Various methods have been employed to investigate different aspects of the brain activity modulation related to the performance of a cycling task. In our study we examined how functional connectivity and brain network efficiency varied during an endurance cycling task. To this purpose, we reconstructed EEG signals at source level: we computed current densities in 28 anatomical regions of interest (ROIs) through the eLORETA algorithm, then we calculated the Lagged Coherence of the 28 current density signals to define the (...) adjacency matrix. To quantify changes of functional network efficiency during an exhaustive cycling task, we computed three graph theoretical indices: local efficiency (LE), global efficiency (GE) and Density (D) in two different frequency bands, Alpha and Beta bands, that indicate alertness processes and motor binding/fatigue respectively. This analysis was conducted for six different task intervals: pre-cycling, initial, intermediate and final stages of cycling, active recovery and passive recovery. Fourteen participants performed an incremental cycling task with simultaneous EEG recording and RPE monitoring to detect participants’ exhaustion. LE remained constant during the endurance cycling task in both bands. Therefore, we speculate that fatigue processes did not affect segregated neural processing. We observed an increase of GE in the Alpha band only during cycling, that could be due to greater alertness processes and preparedness to stimuli during exercise. Conversely, although D did not change significantly over time in the Alpha band, its general reduction in the Beta bands during cycling could be interpreted within the framework of the neural efficiency hypothesis, which posits a reduced neural activity for expert/automated performances. We argue that the use of graph theoretical indices represents a clear methodological advancement in studying endurance performance. (shrink)

In recent years, functional connectivity in the developmental science received increasing attention. Although it has been reported that the anatomical connectivity in the preterm brain develops dramatically during the last months of pregnancy, little is known about how functional and effective connectivity change with maturation. The present study investigated how effective connectivity in premature infants evolves. To assess it, we use EEG measurements and graph-theory methodologies. We recorded data from 25 preterm babies, who underwent (...) long-EEG monitoring at least twice during their stay in the NICU. The recordings took place from 27 weeks postmenstrual age until 42 weeks PMA. Results showed that the EEG-connectivity, assessed using graph-theory indices, moved from a small-world network to a random one, since the clustering coefficient increases and the path length decreases. This shift can be due to the development of the thalamocortical connections and long-range cortical connections. Based on the network indices, we developed different age-prediction models. The best result showed that it is possible to predict the age of the infant with a root mean-squared error equal to 2.11 weeks. These results are similar to the ones reported in the literature for age prediction in preterm babies. (shrink)

In graph theory, the concept of domination is essential in a variety of domains. It has broad applications in diverse fields such as coding theory, computer net work models, and school bus routing and facility lo cation problems. If a fuzzy graph fails to obtain acceptable results, neutrosophic sets and neutrosophic graphs can be used to model uncertainty correlated with indeterminate and inconsistent information in arbitrary real-world scenario. In this study, we consider the concept of domination (...) as it relates to single-valued neutrosophic incidence graphs (SVNIGs). Given the importance of domination and its utilization in numerous fields, we propose the application of dominating sets in SVNIG with valid edges. We present some relevant definitions such as those of valid edges, cardinality, and isolated vertices in SVNIG along with some examples. Furthermore, we also show a few significant sets connected to the dominating set in an SVNIG such as independent and irredundant sets. We also investigate a relationship between the concepts of dominating sets and domination numbers as well as irredundant and independence sets in SVNIGs. Finally, a real-life deployment of domination in SVNIGsis investigated in relation to COVID-19 vaccination locations as a practical application. (shrink)

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

Music-based interventions (MBI) have become increasingly widely adopted for dementia and related disorders. Previous research shows that music engages reward-related regions through functional connectivity with the auditory system, but evidence for the effectiveness of MBI is mixed in older adults with mild cognitive impairment (MCI) and Alzheimer’s disease (AD). This underscores the need for a unified mechanistic understanding to motivate MBIs. The main objective of the present study is to characterize the intrinsic connectivity of the auditory and reward (...) systems in healthy aging individuals with MCI, and those with AD. Using resting-state fMRI data from the Alzheimer’s Database Neuroimaging Initiative, we tested resting-state functional connectivity within and between auditory and reward systems in older adults with MCI, AD, and age-matched healthy controls (N = 105). Seed-based correlations were assessed from regions of interest (ROIs) in the auditory network (i.e., anterior superior temporal gyrus, posterior superior temporal gyrus, Heschl’s Gyrus), and the reward network (i.e., nucleus accumbens, caudate, putamen, and orbitofrontal cortex). AD individuals were lower in both within-network and between-network functional connectivity in the auditory network and reward networks compared to MCI and controls. Furthermore, graph theory analyses showed that the MCI group had higher clustering and local efficiency than both AD and control groups, whereas AD individuals had lo... (shrink)

The human brain shows neuroplastic adaptations induced by motor skill training. However, the description of the plastic architecture of the whole-brain network in resting-state is still limited. In the present study, we aimed to detect how motor training affected the density distribution of whole-brain resting-state functional connectivity (FC) brain in fast-ball student-athletes using resting-state functional magnetic resonance imaging (fMRI) data of student-athletes (SA), and non-athlete healthy controls (NC). The voxel-wise data-driven graph theory approach, namely global functional (...) class='Hi'>connectivity density (gFCD) mapping was applied. The results showed that the SA group exhibited significantly decreased gFCD in brain regions centered at left triangular part of inferior frontal gyrus (IFG), extending to opercular part of left IFG and middle frontal gyrus (MFG) compared with NC group. The findings supported the idea of an increased neural efficiency of athletes’ brain in the brain regions associated with attentional-motor modulation and executive control. Furthermore, the behavioral results demonstrated that in the SA group, faster executive control reaction time was associated with smaller gFCD values in left IFG. The findings implied that the motor skill training would decrease the numbers of FC in IFG to accelerate the execution control with high attentional demands, and focus the attention to the target detection the athletes are interested in. (shrink)

