The issue of proper functioning of operative computing and the utility of program verification, both in general and of specific methods, has been discussed a lot. In many of those discussions, attempts have been made to take mathematics as a model of knowledge and certitude achieving, and accordingly infer about the suitable ways to handle computing. I shortly review three approaches to the subject, and then take a stance by considering social factors which affect the epistemic status (...) of both mathematics and computing. I use the analogy between mathematics and computing in reverse—that is to say, I consider operative computing as a form of making mathematics, and so attempt to learn from computing to mathematics in general. I conclude that mathematics engineering is a field to be both developed for practical improvement of doing mathematics and taken into consideration while philosophizing about mathematics as well. (shrink)

For regular sets in Euclidean space, previous work has identified twelve ‘basic’ computability notions to which many previous notions considered in literature were shown to be equivalent. With respect to those basic notions we now investigate on the computability of natural operations on regular sets: union, intersection, complement, convex hull, image, and pre-image under suitable classes of functions. It turns out that only few of these notions are suitable in the sense of rendering all those operations uniformly computable.

We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable (...) Schauder bases that are well-behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non-uniform result on composition with bounded linear operators from right. (shrink)

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

The functional capabilities that consciousness seems to provide to biological systems can supply valuable principles in the design of more autonomous and robust technical systems. These functional concepts keep a notable similarity to those underlying the notion of operating system in software engineering, which allows us to specialize the computer metaphor for the mind into that of the operating system metaphor for consciousness. In this article, departing from these ideas and a model-based theoretical framework for cognition, we present an architectural (...) proposal for machine consciousness, called the Operative Mind. According to it, machine consciousness could be implemented as a set of services, in an operative system fashion, based on modeling of the own control architecture, that supervise the adequacy of the system architectural structure to the current objectives, triggering and managing adaptativity mechanisms. (shrink)

The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum, product, difference K 2, n ⊖ K 2, and the conjunction of S n with K 2. All our computations have been concluded by implementation of the methods of linear algebra (...) and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article. (shrink)

We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs to solutions of Tx = u ? Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T†u, which is the standard concept of solution in the theory of inverse problems. Typically, T† is discontinuous and hence no computable mapping. However, we will use (...) effective versions of theorems from the theory of regularization to show that the mapping ↦ T†u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T† is considered as an element of an L2-space. We define suitable representations for such spaces and use the results from the first part of the paper to show that ↦ T† is computable. (shrink)

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions (...) by constants, and the degree spectra of relations. (shrink)

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, (...) we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

This paper analyses the impact of representation and search operators on Computational Complexity. A model of computation is introduced based on a directed graph, and representation and search are defined to be the vertices and edges of this graph respectively. Changing either the representation or the search algorithm leads to different possible complexity classes. The final section explores the role of representation in reducing time complexity in Artificial Intelligence.

Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.

Computer models and simulations have become, since the 1960s, an essential instrument for scientific inquiry and political decision making in several fields, from climate to life and social sciences. Philosophical reflection has mainly focused on the ontological status of the computational modeling, on its epistemological validity and on the research practices it entails. But in computational sciences, the work on models and simulations are only two steps of a longer and richer process where operations on data are as important as, (...) and even more time and energy-consuming than modeling itself. Drawing on two study cases—computational embryology and computational epidemiology—this article contributes to filling the gap by focusing on the operations of producing and re-using data in computational sciences. The different phases of the scientific and artisanal work of modelers include data collection, aggregation, homogenization, assemblage, analysis and visualization. The article deconstructs the ideas that data are self-evident informational aggregates and that data-driven approaches are exempted from theoretical work. More importantly, the paper stresses the fact that data are constructed and theory laden not only in their fabrication, but also in their reusing. (shrink)

Quantum probability (QP) theory can be seen as a type of vector symbolic architecture (VSA): mental states are vectors storing structured information and manipulated using algebraic operations. Furthermore, the operations needed by QP match those in other VSAs. This allows existing biologically realistic neural models to be adapted to provide a mechanistic explanation of the cognitive phenomena described in the target article by Pothos & Busemeyer (P&B).

