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Wu, Xinhe. "Boolean-Valued Models and Their Applications." Bulletin of Symbolic Logic 28, no. 4 (December 2022): 533. http://dx.doi.org/10.1017/bsl.2022.34.
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AbstractBoolean-valued models generalize classical two-valued models by allowing arbitrary complete Boolean algebras as value ranges. The goal of my dissertation is to study Boolean-valued models and explore their philosophical and mathematical applications.In Chapter 1, I build a robust theory of first-order Boolean-valued models that parallels the existing theory of two-valued models. I develop essential model-theoretic notions like “Boolean-valuation,” “diagram,” and “elementary diagram,” and prove a series of theorems on Boolean-valued models, including the (strengthened) Soundness and Completeness Theorem, the Löwenheim–Skolem Theorems, the Elementary Chain Theorem, and many more.Chapter 2 gives an example of a philosophical application of Boolean-valued models. I apply Boolean-valued models to the language of mereology to model indeterminacy in the parthood relation. I argue that Boolean-valued semantics is the best degree-theoretic semantics for the language of mereology. In particular, it trumps the well-known alternative—fuzzy-valued semantics. I also show that, contrary to what many have argued, indeterminacy in parthood entails neither indeterminacy in existence nor indeterminacy in identity, though being compatible with both.Chapter 3 (joint work with Bokai Yao) gives an example of a mathematical application of Boolean-valued models. Scott and Solovay famously used Boolean-valued models on set theory to obtain relative consistency results. In Chapter 3, I investigate two ways of extending the Scott–Solovay construction to set theory with urelements. I argue that the standard way of extending the construction faces a serious problem, and offer a new way that is free from the problem.Abstract prepared by Xinhe Wu.E-mail: xinhewu@mit.edu
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Dahn, Bernd I. "Boolean valued models and incomplete specifications." Journal of Logic Programming 12, no. 3 (February 1992): 225–36. http://dx.doi.org/10.1016/0743-1066(92)90025-x.
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OZAWA, MASANAO. "ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY." Review of Symbolic Logic 10, no. 4 (June 5, 2017): 782–807. http://dx.doi.org/10.1017/s1755020317000120.
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AbstractIn 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.
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Hansen, Lars. "On an algebra of lattice-valued logic." Journal of Symbolic Logic 70, no. 1 (March 2005): 282–318. http://dx.doi.org/10.2178/jsl/1107298521.
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AbstractThe purpose of this paper is to present an algebraic generalization of the traditional two-valued logic. This involves introducing a theory of automorphism algebras, which is an algebraic theory of many-valued logic having a complete lattice as the set of truth values. Two generalizations of the two-valued case will be considered, viz., the finite chain and the Boolean lattice. In the case of the Boolean lattice, on choosing a designated lattice value, this algebra has binary retracts that have the usual axiomatic theory of the propositional calculus as suitable theory. This suitability applies to the Boolean algebra of formalized token models [2] where the truth values are, for example, vocabularies. Finally, as the actual motivation for this paper, we indicate how the theory of formalized token models [2] is an example of a many-valued predicate calculus.
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Hernandez, E. G. "Boolean-Valued Models of Set Theory with Automorphisms." Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 32, no. 7-9 (1986): 117–30. http://dx.doi.org/10.1002/malq.19860320704.
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Dobrić, Vladimir, Pavle Milošević, Aleksandar Rakićević, Bratislav Petrović, and Ana Poledica. "Interpolative Boolean Networks." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/2647164.
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Boolean networks are used for modeling and analysis of complex systems of interacting entities. Classical Boolean networks are binary and they are relevant for modeling systems with complex switch-like causal interactions. More descriptive power can be provided by the introduction of gradation in this model. If this is accomplished by using conventional fuzzy logics, the generalized model cannot secure the Boolean frame. Consequently, the validity of the model’s dynamics is not secured. The aim of this paper is to present the Boolean consistent generalization of Boolean networks, interpolative Boolean networks. The generalization is based on interpolative Boolean algebra, the [0,1]-valued realization of Boolean algebra. The proposed model is adaptive with respect to the nature of input variables and it offers greater descriptive power as compared with traditional models. For illustrative purposes, IBN is compared to the models based on existing real-valued approaches. Due to the complexity of the most systems to be analyzed and the characteristics of interpolative Boolean algebra, the software support is developed to provide graphical and numerical tools for complex system modeling and analysis.