We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 . Our motivation partly (...) stems from the fact that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems. (shrink)

Graph-based frames have been introduced as a logical framework which internalises an inherent boundary to knowability (referred to as ‘informational entropy’), due, e.g. to perceptual, evidential or linguistic limits. They also support the interpretation of lattice-based (modal) logics as hyper-constructive logics of evidential reasoning. Conceptually, the present paper proposes graph-based frames as a formal framework suitable for generalising Pawlak's rough set theory to a setting in which inherent limits to knowability exist and need to be considered. Technically, (...) the present paper establishes systematic connections between the first-order correspondents of Sahlqvist modal reduction principles on Kripke frames, and on the more general relational environments of graph-based and polarity-based frames. This work is part of a research line aiming at: (a) comparing and inter-relating the various (first-order) conditions corresponding to a given (modal) axiom in different relational semantics; (b) recognising when first-order sentences in the frame-correspondence languages of different relational structures encode the same ‘modal content’; (c) meaningfully transferring relational properties across different semantic contexts. The present paper develops these results for the graph-based semantics, polarity-based semantics, and all Sahlqvist modal reduction principles. As an application, we study well known modal axioms in rough set theory (such as those corresponding to seriality, reflexivity, and transitivity) on graph-based frames and show that, although these axioms correspond to different first-order conditions on graph-based frames, their intuitive meaning is retained. This allows us to introduce the notion of hyperconstructivist approximation spaces as the subclass of graph-based frames defined by the first-order conditions corresponding to the same modal axioms defining classical generalised approximation spaces, and to transfer the properties and the intuitive understanding of different approximation spaces to the more general framework of graph-based frames. The approach presented in this paper provides a base for systematically comparing and connecting various formal frameworks in rough set theory, and for the transfer of insights across different frameworks. (shrink)

Many do not doubt that Peirce's Existential Graphs are diagrammatic, as opposed to symbolic. However, when we are pressured to draw a distinction between the two different forms of representation, we find ourselves at a loss and our intuition quite vague. In this paper, I locate fundamental differences between two logically equivalent systems, Peirce's Alpha system and propositional languages. Suppose we have only two sentential connectives, ¬ and ^. In spite of its truth-functional completeness, we don't want to use this (...) language for the translation of English sentences or as a deductive calculus. We would adopt this language only when we intend to develop logical theories. That is, it is convenient to have fewer connectives for a meta-theory, but not for practical use. So, there seems to be a trade-off between a language with fewer connectives and a language with more connectives. This view has been accepted without question. In this paper, I will argue that this trade-off is limited to linear symbolic systems and that we could have a diagrammatic system with fewer operations but no need to suffer from problems like those of a sentential language with fewer connectives. How is that possible? A comparison between Peirce's Alpha system and a propositional language is presented to answer this question. The case study identifies the following unique property of a non-linear diagrammatic system as a main source of the discrepancy between two different types of representation: One and the same diagram can be read off in more than one way by carving it up in many ways, but without ambiguity. (shrink)

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier $\sf Q$ in ${\rm FO}({\vec Q}_k)$ , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two ${\rm FO}({\vec Q}_1)$ -equivalent models with certain additional structure, yields a pair of ${\rm FO}({\vec Q}_k)$ -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as (...) class='Hi'>connectivity and planarity. (shrink)

This rare publication continues an exploratory journey in relational biology, a study of biology in terms of the organization of networked connections in living systems. It builds on the author's two earlier monographs which looked at the epistemology of life and the ontogeny of life. Here the emphasis is on the intangibility of life, that the real nature of living systems is conveyed not by their tangible material basis but by their intangible inherent processes. Relational biology is the approach that (...) hails 'function dictates structure'; it is mathematics decoded into biological realizations. Therefore, the work begins with a concise introduction to category theory, equiping the reader with the mathematical metalanguage of relation biology. The book is organized around three parts: Part I is a comprehensive study of the most important functor in relational biology, the power set functor. The author lays the set-theoretic foundations of the functorial connections in relational biology, exploring relations, mappings, and set-valued mappings. In Part II, Natural Law receives a new mathematical formulation founded on two axioms: 'Everything is a set.' and 'Every process is a set-valued mapping.' The reader sees how Metabolism-Repair networks, equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems. Part III expounds the various shades of invertibility in general, and the inversion of encoding to decoding in particular. A plethora of mathematical and biological examples illustrate the category-theoretic concepts of equivalence and adjunction. This book's algebraic approach to biological models will appeal to researchers and graduate students in mathematics, biology, and the philosophy of science. (shrink)