This work focuses on the application of machine learning techniques to assist the clinicians in the administration of analgesic drug during general anaesthesia. Specifically, the main objective is to propose the basis of an intelligent system capable of making decisions to guide the opioid dose changes based on a new nociception monitor, the analgesia nociception index. Clinical data were obtained from 15 patients undergoing cholecystectomy surgery. By means of an off-line study, machine learning techniques were applied to analyse the possible (...) relationship between the analgesic dose changes performed by the physician due to the hemodynamic activity of the patients and the evolution of the ANI. After training different classifiers and testing the results under cross validation, a preliminary relationship between the evolution of ANI and the dosage of remifentanil was found. These results evidence the potential of the ANI as a promising index to guide the infusion of analgesia. (shrink)

Computational models of infant word‐finding typically operate over transcriptions of infant‐directed speech corpora. It is now possible to test models of word segmentation on speech materials, rather than transcriptions of speech. We propose that such modeling efforts be conducted over the speech of the experimental stimuli used in studies measuring infants' capacity for learning from spoken sentences. Correspondence with infant outcomes in such experiments is an appropriate benchmark for models of infants. We demonstrate such an analysis by applying the DP‐Parser (...) model of Algayres and colleagues to auditory stimuli used in infant psycholinguistic experiments by Pelucchi and colleagues. The DP‐Parser model takes speech as input, and creates multiple overlapping embeddings from each utterance. Prospective words are identified as clusters of similar embedded segments. This allows segmentation of each utterance into possible words, using a dynamic programming method that maximizes the frequency of constituent segments. We show that DP‐Parse mimics American English learners' performance in extracting words from Italian sentences, favoring the segmentation of words with high syllabic transitional probability. This kind of computational analysis over actual stimuli from infant experiments may be helpful in tuning future models to match human performance. (shrink)

According to structuralism in philosophy of mathematics, arithmetic is about a single structure. First-order theories are satisfied by models that do not instantiate this structure. Proponents of structuralism have put forward various accounts of how we succeed in fixing one single structure as the intended interpretation of our arithmetical language. We shall look at a proposal that involves Tennenbaum's theorem, which says that any model with addition and multiplication as recursive operations is isomorphic to the standard model of arithmetic. On (...) this account, the intended models of arithmetic are the notation systems with recursive operations on them satisfying the Peano axioms. [A]m Anfang […] ist das Zeichen. (shrink)

Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.

Computational explanations focus on information processing required in specific cognitive capacities, such as perception, reasoning or decision-making. These explanations specify the nature of the information processing task, what information needs to be represented, and why it should be operated on in a particular manner. In this article, the focus is on three questions concerning the nature of computational explanations: What type of explanations they are, in what sense computational explanations are explanatory and to what extent they involve a special, “independent” (...) or “autonomous” level of explanation. In this paper, we defend the view computational explanations are genuine explanations, which track non-causal/formal dependencies. Specifically, we argue that they do not provide mere sketches for explanation, in contrast to what for example Piccinini and Craver :283–311, 2011) suggest. This view of computational explanations implies some degree of “autonomy” for the computational level. However, as we will demonstrate that does not make this view “computationally chauvinistic” in a way that Piccinini or Kaplan :339–373, 2011) have charged it to be. (shrink)