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Butz, C., and I. Moerdijk. "An elementary definability theorem for first order logic." Journal of Symbolic Logic 64, no. 3 (September 1999): 1028–36. http://dx.doi.org/10.2307/2586617.
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In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.
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Molchanov, I. S. "Set-Valued Estimators for Mean Bodies Related to Boolean Models." Statistics 28, no. 1 (January 1996): 43–56. http://dx.doi.org/10.1080/02331889708802547.
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Trinh, Van-Giang, Belaid Benhamou, Thomas Henzinger, and Samuel Pastva. "Trap spaces of multi-valued networks: definition, computation, and applications." Bioinformatics 39, Supplement_1 (June 1, 2023): i513—i522. http://dx.doi.org/10.1093/bioinformatics/btad262.
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Abstract Motivation Boolean networks are simple but efficient mathematical formalism for modelling complex biological systems. However, having only two levels of activation is sometimes not enough to fully capture the dynamics of real-world biological systems. Hence, the need for multi-valued networks (MVNs), a generalization of Boolean networks. Despite the importance of MVNs for modelling biological systems, only limited progress has been made on developing theories, analysis methods, and tools that can support them. In particular, the recent use of trap spaces in Boolean networks made a great impact on the field of systems biology, but there has been no similar concept defined and studied for MVNs to date. Results In this work, we generalize the concept of trap spaces in Boolean networks to that in MVNs. We then develop the theory and the analysis methods for trap spaces in MVNs. In particular, we implement all proposed methods in a Python package called trapmvn. Not only showing the applicability of our approach via a realistic case study, we also evaluate the time efficiency of the method on a large collection of real-world models. The experimental results confirm the time efficiency, which we believe enables more accurate analysis on larger and more complex multi-valued models. Availability and implementation Source code and data are freely available at https://github.com/giang-trinh/trap-mvn.
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Pantle, Ursa, Volker Schmidt, and Evgueni Spodarev. "Central limit theorems for functionals of stationary germ-grain models." Advances in Applied Probability 38, no. 1 (March 2006): 76–94. http://dx.doi.org/10.1239/aap/1143936141.
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Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.
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Pantle, Ursa, Volker Schmidt, and Evgueni Spodarev. "Central limit theorems for functionals of stationary germ-grain models." Advances in Applied Probability 38, no. 01 (March 2006): 76–94. http://dx.doi.org/10.1017/s0001867800000811.
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Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.
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Monro, G. P. "A category-theoretic approach to boolean-valued models of set theory." Journal of Pure and Applied Algebra 42, no. 3 (October 1986): 245–74. http://dx.doi.org/10.1016/0022-4049(86)90010-1.
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Jech, Thomas. "First Order Theory of Complete Stonean Algebras (Boolean-Valued Real and Complex Numbers)." Canadian Mathematical Bulletin 30, no. 4 (December 1, 1987): 385–92. http://dx.doi.org/10.4153/cmb-1987-057-7.
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AbstractWe axiomatize the theory of real and complex numbers in Boolean-valued models of set theory, and prove that every Horn sentence true in the complex numbers is true in any complete Stonean algebra, and provable from its axioms.
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Scedrov, Andre. "Embedding sheaf models for set theory into boolean-valued permutation models with an interior operator." Annals of Pure and Applied Logic 32 (1986): 103–9. http://dx.doi.org/10.1016/0168-0072(86)90046-1.
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Ozawa, Masanao. "From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti." Mathematics 9, no. 4 (February 17, 2021): 397. http://dx.doi.org/10.3390/math9040397.
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Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.
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Milosevic, Pavle, Ivan Nesic, Ana Poledica, Dragan Radojevic, and Branislav Petrovic. "Logic-based aggregation methods for ranking student applicants." Yugoslav Journal of Operations Research 27, no. 4 (2017): 463–79. http://dx.doi.org/10.2298/yjor161110007m.