Cognitive functions are directly related to interactions between the brain's functional networks. This functional organization changes in the autism spectrum disorder (ASD). However, the heterogeneous nature of autism brings inconsistency in the findings, and specific pattern of changes based on the cognitive theories of ASD still requires to be well-understood. In this study, we hypothesized that the theory of mind (ToM), and the weak central coherence theory must follow an alteration pattern in the network level of functional interactions. (...) The main aim is to understand this pattern by evaluating interactions between all the brain functional networks. Moreover, the association between the significantly altered interactions and cognitive dysfunctions in autism is also investigated. We used resting-state fMRI data of 106 subjects (5–14 years, 46 ASD: five female, 60 HC: 18 female) to define the brain functional networks. Functional networks were calculated by applying four parcellation masks and their interactions were estimated using Pearson's correlation between pairs of them. Subsequently, for each mask, a graph was formed based on the connectome of interactions. Then, the local and global parameters of the graph were calculated. Finally, statistical analysis was performed using a two-sample t-test to highlight the significant differences between autistic and healthy control groups. Our corrected results show significant changes in the interaction of default mode, sensorimotor, visuospatial, visual, and language networks with other functional networks that can support the main cognitive theories of autism. We hope this finding sheds light on a better understanding of the neural underpinning of autism. (shrink)

We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work connects developments in stability theory, finite model theory, abstract model theory, and probability. We conclude by linking this context with work on the Urysohn space.

We use methods of reverse mathematics to analyze the proof theoretic strength of a theorem involving the notion of coloring number. Classically, the coloring number of a graph $G=$ is the least cardinal $\kappa$ such that there is a well-ordering of $V$ for which below any vertex in $V$ there are fewer than $\kappa$ many vertices connected to it by $E$. We will study a theorem due to Komjáth and Milner, stating that if a graph is the union (...) of $n$ forests, then the coloring number of the graph is at most $2n$. We focus on the case when $n=1$. (shrink)

Previous work employing graph theory and nonlinear analysis has found increased spatial and temporal disorder, respectively, of functional brain connectivity in schizophrenia. We present a new method combining graph theory and nonlinear techniques that measures the temporal disorder of functional brain connections. Multichannel electroencephalographic data were windowed and functional networks were reconstructed using the minimum spanning trees of correlation matrices. Using a method based on Shannon entropy, we found elevated connection entropy in gamma activity of (...) patients with schizophrenia; however, gamma connection entropy remained elevated in patients with schizophrenia even after a reduction in symptoms due to treatment with antipsychotics. Our results are consistent with several possibilities: aberrant functional connectivity is epiphenomenal to schizophrenia, aberrant functional connectivity is a central feature but antipsychotics reduce symptoms by an independent mechanism, or connection entropy is not an appropriately sensitive measure of brain abnormalities in schizophrenia. (shrink)

A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The (...) basic idea underlying the results and examples presented here is that it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset T′ of T has a model which is rigid. (shrink)

ObjectiveTo explore alterations in white matter network topology in de novo Parkinson's disease patients with rapid eye movement sleep behavior disorder.Materials and MethodsThis study included 171 de novo PD patients and 73 healthy controls recruited from the Parkinson's Progression Markers Initiative database. The patients were divided into two groups, PD with probable RBD and PD without probable RBD, according to the RBD screening questionnaire. Individual structural network of brain was constructed based on deterministic fiber tracking and analyses were performed using (...) graph theory. Differences in global and nodal topological properties were analyzed among the three groups. After that, post hoc analyses were performed to explore further differences. Finally, correlations between significant different properties and RBDSQ scores were analyzed in PD-pRBD group.ResultsAll three groups presented small-world organization. PD-pRBD patients exhibited diminished global efficiency and increased shortest path length compared with PD-npRBD patients and HCs. In nodal property analyses, compared with HCs, the brain regions of the PD-pRBD group with changed nodal efficiency were widely distributed mainly in neocortical and paralimbic regions. While compared with PD-npRBD group, only increased Ne in right insula, left middle frontal gyrus, and decreased Ne in left temporal pole were discovered. In addition, significant correlations between Ne in related brain regions and RDBSQ scores were detected in PD-pRBD patients.ConclusionsPD-pRBD patients showed disrupted topological organization of white matter in the whole brain. The altered Ne of right insula, left temporal pole and left middle frontal gyrus may play a key role in the pathogenesis of PD-RBD. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and (...) finite-universe logic. We review the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

The critical node detection problem is a central task in computational graph theory due to its large applicability, consisting in deleting $k$ nodes to minimize a certain graph measure. In this article, we propose a new Extremal Optimization-based approach, the Pseudo-Deterministic Noisy Extremal Optimization (PDNEO) algorithm, to solve the Critical Node Detection variant in which the pairwise connectivity is minimized. PDNEO uses an adaptive pseudo-deterministic parameter to switch between random nodes and articulation points during the search, (...) as well as other features, such as noise induction to preserve diversity, greedy search to better exploit the search space and a greater search space exploration mechanism. Numerical experiments on synthetic and real-world networks show the effectiveness of the proposed algorithm compared with existing methods. (shrink)