To explain why, in scientific problem solving, a diagram can be “worth ten thousand words,” Jill Larkin and Herbert Simon (1987) relied on a computer model: two representations can be “informationally” equivalent but differ “computationally,” just as the same data can be encoded in a computer in multiple ways, more or less suited to different kinds of processing. The roots of this proposal lay in cognitive psychology, more precisely in the “imagery debate” of the 1970s on whether there are image-like (...) mental representations. Simon (1972, 1978) hoped to solve this debate by thoroughly reducing the differences between forms of mental representations (e.g., between images and sentences) to differences in computational efficiency; to carry out this reduction, he borrowed from computer science the concepts of data type and of data structure. I argue that, in the end, his account amounted to nothing more than characterizing representations by the fast operations on them. This analysis then allows me to assess what Simon’s approach actually achieves when transported from psychology to the study of scientific representations, as in Larkin and Simon (1987): it allows comparing, not representations in and of themselves, but rather the computational roles they play in particular problem-solving processes—that is, representations together with a particular way of using them. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Using P300-based brain–computer interfaces in daily life should take into account the user’s emotional state because various emotional conditions are likely to influence event-related potentials and consequently the performance of P300-based BCIs. This study aimed at investigating whether external emotional stimuli affect the performance of a P300-based BCI, particularly built for controlling home appliances. We presented a set of emotional auditory stimuli to subjects, which had been selected for each subject based on individual valence scores evaluated a priori, while they (...) were controlling an electric light device using a P300-based BCI. There were four conditions regarding the auditory stimuli, including high valence, low valence, noise, and no sound. As a result, subjects controlled the electric light device using the BCI in real time with a mean accuracy of 88.14%. The overall accuracy and P300 features over most EEG channels did not show a significant difference between the four auditory conditions. When we measured emotional states using frontal alpha asymmetry and compared FAA across the auditory conditions, we also found no significant difference. Our results suggest that there is no clear evidence to support a hypothesis that external emotional stimuli influence the P300-based BCI performance or the P300 features while people are controlling devices using the BCI in real time. This study may provide useful information for those who are concerned with the implementation of a P300-based BCI in practice. (shrink)

Currently, there is widespread skepticism that higher cognitive processes, given their apparent flexibility and globality, could be carried out by specialized computational devices, or modules. This skepticism is largely due to Fodor’s influential definition of modularity. From the rather flexible catalogue of possible modular features that Fodor originally proposed has emerged a widely held notion of modules as rigid, informationally encapsulated devices that accept highly local inputs and whose opera- tions are insensitive to context. It is a mistake, however, to (...) equate such features with computational devices in general and therefore to assume, as Fodor does, that higher cognitive processes must be non-computational. Of the many possible non-Fodorean architectures, one is explored here that offers possible solutions to computational problems faced by conventional modular systems: an ‘enzymatic’ architecture. Enzymes are computational devices that use lock-and-key template matching to iden- tify relevant information (substrates), which is then operated upon and returned to a common pool for possible processing by other devices. Highly specialized enzymes can operate together in a common pool of information that is not pre-sorted by information type. Moreover, enzymes can use molecular ‘tags’ to regulate the operations of other devices and to change how particular substrates are construed and operated upon, allowing for highly interactive, context-specific processing. This model shows how specialized, modular processing can occur in an open system, and suggests that skepti- cism about modularity may largely be due to failure to consider alternatives to the standard model. (shrink)

Computation is interpretable symbol manipulation. Symbols are objects that are manipulated on the basis of rules operating only on theirshapes, which are arbitrary in relation to what they can be interpreted as meaning. Even if one accepts the Church/Turing Thesis that computation is unique, universal and very near omnipotent, not everything is a computer, because not everything can be given a systematic interpretation; and certainly everything can''t be givenevery systematic interpretation. But even after computers and computation have been successfully distinguished (...) from other kinds of things, mental states will not just be the implementations of the right symbol systems, because of the symbol grounding problem: The interpretation of a symbol system is not intrinsic to the system; it is projected onto it by the interpreter. This is not true of our thoughts. We must accordingly be more than just computers. My guess is that the meanings of our symbols are grounded in the substrate of our robotic capacity to interact with that real world of objects, events and states of affairs that our symbols are systematically interpretable as being about. (shrink)

