The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted the existence of non-empty co-r. e. closed sets devoid of computable points: sets which are even “large” in the sense of positive Lebesgue measure.This leads us to investigate for various classes of computable real subsets whether they always contain a computable point.

We prove uniformly computable versions of the Implicit Function Theorem in its differentiable and non-differentiable forms. We show that the resulting operators are not computable if information about some of the partial derivatives of the implicitly defining function is omitted. Finally, as a corollary, we obtain a uniformly computable Inverse Function Theorem, first proven by M. Ziegler.

We propose a fast and reliable corner detector that can detect corners under non-uniform illumination and fuzzy mineshaft images effectively. First, we presented an inner mask that used only four pixels to determine the flat and corner regions of an image, which could eliminate unnecessary computation of flat regions, thus reducing computing cost. Second, we separated the corner regions into background and foreground and computed the separate corner threshold to settle non-uniform illumination. Third, we proposed a fast corner-detection algorithm to (...) compute the nucleus continuous contributive segment based on the corner state. Finally, we proposed two effective methods to remove the false corners. Experimental results showed that our approach has a better detection quality and is less time consuming than three other algorithms on an artificial image, a noisy image, and non-uniform images and could meet the real-time requirement of mineshaft applications. (shrink)

In this paper, a proof-theoretic method to prove uniform Lyndon interpolation for non-normal modal logics is introduced and applied to show that the logics E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {E}$$\end{document}, M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {M}$$\end{document}, MC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MC}$$\end{document}, EN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EN}$$\end{document}, MN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MN}$$\end{document} have that (...) property. In particular, these logics have uniform interpolation. Although for some of them the latter is known, the fact that they have uniform Lyndon interpolation is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. It is also shown that the non-normal modal logics EC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EC}$$\end{document} and ECN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ECN}$$\end{document} do not have Craig interpolation, and whence no uniform interpolation. (shrink)

We investigate in detail the electromagnetic fields of a uniformly accelerated charge, in order to ascertain whether such a charge does ‘emit’ radiation, especially in view of the Poynting flow computed at large distances and taken as an evidence of radiation emitted by the charge. In this context, certain important aspects of the fields need to be taken into account. First and foremost is the fact that in the case of a uniformly accelerated charge, one cannot ignore the velocity fields. (...) This then leads to other equally vital points. The net field energy turns out to be exactly the same as that of a non-accelerated charge having a uniform velocity equal to the instantaneous velocity of the uniformly accelerated charge. Further, the Poynting vector, seen with respect to the ’present’ location of the uniformly accelerated charge, during the deceleration phase, possesses everywhere a radial component pointing inward toward the charge, becoming nil when the charge becomes momentarily stationary, and during the acceleration phase, points away from the charge position. Last, but not least, when the leading spherical front of the relativistically beamed Poynting flux, advances forward at a large time t to a far-off distance \, the charge too is not lagging far behind. In fact, these relativistically beamed fields, increasingly resemble fields of a charge moving in an inertial frame with a uniform velocity \, with a convective flow of fields in that frame along with the movement of the charge. There is no other Poynting flow in the far-zones that could be termed as radiation emitted by the charge which, in turn, is fully consistent with the absence of radiation reaction and is also fully conversant with the strong principle of equivalence. (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)

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)

Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T−1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T−1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based (...) theory of computability on Euclidean space and other topological spaces. Using a computable version of Banach’s Inverse Mapping Theorem we can answer the first question positively. Hence, the non-uniform version of the inversion problem is solvable, while a topological argument shows that the uniform version is not. Thus, we are in the striking situation that any computable linear operator has a computable inverse while there exists no general algorithmic procedure to transfer a program of the operator into a program of its inverse. As a consequence, the computable version of Banach’s Inverse Mapping Theorem is a powerful tool which can be used to produce highly non-constructive existence proofs of algorithms. We apply this method to prove that a certain initial value problem admits a computable solution. As a preparation of Banach’s Inverse Mapping Theorem we also study the Open Mapping Theorem and we show that the uniform versions of both theorems are limit computable, which means that they are effectively -measurable with respect to the effective Borel hierarchy. (shrink)