In this paper, I construe scientific understanding not only as understanding the phenomena by means of some theoretical material (theory, law or model), but more fundamentally as understanding the theoretical material itself that is supposed to explain the phenomena. De Regt and Dieks (2005) emphasise the contextual aspects of the intelligibility of theories, showing that it depends on their "virtues", on the historical standards of intelligibility, and on the particular "skills" of their users. My paper aims at continuing this (...) proposal, first by giving a more precise definition of one's understanding of a theory and then by emphasising the importance, for this issue, of the particular formats in which a theory is expressed and hence grasped by its users. To defend this, I take the example of the versions of classical mechanics (variational versus vectorial) and the various formats of representation of its main principles and models. What does "understanding a theory" mean? At first sight, we could say that it amounts to having a clear view of the logical relations between its core principles and theorems. This kind of understanding, though global, is quite abstract: one can understand the logical structure of a theory without being able to connect it to the phenomena. Moreover, this definition depends on how one 115 construes the structure of theories: it will vary according to whether one defines theories as logical sets of statements with interpretative rules (following the "syntactic conception" of theories) or as families of models ("semantic conception"). I thus suggest that there is another sense of "understanding a theory" that itself has two aspects. To understand a theory, one has to understand both what the theory says or means and how it works; in other words, one has to grasp the phenomena by means of its conceptual apparatus (representational aspect) and to be able to manipulate it and make it fit the phenomena (computational aspect). I claim that these are essentially contextual and practical matters, and that the particular format in which the theoretical content is displayed is crucial to them. Following Humphreys' proposal (2004), I claim that one never accesses to a theory as a whole. Be it a set of statements or a class of models, in practice, it is always displayed in some particular equations, statements, images, graphs, diagrams. Humphreys' proposal of the notion of "template" to complement the classical "units of analysis" of science, like theories and models, may be a good candidate to study the relationship between the representational and computational aspects of understanding: a template is a "concrete piece of syntax" (most of the time an equation, but I suggest that Humphreys' claim could be extended to other formats) that has both a representational and computational function. With the example of classical mechanics, I show how these two functions are interrelated and, as Humphreys suggests, sometimes in tension with each other. Adressing these issues by focusing on the particular formats that are dealt with in practice may enlight this problematic relationship. (shrink)

In this paper, I construe scientific understanding not only as understanding the phenomena by means of some theoretical material (theory, law or model), but more fundamentally as understanding the theoretical material itself that is supposed to explain the phenomena. De Regt and Dieks (2005) emphasise the contextual aspects of the intelligibility of theories, showing that it depends on their ―virtues‖, on the historical standards of intelligibility, and on the particular ―skills‖of their users. My paper aims at continuing this proposal, (...) first by giving a more precise definition of one's understanding of a theory and then by emphasising the importance, for this issue, of the particular formats in which a theory is expressed and hence grasped by its users. To defend this, I take the example of the versions of classical mechanics and the various formats of representation of its main principles and models. What does ―understanding a theory‖ mean? At first sight, we could say that it amounts to having a clear view of the logical relations between its core principles and theorems. This kind of understanding, though global, is quite abstract: one can understand the logical structure of a theory without being able to connect it to the phenomena. Moreover, this definition depends on how one construes the structure of theories: it will vary according to whether one defines theories as logical sets of statements with interpretative rules (following the ―syntactic conception‖ of theories) or as families of models (―semantic conception‖). I thus suggest that there is another sense of ―understanding a theory‖ that itself has two aspects. To understand a theory, one has to understand both what the theory says or means and how it works; in other words, one has to understand it as representing the phenomena (representational aspect) and to be able to manipulate it and make it fit the phenomena (computational aspect). I claim that these are essentially contextual and practical matters, and that the particular format in which the theoretical content is displayed is crucial to them. Following Humphreys' proposal (2004), I claim that one never accesses to a theory as a whole. Be it a set of statements or a class of models, in practice, it is always displayed in some particular equations, statements, images, graphs, diagrams. Humphreys' proposal of the notion of ―template‖ to complement the classical ―units of analysis‖ of science, like theories and models, may be a good candidate to study the relationship between the representational and computational aspects of understanding: a template is a ―concrete piece of syntax‖ (most of the time an equation, but I suggest that Humphreys' claim could be extended to other formats) that has both a representational and computational function. With the example of classical mechanics, I show how these two functions are interrelated and, as Humphreys suggests, are sometimes in tension with each other. Adressing these issues by focusing on the particular formats that are dealt with in practice may enlight this problematic relationship. (shrink)

Model theory is concerned mainly, although not exclusively, with infinite structures. In recent years, finite structures have risen to greater prominence, both within the context of mainstream model theory, e.g., in work of Lachlan, Cherlin, Hrushovski, and others, and with the advent of finite model theory, which incorporates elements of classical model theory, combinatorics, and complexity theory. The purpose of this survey is to provide an overview of what might be called the model theory (...) of finite structures. Some topics in finite model theory have strong connections to theoretical computer science, especially descriptive complexity theory. In fact, it has been suggested that finite model theory really is, or should be, logic for computer science. These connections with computer science will, however, not be treated here.It is well-known that many classical results of ‘infinite model theory’ fail over the class of finite structures, including the compactness and completeness theorems, as well as many preservation and interpolation theorems. The failure of compactness in the finite, in particular, means that the standard proofs of many theorems are no longer valid in this context. At present, there is no known example of a classical theorem that remains true over finite structures, yet must be proved by substantially different methods. It is generally concluded that first-order logic is ‘badly behaved’ over finite structures.From the perspective of expressive power, first-order logic also behaves badly: it is both too weak and too strong. Too weak because many natural properties, such as the size of a structure being even or a graph being connected, cannot be defined by a single sentence. Too strong, because every class of finite structures with a finite signature can be defined by an infinite set of sentences. Even worse, every finite structure is defined up to isomorphism by a single sentence. In fact, it is perhaps because of this last point more than anything else that model theorists have not been very interested in finite structures. Modern model theory is concerned largely with complete first-order theories, which are completely trivial here. (shrink)