The development of technology is unbelievably rapid. From limited local networks to high speed Internet, from crude computing machines to powerful semi-conductors, the world had changed drastically compared to just a few decades ago. In the constantly renewing process of adapting to such an unnaturally high-entropy setting, innovations as well as entirely new concepts, were often born. In the business world, one such phenomenon was the creation of a new type of entrepreneurship. This paper proposes a new academic discipline (...) of computational entrepreneurship, which centers on: (i) an exponentially growing (and less expensive) computing power, to the extent that almost everybody in a modern society can own and use that; (ii) omnipresent high-speed Internet connectivity, wired or wireless, representing our modern day’s economic connectomics; (iii) growing concern of exploiting “serendipity” for a strategic commercial advantage; and (iv) growing capabilities of lay people in performing calculations for their informed decisions in taking fast-moving entrepreneurial opportunities. Computational entrepreneurship has slowly become a new mode of operation for business ventures and will likely bring the academic discipline of entrepreneurship back to mainstream economics. (shrink)

The medium‐independence of computational descriptions has shaped common conceptions of computational explanation. So long as our goal is to explain how a system successfully carries out its computations, then we only need to describe the abstract series of operations that achieve the desired input–output mapping, however they may be implemented. It is argued that this abstract conception of computational explanation cannot be applied to so‐called real‐time computing systems, in which meeting temporal deadlines imposed by the systems with which a (...) device interfaces are constitutive of the computing tasks that a device performs. Instead, real‐time computing reveals the need for alternative conceptions of computational explanation, as well as computational implementation, that eschew medium‐independence. (shrink)

Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations—whatever they are—must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise (...) to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This theory implements a version of conceptual role semantics by positing an internal universal representation language in which learners may create mental models to capture dynamics they observe in the world. The theory formalizes one account of how truly novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned from a more primitive basis. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. This account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in a connectionist framework. The resulting theory provides an “assembly language” for cognition, where high-level theories of symbolic computation can be implemented in simple dynamics that themselves could be encoded in biologically plausible systems. (shrink)

One of Noam Chomsky's earliest contributions is the idea that a theory of the unbounded construction of hierarchical structures should incorporate a computational system that generates the structures. This chapter focuses on the structure building system, what is sometimes called the computational system, as a source of explanation. In some sense it is the fundamental source of explanation in generative grammar, as it accounts for the central question of the unbounded hierarchical nature of the syntax of human language. The architecture (...) of the computational system of early generative grammar involves a device that carries out elementary operations following a procedure that maps strings of symbols to strings of symbols in two passes (a phrase structure sub‐procedure and a transformational sub‐procedure). (shrink)

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic (...) distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

In this paper I criticize a view of functional localization in neuroscience, which I call “computational absolutism”. “Absolutism” in general is the view that each part of the brain should be given a single, univocal function ascription. Traditional varieties of absolutism posit that each part of the brain processes a particular type of information and/or performs a specific task. These function attributions are currently beset by physiological evidence which seems to suggest that brain areas are multifunctional—that they process distinct information (...) and perform different tasks depending on context. Many theorists take this contextual variation as inimical to successful localization, and claim that we can avoid it by changing our functional descriptions to computational descriptions. The idea is that we can have highly generalizable and predictive functional theories if we can discover a single computation performed by each area regardless of the specific context in which it operates. I argue, drawing on computational models of perceptual area MT, that this computational version of absolutism fails to come through on its promises. In MT, the modeling field has not produced a univocal computational description, but instead a plurality of models analyzing different aspects of MT function. Moreover, CA cannot appeal to theoretical unification to solve this problem, since highly general models, on their own, neither explain nor predict what MT does in any particular context. I close by offering a perspective on neural modeling inspired by Nancy Cartwright’s and Margaret Morrison’s views of modeling in the physical sciences. (shrink)

In the book, I argue that the mind can be explained computationally because it is itself computational—whether it engages in mental arithmetic, parses natural language, or processes the auditory signals that allow us to experience music. All these capacities arise from complex information-processing operations of the mind. By analyzing the state of the art in cognitive science, I develop an account of computational explanation used to explain the capacities in question.