Using computable reducibility ⩽ on equivalence relations, we investigate weakly precomplete ceers (a “ceer” is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers, that are not precomplete; and there are infinitely many isomorphism types of non‐universal weakly precomplete ceers. Whereas the Visser space of a precomplete ceer always (...) contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete ceers is Σ3‐complete, whereas the index set of the weakly precomplete ceers is Π3‐complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the e‐complete ceers are Σ3‐complete. (shrink)

These are the proceedings of the First International Conference on Compu- tional Logic (CL 2000) which was held at Imperial College in London from 24th to 28th July, 2000. The theme of the conference covered all aspects of the theory, implementation, and application of computational logic, where computational logic is to be understood broadly as the use of logic in computer science. The conference was collocated with the following events: { 6th International Conference on Rules and Objects in Databases (DOOD (...) 2000) { 10th International Workshop on Logic-based Program Synthesis and Tra- formation (LOPSTR 2000) { 10th International Conference on Inductive Logic Programming (ILP 2000). CL 2000 consisted of seven streams: { Program Development (LOPSTR 2000) { Logic Programming: Theory and Extensions { Constraints { Automated Deduction: Putting Theory into Practice { Knowledge Representation and Non-monotonic Reasoning { Database Systems (DOOD 2000) { Logic Programming: Implementations and Applications. The LOPSTR 2000 workshop constituted the program development stream and the DOOD 2000 conference constituted the database systems stream. Each stream had its own chair and program committee, which autonomously selected the papers in the area of the stream. Overall, 176 papers were submitted, of which 86 were selected to be presented at the conference and appear in these proceedings. The acceptance rate was uniform across the streams. In addition, LOPSTR 2000 accepted about 15 extended abstracts to be presented at the conference in the program development stream. (shrink)

We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the (...) degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers. (shrink)

This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and (...) clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility.We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $\mathbb {N}$ or is single-valued.These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $\mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $\mathsf {DS}$, and the Weihrauch equivalent problem $\mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level.Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $\mathbf {K}$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol {\Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf {K}$. We also generalize the results by relaxing the compactness of the ambient space and show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}$ with the Fell topology. A similar result holds also for the Vietoris topology.We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.prepared by Manlio Valenti.E-mail:[email protected]. (shrink)

The -logic of Terwijn is a variant of first-order logic with the same syntax in which the models are equipped with probability measures and the quantifier is interpreted as ‘there exists a set A of a measure such that for each,...’. Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational, respectively -complete and -hard, and ii) for, respectively decidable and -complete. The adjective ‘general’ here means ‘uniformly over all languages’. We (...) extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability and validity with respect to finite models in E-logic are, i) for rational, respectively -complete and -complete, and ii) for, respectively decidable and -complete. Although partial results toward the countable case are also achieved, the computability of E-logic over countable models still remains largely unsolved. In addition, most o... (shrink)

The domain of “procedural ethics” is the set of reflective and deliberative methods that maximize the reliability of moral judgment. While no general algorithmic method exists that will guarantee the validity of ethical deliberation, non‐algorithmic “heuristic” methods can guide and inform the process, making it significantly more robust and dependable. This essay examines various representative heuristic procedures commonly recommended for use in applied ethics, maps them into a uniform set of twelve stages, identifies common faults, then shows how the resulting (...) stage‐by‐stage decision‐making model could be adapted for general use and for use in computer ethics. (shrink)