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In this paper, we present logic-based aggregation models used for ranking student applicants and we compare them with a number of existing aggregation methods, each more complex than the previous one. The proposed models aim to include depen- dencies in the data using Logical aggregation (LA). LA is a aggregation method based on interpolative Boolean algebra (IBA), a consistent multi-valued realization of Boolean algebra. This technique is used for a Boolean consistent aggregation of attributes that are logically dependent. The comparison is performed in the case of student applicants for master programs at the University of Belgrade. We have shown that LA has some advantages over other presented aggregation methods. The software realization of all applied aggregation methods is also provided. This paper may be of interest not only for student ranking, but also for similar problems of ranking people e.g. employees, team members, etc.
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Bykovsky, Alexey Yu. "Heterogeneous Network Architecture for Integration of AI and Quantum Optics by Means of Multiple-Valued Logic." Quantum Reports 2, no. 1 (January 31, 2020): 126–65. http://dx.doi.org/10.3390/quantum2010010.
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Quantum optics is regarded as the acknowledged method to provide network quantum keys distribution and in the future secure distributed quantum computing, but it should also provide cryptography protection for mobile robots and the Internet of Things (IoT). This task requires the design of new secret coding schemes, which can be also based on multiple-valued logic (MVL). However, this very specific logic model reveals new possibilities for the hierarchical data clustering of arbitrary data sets. The minimization of multiple-valued logic functions is proposed for the analysis of aggregated objects, which is possible for an arbitrary number of variables. In order to use all the useful properties of the multiple-valued logic, the heterogeneous network architecture is proposed, which includes three allocated levels of artificial intelligence (AI) logic modeling for discrete multiple-valued logic, Boolean logic, and fuzzy logic. Multiple-valued logic is regarded as the possible platform for additional secret coding, data aggregation, and communications, which are provided by the united high dimensional space for network addressing and the targeted control of robotic devices. Models of Boolean and fuzzy logic are regarded as separate logic levels in order to simplify the integration of various algorithms and provide control of additional data protection means for robotic agents.
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Truss, J. K. "BOOLEAN-VALUED MODELS AND INDEPENDENCE PROOFS IN SET THEORY Second Edition (Oxford Logic Guides, 12)." Bulletin of the London Mathematical Society 18, no. 1 (January 1986): 82–83. http://dx.doi.org/10.1112/blms/18.1.82.
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Nikolaidis, Pavlos. "Mixed Thermal and Renewable Energy Generation Optimization in Non-Interconnected Regions via Boolean Mapping." Thermo 4, no. 4 (October 23, 2024): 445–60. http://dx.doi.org/10.3390/thermo4040024.
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Global efforts aiming to shift towards renewable energy and smart grid configurations require accurate unit commitment schedules to guarantee power balance and ensure feasible operation under different complex constraints. Intelligent systems utilizing hybrid and high-level techniques have arisen as promising solutions to provide optimum exploration–exploitation trade-offs at the expense of computational complexity. To ameliorate this requirement, which is extremely expensive in non-interconnected renewable systems, radically different approaches based on enhanced priority schemes and Boolean encoding/decoding have to take place. This compilation encompasses various mappings that convert multi-valued clausal forms into Boolean expressions with equivalent satisfiability. Avoiding any need to introduce prior parameter settings, the solution utilizes state-of-the-art advancements in the field of artificial intelligence models, namely Boolean mapping. It allows for the efficient identification of the optimal configuration of a non-convex system with binary and discontinuous dynamics in the fewest possible trials, providing impressive performance. In this way, Boolean mapping becomes capable of providing global optimum solutions to unit commitment utilizing fully tractable procedures without deteriorating the computational time. The results, considering a non-interconnected power system, show that the proposed model based on artificial intelligence presents advantageous performance in terms of generating cost and complexity. This is particularly important in isolated networks, where even a-not-so great deviation between production and consumption may reflect as a major disturbance in terms of frequency and voltage.
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Allwein, Gerard, and J. Michael Dunn. "Kripke models for linear logic." Journal of Symbolic Logic 58, no. 2 (June 1993): 514–45. http://dx.doi.org/10.2307/2275217.
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AbstractWe present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutativity and associativity are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is residuated with Linear Logic's cotensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic.