In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$ : strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding (...) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms, and (iii) that we can use strong $\mathcal {L}_{\mathsf { P}}$ -homomorphisms to provide a simple demonstration of the paradoxical nature of a well-known paradox that has not received much attention in this context: the McGee paradox. In addition, along the way we will highlight how strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms highlight novel connections between the graph-theoretic analyses of paradoxes mobilized in the $\mathcal {L}_{\mathsf {P}}$ framework and the methods and tools of category theory. (shrink)

Coherence is a key concept in many accounts of epistemic justification within ‘traditional’ analytic epistemology. Within formal epistemology, too, there is a substantial body of research on coherence measures. However, there has been surprisingly little interaction between the two bodies of literature. The reason is that the existing formal literature on coherence measure operates with a notion of belief system that is very different from—what we argue is—a natural Bayesian formalisation of the concept of belief system from traditional epistemology. Therefore, (...) formal epistemology has so far only been concerned with one particular—arguably not even very natural—way of formalising coherence of belief systems; it has by no means refuted the viability of coherentism. In contrast to the existing literature, we formalise belief systems as families of assignments of (conditional) degrees of belief (which may be compatible with several subjective probability measures). Within this framework, we propose a Bayesian formalisation of the thrust of BonJour’s coherence concept in The structure of empirical knowledge (Harvard University Press, Cambridge, 1985), using a combination of Bayesian confirmation theory and basic graph theory. In excursions, we introduce graded notions for both logical and probabilistic consistency of belief systems—the latter being based on certain geometrical structures induced by probabilistic belief systems. For illustration, we reconsider BonJour’s “ravens” challenge (op. cit., p. 95f.). Finally, potential objections to our proposed formal coherence notion are explored. (shrink)

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small (...) model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs. (shrink)

Magnetoencephalography (MEG) and electroencephalography (EEG) are invaluable neuroscientific tools for unveiling human neural dynamics in three dimensions (space, time, and frequency), which are associated with a wide variety of perceptions, cognition, and actions. MEG/EEG also provides different categories of neuronal indices including activity magnitude, connectivity, and network properties along the three dimensions. In the last 20 years, interest has increased in inter-regional connectivity and complex network properties assessed by various sophisticated scientific analyses. We herein review the definition, computation, (...) short history, and pros and cons of connectivity and complex network (graph-theory) analyses applied to MEG/EEG signals. We briefly describe recent developments in source reconstruction algorithms essential for source-space connectivity and network analyses. Furthermore, we discuss a relatively novel approach used in MEG/EEG studies to examine the complex dynamics represented by human brain activity. The correct and effective use of these neuronal metrics provides a new insight into the multi-dimensional dynamics of the neural representations of various functions in the complex human brain. (shrink)

nature of modern data collection and storage techniques, and the increases in the speed and storage capacities of computers. Statistics books from 30 years ago often presented examples with fewer than 10 variables, in domains where some background knowledge was plausible. In contrast, in new domains, such as climate research where satellite data now provide daily quantities of data unthinkable a few decades ago, fMRI brain imaging, and microarray measurements of gene expression, the number of variables can range into the (...) tens of thousands, and there is often limited background knowledge to reduce the space of alternative causal hypotheses. In such domains, non-automated causal discovery techniques appear to be hopeless, while the availability of faster computers with larger memories and disc space allow for the practical implementation of computationally intensive automated search algorithms over large search spaces. Contemporary science is not your grandfather’s science, or Karl Popper’s. Causal inference without experimental controls has long seemed as if it must somehow be capable of being cast as a kind of statistical inference involving estimators with some kind of convergence and accuracy properties under some kind of assumptions. Until recently, the statistical literature said not. While parameter estimation and experimental design for the effective use of data developed throughout the 20th century, as recently as 20 years ago the methodology of causal inference without experimental controls remained relatively primitive. Besides a cessation of hostilities from the majority of the statistical and philosophical communities (which has still only partially happened), several things were needed for theories of causal estimation to appear and to flower: well defined mathematical objects to represent causal relations; well defined connections between aspects of these objects and sample data; and a way to compute those connections. A sequence of studies beginning with Dempster’s work on the factorization of probability distributions [Dempster 1972] and culminating with Kiiveri and Speed’s [Kiiveri & Speed 1982] study of linear structural equation models, provided the first, in the form of directed acyclic graphs, and the second, in the form of the “local” Markov condition.. (shrink)

Several papers (e.g. [7, 23, 42]) have recently sought to give general frameworks for online structures and algorithms ([4]), and seeking to connect, if only by analogy, online and computable structure theory. These initiatives build on earlier work on online colouring and other combinatorial algorithms by Bean [10], Kierstead, Trotter et al. [48, 54, 57] and others, as we discuss below. In this paper we will look at such frameworks and illustrate them with examples from the first author’s MSc (...) Thesis ([58]). In this thesis, Askes looks at online, adversarial online and strongly online algorithms and structures. Additionally, we prove some new theorems about online algorithms on graphs of bounded pathwidth as well as some new results on punctually 1-decidable structures. (shrink)

Proponents of mechanistic explanation all acknowledge the importance of organization. But they have also tended to emphasize specificity with respect to parts and operations in mechanisms. We argue that in understanding one important mode of organization—patterns of causal connectivity—a successful explanatory strategy abstracts from the specifics of the mechanism and invokes tools such as those of graph theory to explain how mechanisms with a particular mode of connectivity will behave. We discuss the connection between organization, abstraction, (...) and mechanistic explanation and illustrate our claims by looking at an example from recent research on so-called network motifs. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the (...) predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