Every artificial-intelligence research project needs a working definition of "intelligence", on which the deepest goals and assumptions of the research are based. In the project described in the following chapters, "intelligence" is defined as the capacity to adapt under insufficient knowledge and resources. Concretely, an intelligent system should be finite and open, and should work in real time. ;If these criteria are used in the design of a reasoning system, the result is NARS, a non-axiomatic reasoning system. ;NARS uses a (...) term-oriented formal language, characterized by the use of subject-predicate sentences. The language has an experience-grounded semantics, according to which the truth value of a judgment is determined by previous experience, and the meaning of a term is determined by its relations with other terms. Several different types of uncertainty, such as randomness, fuzziness, and ignorance, can be represented in the language in a single way. ;The inference rules of NARS are based on three inheritance relations between terms. With different combinations of premises, revision, deduction, induction, abduction, exemplification, comparison, and analogy can all be carried out in a uniform format, the major difference between these types of inference being that different functions are used to calculate the truth value of the conclusion from the truth values of the premises. ;Since it has insufficient space-time resources, the system needs to distribute them among its tasks very carefully, and to dynamically adjust the distribution as the situation changes. This leads to a "controlled concurrency" control mechanism, and a "bag-based" memory organization. ;A recent implementation of the NARS model, with examples, is discussed. The system has many interesting properties that are shared by human cognition, but are absent from conventional computational models of reasoning. ;This research sheds light on several notions in artificial intelligence and cognitive science, including symbol-grounding, induction, categorization, logic, and computation. These are discussed to show the implications of the new theory of intelligence. ;Finally, the major results of the research are summarized, a preliminary evaluation of the working definition of intelligence is given, and the limitations and future extensions of the research are discussed. (shrink)

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness and (...) proof normalization. We have implemented our work in the Isabelle Logical Framework. (shrink)

Philosophers often make exotic-sounding modal claims, such as: “A timeless world is impossible”, “The laws of physics could have been different from what they are”, “There could have been an additional phenomenal colour”. Otherwise popular empiricist modal epistemologies in the contemporary literature cannot account for whatever epistemic justification we might have for making such modal claims. Those who do not, as a result of this, endorse scepticism with respect to their epistemic status typically suggest that they can be justified but (...) have yet to develop some distinct, workable theory of how. That is, they endorse a form of non-uniformism about the epistemology of modality, according to which claims about philosophically interesting modal matters need to be justified differently from e.g. everyday or scientific modal claims, but they fail to provide any more detail. This article aims to fill this gap by outlining how such a non-uniformist view could be spelled out and what story about philosophically interesting modal justification it could contain. (shrink)

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and positive logics,and (...) also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. (shrink)

Uniformism about the epistemology of modality is the view that there is only one basic route to modal knowledge; non-uniformism is the view that there are several. Non-uniformism is becoming an increasingly popular stance, but how can it be defended? I prise apart two ways of understanding the uniformism/non-uniformism conflict that are mixed up in the literature. I argue that once separated, it is evident that they lead up to two different non-uniformist theses that need to be argued for in (...) very different ways. In particular, one construal allows only for a weak thesis about the actual lay of the current theoretical land in modal epistemology, while the other allows for a strong thesis that can be defended without reliance on claims about the alleged goodness or badness of individual accounts of modal justification. I argue that philosophers inclined towards non-uniformism ought to go for the strong, rather than the weak, thesis. (shrink)

Depending on what one means by the main connective of logic, the -if..., then... -, several systems of logic result: classic and modal logics, intuitionistic logic or relevance logic. This book presents the underlying ideas, the syntax and the semantics of these logics. Soundness and completeness are shown constructively and in a uniform way. Attention is paid to the interdisciplinary role of logic: its embedding in the foundations of mathematics and its intimate connection with philosophy, in particular the philosophy of (...) language. Set theory is presented both as a conditio sine qua non for logic and as a interesting exact ontology. The study of infinite sets yields perplexing results. Formalization of informal number theory results in formal number theory; Godel's incompleteness is treated. At appropriate places attention is paid to paradoxes, intuitionism, conditionals, the historical development of logic, to logic programming and automated theorem proving for classical logic.". (shrink)

The theory of inner functions plays an important role in the study of bounded analytic functions. Inner functions are also useful in applied mathematics. Two foundational results in this theory are Frostman's Theorem and the Factorization Theorem. We prove a uniformly computable version of Frostman's Theorem. We then show that the Factorization Theorem is not uniformly computably true. We then show that for an inner function u with infinitely many zeros, the Blaschke sum of u provides the exact amount of (...) information necessary to compute the factorization of u. Along the way, we prove some uniform computability results for Blaschke products; these results play a key role in the analysis of factorization. We also give some computability results concerning zeros and singularities of analytic functions. We use Type-Two Effectivity as our foundation. (shrink)

We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a (...) semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant. (shrink)

