on the composite structure layer. The proposed composite structure and prediction of radiative properties can be helpful for designing the structure of micron porous materials and regulating the radiative transfer.

Compounding is the most common way in which languages can create new words from existing ones. The investigation of how compound words are represented and processed in the mind can therefore reveal fundamental properties of lexical representation and processing. Because of the properties of its morphology and its writing system, Chinese offers a special opportunity for such investigation. In Chinese, it is common for compound words to contain three or more constituents, which are organized into hierarchical structures. Using a lexical decision task, we examined whether the arrangement of such sublexical hierarchies within triconstituent compound words would affect their ease of visual processing. We found that left-branching structures were processed more accurately and more quickly than right-branching ones (left-branching triconstituent compounds are those in which an embedded two-character compound is the modifier and a single character is as the head, as in the English left-branching structure hot dog bun). This processing advantage of left-branching structures was supported by the results of a metalinguistic segmentation task in which participants showed left-branching preferences when presented with special triconstituent compounds which have no feasible branching structure (loanwords and flat structure compounds). In the Chinese character-based writing system, left-branching and right-branching compounds are written exactly the same way. This is in contrast to English in which it is common for the branching direction of a triconstituent compound to be indicated by the presence or absence of a space within the compound structure (e.g., watchdog group, consumer watchdog). Thus, the results from Chinese support the view that hierarchical constituent structure plays a role in processing in the absence of orthographic support and that the observed left-branching preference reflects central aspects of lexical representation and processing.

Electrochemical deposition has continued to grow as an important industry tool not only for depositing traditional metal structures, but also for manufacturing more complex composite thin films. Previous work has demonstrated the ability to manufacture metal matrix composites with precisely controlled oxide nanoparticle incorporation using a rotating disk electrode (RDE) setup. In this method the speed of the RDE is used to directly control the rate of particle incorporation. It is feasible to manufacture higher order composite structures using this novel technique by simply varying the rotation rate of the RDE during deposition. Specific attention is given to periodically layered composite structures. These hierarchical structures are demonstrated with nanometer scale interface resolution and sub-micron to millimeter characteristic length scales (layer thicknesses). These ordered composites should yield unique mechanical properties compared to those of randomly dispersed composite structures, and have immediate industry applications.