Multiple -conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations. This is a sustained study of the subject and is certain to stimulate further research. Part I reworks (...) the fundamental ideas of logic to take account of multiple conclusions, and investigates the connections between multiple - and single - conclusion calculi. Part II draws on graph theory to discuss the form and validity of arguments independently of particular logical systems. Part III contrasts the multiple - and the single - conclusion treatment of one and the same subject, using many-valued logic as the example; and Part IV shows how the methods of 'natural deduction' can be matched by direct proofs using multiple conclusions. (shrink)

The idea that intelligence is embedded not only in a single brain network, but instead in a complex, well-optimized system of complementary networks, has led to the development of whole brain network analysis. Using graph theory to analyze resting-state functional MRI data, we investigated the brain graph networks (or brain networks) of high intelligence quotient (HIQ) children. To this end, we computed the “hub disruption index κ”, an index sensitive to graph network modifications. We found significant (...) topological differences in the integration and segregation properties of brain networks in HIQ compared to standard IQ children, not only for the whole brain graph, but also for each hemispheric graph, and for the homotopic connectivity. Moreover, two profiles of HIQ children, homogenous and heterogeneous, based on the differences between the two main IQ subscales (verbal comprehension index (VCI) and perceptual reasoning index (PRI)), were compared. Brain networks changes were more pronounced in the heterogeneous HIQ than in the homogeneous HIQ subgroups. Finally, we found significant correlations between the graph networks’ changes and the full-scale IQ (FSIQ), as well as the subscales VCI and PRI. Specifically, the higher the FSIQ the greater was the brain organization modification in the whole brain, the left hemisphere, and the homotopic connectivity. These results shed new light on the relation between functional connectivity topology and high intelligence, as well as on different intelligence profiles. (shrink)

The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the (...) point of view ofAlgebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies. (shrink)

We examine recent work in cognitive neuroscience that investigates brain networks. Brain networks are characterized by the ways in which brain regions are functionally and anatomically connected to one another. Cognitive neuroscientists use various noninvasive techniques (e.g., fMRI) to investigate these networks. They represent them formally as graphs. And they use various graph theoretic techniques to analyze them further. We distinguish between knowledge of the graph theoretic structure of such networks (structural knowledge) and knowledge of what instantiates that (...) structure (nonstructural knowledge). And we argue that this work provides structural knowledge of brain networks. We explore the significance of this conclusion for the scientific realism debate. We argue that our conclusion should not be understood as an instance of a global structural realist claim regarding the structure of the unobservable part of the world, but instead, as a local structural realist attitude towards brain networks in particular. And we argue that various local approaches to the realism debate, i.e., approaches that restrict realist commitments to particular theories and/or entities, are problematic insofar as they don't allow for the possibility of such a local structural realist attitude. (shrink)

Topological indices are of incredible significance in the field of graph theory. Convex polytopes play a significant role both in various branches of mathematics and also in applied areas, most notably in linear programming. We have calculated some topological indices such as atom-bond connectivity index, geometric arithmetic index, K-Banhatti indices, and K-hyper-Banhatti indices and modified K-Banhatti indices from some families of convex polytopes through M-polynomials. The M-polynomials of the graphs provide us with a great help to calculate (...) the topological indices of different structures. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: (...) the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

In contemporary human brain mapping, it is commonly assumed that the “mind is what the brain does”. Based on that assumption, task-based imaging studies of the last three decades measured differences in brain activity that are thought to reflect the exercise of human mental capacities (e.g., perception, attention, memory). With the advancement of resting state studies, tractography and graph theory in the last decade, however, it became possible to study human brain connectivity without relying on cognitive tasks (...) or constructs. It therefore is currently an open question whether the assumption that “the mind is what the brain does” is an indispensable working hypothesis in human brain mapping. This paper argues that the hypothesis is, in fact, dispensable. If it is dropped, researchers can “meet the brain on its own terms” by searching for new, more adequate concepts to describe human brain organization. Neuroscientists can establish such concepts by conducting exploratory experiments that do not test particular cognitive hypotheses. The paper provides a systematic account of exploratory neuroscientific research that would allow researchers to form new concepts and formulate general principles of brain connectivity, and to combine connectivity studies with manipulation methods to identify neural entities in the brain. These research strategies would be most fruitful if applied to the mesoscopic scale of neuronal assemblies, since the organizational principles at this scale are currently largely unknown. This could help researchers to link microscopic and macroscopic evidence to provide a more comprehensive understanding of the human brain. The paper concludes by comparing this account of exploratory neuroscientific experiments to recent proposals for large-scale, discovery-based studies of human brain connectivity. (shrink)

It is not only difficult to be a sports expert but also difficult to grow from a sports expert to a sports elite. Professional athletes are often concerned about the differences between an expert and an elite and how to eventually become an elite athlete. To explore the differences in brain neural mechanism between experts and elites in the process of motor behavior and reveal the internal connection between motor performance and brain activity, we collected and analyzed the electroencephalography findings (...) of 14 national archers and 14 provincial archers during aiming and resting states and constructed the EEG brain network of the two archer groups based on weighted phase lag index; the graph theory was used to analyze and compare the network characteristics via local network and global network topologies. The results showed that compared with the expert archers, the elite archers had stronger functional coupling in beta1 and beta2 bands, and the difference was evident in the frontal and central regions; in terms of global characteristics of brain network topology, the average clustering coefficient and global efficiency of elite archers were significantly higher than that of expert archers, and the eigenvector centrality of expert archers was higher; for local characteristics, elite archers had higher local efficient; and the brain network characteristics of expert archers showed a strong correlation with archery performance. This suggests that compared with expert archers, elite archers showed stronger functional coupling, higher integration efficiency of global and local information, and more independent performance in the archery process. These findings reveal the differences in brain electrical network topologies between elite and expert archers in the archery preparation stage, which is expected to provide theoretical reference for further training and promotion of professional athletes. (shrink)