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework for (...) the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics. (shrink)

We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete metric space cannot be decomposed into countably many nowhere dense pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of (...) the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets. (shrink)

It is only when mixing two or more pure substances along a reversible path that the entropy of the mixing can be made physically manifest. It is not, in this case, a mere mathematical artifact. This mixing requires a process of successive stages. In any finite number of stages, the external manifestation of the entropy change, as a definite and measurable quantity of heat, isa fully continuous function of the relevant variables. It is only at an infinite and unattainable limit (...) thata non-uniform convergence occurs. And this occurs when considered in terms of the number of stages together with a distinguishability parameter appropriate to the particular device which is used to achieve reversibility. These considerations, which are of technological interest to chemical engineers, resolve a paradox derived in chemical theory called Gibbs'' Paradox. (shrink)

A true Turing machine requires an infinitely long paper tape. Thus a TM can be housed in the infinite world of Newtonian spacetime, but not necessarily in our world, because our world-at least according to our best spacetime theory, general relativity-may be finite. All the same, one can argue for the "existence" of a TM on the basis that there is no such housing problem in some other relativistic worlds that are similar to our world. But curiously enough-and this is (...) the main point of this paper-some of these close worlds have a special spacetime structure that allows TMs to perform certain Turing unsolvable tasks. For example, in one kind of spacetime a TM can be used to solve first-order predicate logic and the halting problem. And in a more complicated spacetime, TMs can be used to decide arithmetic. These new computers serve to show that Church's thesis is a thoroughly contingent claim. Moreover, since these new computers share the fundamental properties of a TM in ordinary operation, a computability theory based on these non-Turing computers is no less worthy of investigation than orthodox computability theory. Some ideas about this new mathematical theory are given. (shrink)

A true Turing machine (TM) requires an infinitely long paper tape. Thus a TM can be housed in the infinite world of Newtonian spacetime (the spacetime of common sense), but not necessarily in our world, because our world-at least according to our best spacetime theory, general relativity-may be finite. All the same, one can argue for the "existence" of a TM on the basis that there is no such housing problem in some other relativistic worlds that are similar ("close") to (...) our world. But curiously enough-and this is the main point of this paper-some of these close worlds have a special spacetime structure that allows TMs to perform certain Turing unsolvable tasks. For example, in one kind of spacetime a TM can be used to solve first-order predicate logic and the halting problem. And in a more complicated spacetime, TMs can be used to decide arithmetic. These new computers serve to show that Church's thesis is a thoroughly contingent claim. Moreover, since these new computers share the fundamental properties of a TM in ordinary operation (e.g. intuitive, finitely programmed, limited in computational capability), a computability theory based on these non-Turing computers is no less worthy of investigation than orthodox computability theory. Some ideas about this new mathematical theory are given. (shrink)

In this paper, I argue that the temporal connective prima (‘before’) is a comparative adverb. The argument is based on a number of grammatical facts from Italian, showing that there is an asymmetry between prima and dopo (‘after’). On the ground of their divergent behaviour, I suggest that dopo has a different grammatical status from prima. I propose a semantic treatment for prima that is based on an independently motivated analysis of comparatives which can be traced back to Seuren (in: (...) Kiefer and Ruwet (eds.) Generative grammar in Europe, 1973). Dopo is analyzed instead as an atomic two-place predicate which contributes a binary relation over events to the sentence meaning. The different semantic treatments of the two connectives provide an explanation for the grammatical asymmetries considered at the outset; interestingly, they also shed some light on other asymmetries between prima and dopo, which are known to hold for the English temporal connectives before and after as well: these asymmetries are related to the veridicality properties, the distribution of NPIs, and the logical properties of these connectives first described in Anscombe (Philos Rev 73:3–24, 1964). (shrink)

The brain is a very complex structure. Over the past several decades, many studies have aimed to understand how various non-uniform variables relate to each other. The current study compared the whole-brain network organization and global spatial distribution of cell densities in the monkey brain. Wide comparisons between 27 graph theoretical measures and cell densities revealed that only participation coefficients (PCs) significantly correlated with cell densities. Interestingly, PCs did not show a significant correlation with spatial coordinates. Furthermore, the significance of (...) the correlation between cell densities and spatial coordinates disappeared only with the removal of the visual module, while the significance of the correlation between cell densities and PCs disappeared with the removal of any one module. Taken together, these results suggested the presence of a combinatorial effect of modular architectures in the network organization related to the non-uniformity of cell densities additional to the spatially monotonic change. (shrink)

It has already been discussed extensively in the literature that wh-elements in Chinese (as in Japanese and Korean) can have non-interrogative interpretations, i.e., they are the socalled “wh-indeterminates” à la Kuroda 1965 (see Huang 1982, Cheng 1991, Li 1992, Lin 1998 among others). (1a-c) illustrate the typical examples.

Imagine cells that live in a high‐gradient magnetic field (HGMF). Through what mechanisms do the cells sense a non‐uniform magnetic field and how such a field changes the cell fate? We show that magnetic forces generated by HGMFs can be comparable to intracellular forces and therefore may be capable of altering the functionality of an individual cell and tissues in unprecedented ways. We identify the cellular effectors of such fields and propose novel routes in cell biology predicting new biological (...) effects such as magnetic control of cell‐to‐cell communication and vesicle transport, magnetic control of intracellular ROS levels, magnetically induced differentiation of stem cells, magnetically assisted cell division, or prevention of cells from dividing. On the basis of experimental facts and theoretical modeling we reveal timescales of cellular responses to high‐gradient magnetic fields and suggest an explicit dependence of the cell response time on the magnitude of the magnetic field gradient. (shrink)

La présente étude a pour objet l’étude des concepts d’uniforme et de non-uniforme dans la physique du Timée. Comme le montre un examen des passages du corpus hippocratique où ces concepts apparaissent avant Platon, ces concepts désignent à la fois des propriétés sensibles et une formalisation de l’expérience perceptive susceptible d’une interprétation géométrique. Platon trouve ainsi dans ces schèmes de la pratique médicale un élément médiateur idéal au sein d’une physique qui a pour projet de reconstituer l’ensemble de l’expérience perceptive (...) à partir d’une construction géométrique des éléments premiers. (shrink)

The important roles of prediction and prior experience are well established in music research and fit well with Clark's concept of unified perception, cognition, and action arising from hierarchical, bidirectional predictive processing. However, in order to fully account for human musical intelligence, Clark needs to further consider the powerful and variable role of affect in relation to prediction error.

. This paper follows the journey of two small fluid units throughout the nineteenth century in Anglo-American medicine and pharmacy, explaining how the non-uniform “drop” survived while the standardized minim became obsolete. I emphasize two roles these units needed to fulfill: that of a physical measuring device, and that of a rhetorical communication device. First, I discuss the challenges unique to measuring small amounts of fluid, outlining how the modern medicine dropper developed out of an effort to resolve problems with (...) the “minimometer,” which measured minims. Second, I explain how drops, utilized in “the open drop method” of administering general anesthesia, effectively communicated a gradual process and epistemically valuable heuristic to the audience of practitioners, whose attention to individual medical outcomes was important for verifying the proper dosage. The standardized minim was never able to achieve success as the drop’s intended replacement; the non-uniform drop better served the relevant epistemic goals within the practical contexts for which these units were designed. The surprising historical trajectory of drops should cause us to question the common equivocation of “standardization” with “progress” in the history and philosophy of measurement. This study also exemplifies how examining non-standard measurement practices can be instructive for better understanding the role and function of standardization within epistemology of measurement. (shrink)

Recently, many philosophers have been inclined to ascribe mentality to animals on the main grounds that they possess certain complex computational abilities. In this paper I contend that this view is misleading, since it wrongly assumes that those computational abilities demand a psychological explanation. On the contrary, they can be just characterised from a computational level of explanation, which picks up a domain of computation and information processing that is common to many computing systems but is autonomous from the domain (...) of psychology. Thus, I propose that it is possible to conceive insects and other animals as mere computing agents, without having any commitment to ascribe mentality to them. I conclude by sketching a proposal about how to draw the line between mere computing and genuine mentality. (shrink)

This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition (...) on so-called focal points . As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. (shrink)