How many words—and which ones—are sufficient to define all other words? When dictionaries are analyzed as directed graphs with links from defining words to defined words, they reveal a latent structure. Recursively removing all words that are reachable by definition but that do not define any further words reduces the dictionary to a Kernel of about 10% of its size. This is still not the smallest number of words that can define all the rest. About 75% of the Kernel turns (...) out to be its Core, a “Strongly Connected Subset” of words with a definitional path to and from any pair of its words and no word's definition depending on a word outside the set. But the Core cannot define all the rest of the dictionary. The 25% of the Kernel surrounding the Core consists of small strongly connected subsets of words: the Satellites. The size of the smallest set of words that can define all the rest—the graph's “minimum feedback vertex set” or MinSet—is about 1% of the dictionary, about 15% of the Kernel, and part-Core/part-Satellite. But every dictionary has a huge number of MinSets. The Core words are learned earlier, more frequent, and less concrete than the Satellites, which are in turn learned earlier, more frequent, but more concrete than the rest of the Dictionary. In principle, only one MinSet's words would need to be grounded through the sensorimotor capacity to recognize and categorize their referents. In a dual-code sensorimotor/symbolic model of the mental lexicon, the symbolic code could do all the rest through recombinatory definition. (shrink)

Consideration of the amazing organized intricacy of human cortical anatomy entails a deeper appreciation of nature that is fully consistent with a mature religious spirit. A brain seems at first glance to be a mere lump of grayish claylike stuff, but facts of basic neuroanatomy compel us to consider that this particular kind of stuff may really contain all the richly tangible and richly ghostly inner essences of emotion, thought, and behavior. Humans are the “college graduates” of evolution. The human (...) cortex is 3,400 times the volume of, yet only slightly thicker (about 3 millimeters) than, that of the mouse. This remarkable sheet is as thin as a graduation‐day “mortarboard” cap, but its 2,600 square centimeter area is four times as large (about 20 × 20 inches if a square; both metric and English units used deliberately). Zooming in, there are about 50 billion cortical neurons; though named after “pyramids,” they are more like tiny “magic trees,” with branches and roots so long and fine that there are 1 or 2 miles of these electrically scintillating fibers within each cubic millimeter of cortex. Cortical neurons communicate intimately: viewed from above, beneath a single square millimeter 100,000 nerve cells intertwine; each such neuron makes 5,000 or more connections with others. These and many additional amazing facts about brain tissue, together with some conjectures about dense connectedness in the mathematics of graph theory, help to bear out the groundwork prepared by such pioneers as Ralph Wendell Burhoe that the spirit and knowledge of science might rejoin that of religion. If it takes enchanted matter to contain consciousness, this is a kind of enchantment that science may well be able to penetrate for eventual thoroughgoing understanding. Inevitable by‐products will be greater reverence for nature and greater awe at the mystery of nature's origin. (shrink)

Purpose: We aimed to find out the distributed functional connectome of white matter in patients with functional dyspepsia.Methods: 20 patients with FD and 24 age- and gender-matched healthy controls were included into the study. The functional connectome of white matter and graph theory were used to these participants. Two-sample t-test was used for the detection the abnormal graph properties in FD. Pearson correlation was used for the relationship between properties and the clinical and neuropshychological information.Results: Patients with (...) FD and healthy controls showed small-world properties in functional connectome of white matter. Compared with healthy controls, the FD group showed decreased global properties. Four pairs of fiber bundles that are connected to the frontal lobe, insula, and thalamus were affected in the FD group. Duration and Pittsburgh Sleep Quality Index positively correlated with the betweenness centrality of white matter regions of interest.Conclusion: FD patients turned to a non-optimized functional organization of WM brain network. Frontal lobe, insula, and thalamus were key regions in brain information exchange of FD. It provided some novel imaging evidences for the mechanism of FD. (shrink)

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite tree with a (...) root has a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree. (shrink)

Micro-expressions can reflect an individual’s subjective emotions and true mental state, and they are widely used in the fields of mental health, justice, law enforcement, intelligence, and security. However, one of the major challenges of working with MEs is that their neural mechanism is not entirely understood. To the best of our knowledge, the present study is the first to use electroencephalography to investigate the reorganizations of functional brain networks involved in MEs. We aimed to reveal the underlying neural mechanisms (...) that can provide electrophysiological indicators for ME recognition. A real-time supervision and emotional expression suppression experimental paradigm was designed to collect video and EEG data of MEs and no expressions of 70 participants expressing positive emotions. Based on the graph theory, we analyzed the efficiency of functional brain network at the scalp level on both macro and micro scales. The results revealed that in the presence of MEs compared with NEs, the participants exhibited higher global efficiency and nodal efficiency in the frontal, occipital, and temporal regions. Additionally, using the random forest algorithm to select a subset of functional connectivity features as input, the support vector machine classifier achieved a classification accuracy for MEs and NEs of 0.81, with an area under the curve of 0.85. This finding demonstrates the possibility of using EEG to recognize MEs, with a wide range of application scenarios, such as persons wearing face masks or patients with expression disorders. (shrink)

Scientists have attempted several times to define the notion of complexity. A proper definition uses elements of three sets: a set of sites, as set of connections, and a set of nodes coincides with the set. Sites and connections can be translated into terms of graph theory as vertices and edges, which enables to consider complexity as an associated graph.Thus complexity of a system (or a structure) will be defined as the number of possible figures and aspects (...) which are obtained by combining vertices and edges. Complexity is the product of two factors, the first factor is tied to the combination of nodes called mutability and the second is tied to the combination of edges called liability. (shrink)

Network theory arguably has its origins in Euler’s (1741) graph theory, which was first developed in the mid-1700s to solve the Königsberg bridge problem. Since then, the basic units of graph theory—vertices and edges—have been utilized by a number of scientific disciplines to describe and analyze a wide variety of phenomena. Mark Newman begins his clear and comprehensive introduction to networks with a sampling of various kinds that have been studied: information networks such as the (...) World Wide Web, biological networks such as neural connections, and social networks such as friendships among members of a club. As Newman highlights, network theory and methods are utilized not only within but also across scientific disciplines. The interdisciplinary nature of the scientific study of networks is a double-edged sword. On the one hand, it fosters the cross-pollination of ideas and methods across fields—for example, applying methods used to study computer networks to the investigation of brain networks (cf., Sporns, 2011). On the other hand, such interdisciplinarity can result in confusion and misapplication of theory and methods. Accordingly, Newman’s goal in “Networks: An Introduction” is to synthesize the current state of network theory into a “consistent language and notation” (2010, p. x, Preface). This is no easy task, but Newman gives an admirable attempt that is successful in many ways. (shrink)

This article discusses some philosophical theories of causation and their application to several areas of science. Topics addressed include regularity, counterfactual, and causal process theories of causation; the causal interpretation of structural equation models and directed graphs; independence assumptions in causal reasoning; and the role of causal concepts in physics. In connection with this last topic, this article focuses on the relationship between causal asymmetries, the time-reversal invariance of most fundamental physical laws, and the significance of differences among varieties of (...) differential equations in causal interpretation. It concludes with some remarks about “grounding” special science causal generalizations in physics. (shrink)

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs—describing the quantum (...) correlations of the states—are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Gödel’s incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven. (shrink)

Theory of networks serves as a mathematical foundation for the construction and modeling of chemical structures and complicated networks. In particular, chemical networking theory has a wide range of utilizations in the study of chemical structures, where examination and manipulation of chemical structural information are made feasible by utilizing the numerical graph invariants. A network invariant or a topological index is a numerical measure of a chemical compound which is capable to describe the chemical structural properties such (...) as melting point, freezing point, density, pressure, tension, and temperature of chemical compounds. Wiener initiated the first distance-based TI which is considered to be the most important TI to preserve the chemical and physical properties of chemical structures. Later on, degree-based TI was introduced to find the π -electron energy of molecules. Recently, connection number-based TIs are studied which are more efficient than degree and distance-based TIs. In this paper, we compute the connection number-based TIs of the structure of crystal cubic carbons which are one of the most significant and interesting composites in modern resources of science due to the involvement of carbon atoms. (shrink)

This book consolidates and extends the authors’ work on the connection between iconicity and abductive inference. It emphasizes a pragmatic, experimental and fallibilist view of knowledge without sacrificing formal rigor. Within this context, the book focuses particularly on scientific knowledge and its prevalent use of mathematics. To find an answer to the question “What kind of experimental activity is the scientific employment of mathematics?” the book addresses the problems involved in formalizing abductive cognition. For this, it implements the concept and (...) method of iconicity, modeling this theoretical framework mathematically through category theory and topoi. Peirce's concept of iconic signs is treated in depth, and it is shown how Peirce's diagrammatic logical notation of Existential Graphs makes use of iconicity and how important features of this iconicity are representable within category theory. Alain Badiou’s set-theoretical model of truth procedures and his relational sheaf-based theory of phenomenology are then integrated within the Peircean logical context. Finally, the book opens the path towards a more naturalist interpretation of the abductive models developed in Peirce and Badiou through an analysis of several recent attempts to reformulate quantum mechanics with categorical methods. Overall, the book offers a comprehensive and rigorous overview of past approaches to iconic semiotics and abduction, and it encompasses new extensions of these methods towards an innovative naturalist interpretation of abductive reasoning. (shrink)

Directed acyclic graphs (DAGs) constitute a qualitative representation for conditional independence (CI) properties of a probability distribution. It is known that every CI statement implied by the topology of a DAG is witnessed over it under a graph-theoretic criterion of d-separation. Alternatively, all such implied CI statements are derivable from the local independencies encoded by a DAG using the so-called semi-graphoid axioms. We consider Labeled Directed Acyclic Graphs (LDAGs) modeling graphically scenarios exhibiting context-specific independence (CSI). Such CSI statements are (...) modeled by labeled edges, where labels encode contexts in which the edge vanishes. We study the problem of identifying all independence statements implied by the structure and the labels of an LDAG. We show that this problem is coNP-hard for LDAGs and formulate a sound extension of the semi-graphoid axioms for the derivation of such implied independencies. Finally we connect our study to certain qualitative versions of independence ubiquitous in database theory and teams semantics. (shrink)