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Flagg, R. C. "Epistemic set theory is a conservative extension of intuitionistic set theory." Journal of Symbolic Logic 50, no. 4 (December 1985): 895–902. http://dx.doi.org/10.2307/2273979.
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In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.There is one question quite central to the above program that has remained open. Namely:Does Ščedrov's extension of the Gödel translation to set theory provide a faithful interpretation of intuitionistic set theory into epistemic set theory?In the present paper we give an affirmative answer to this question.The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.
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Chaouiya, Claudine, Sarah M. Keating, Duncan Berenguier, Aurélien Naldi, Denis Thieffry, Martijn P. van Iersel, Nicolas Le Novère, and Tomáš Helikar. "SBML Level 3 package: Qualitative Models, Version 1, Release 1." Journal of Integrative Bioinformatics 12, no. 2 (June 1, 2015): 691–730. http://dx.doi.org/10.1515/jib-2015-270.
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Summary Quantitative methods for modelling biological networks require an in-depth knowledge of the biochemical reactions and their stoichiometric and kinetic parameters. In many practical cases, this knowledge is missing. This has led to the development of several qualitative modelling methods using information such as, for example, gene expression data coming from functional genomic experiments. The SBML Level 3 Version 1 Core specification does not provide a mechanism for explicitly encoding qualitative models, but it does provide a mechanism for SBML packages to extend the Core specification and add additional syntactical constructs.The SBML Qualitative Models package for SBML Level 3 adds features so that qualitative models can be directly and explicitly encoded. The approach taken in this package is essentially based on the definition of regulatory or influence graphs. The SBML Qualitative Models package defines the structure and syntax necessary to describe qualitative models that associate discrete levels of activities with entity pools and the transitions between states that describe the processes involved. This is particularly suited to logical models (Boolean or multi-valued) and some classes of Petri net models can be encoded with the approach.
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Athreya, Siva, Rahul Roy, and Anish Sarkar. "On the coverage of space by random sets." Advances in Applied Probability 36, no. 1 (March 2004): 1–18. http://dx.doi.org/10.1239/aap/1077134461.
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Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞)d, d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃i≥1 {ξi + [0,ρi]d}. If, for some t > 0, (0,∞)d ⊆ C, then we say that (0,∞)d is eventually covered by C. We show that the eventual coverage of (0,∞)d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝd by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:Xi=1} [i,i+ρi], where X1, X2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.
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Athreya, Siva, Rahul Roy, and Anish Sarkar. "On the coverage of space by random sets." Advances in Applied Probability 36, no. 01 (March 2004): 1–18. http://dx.doi.org/10.1017/s0001867800012842.
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Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞) d , d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃ i≥1 {ξ i + [0,ρ i ] d }. If, for some t > 0, (0,∞) d ⊆ C, then we say that (0,∞) d is eventually covered by C. We show that the eventual coverage of (0,∞) d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝ d by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:X i =1} [i,i+ρ i ], where X 1, X 2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.
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Varela, Pedro L., Camila V. Ramos, Pedro T. Monteiro, and Claudine Chaouiya. "EpiLog: A software for the logical modelling of epithelial dynamics." F1000Research 7 (July 27, 2018): 1145. http://dx.doi.org/10.12688/f1000research.15613.1.
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Cellular responses are governed by regulatory networks subject to external signals from surrounding cells and to other micro-environmental cues. The logical (Boolean or multi-valued) framework proved well suited to study such processes at the cellular level, by specifying qualitative models of involved signalling pathways and gene regulatory networks. Here, we describe and illustrate the main features of EpiLog, a computational tool that implements an extension of the logical framework to the tissue level. EpiLog defines a collection of hexagonal cells over a 2D grid, which embodies a mono-layer epithelium. Basically, it defines a cellular automaton in which cell behaviours are driven by associated logical models subject to external signals. EpiLog is freely available on the web at http://epilog-tool.org. It is implemented in Java (version ≥1.7 required) and the source code is provided at https://github.com/epilog-tool/epilog under a GNU General Public License v3.0.
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Varela, Pedro L., Camila V. Ramos, Pedro T. Monteiro, and Claudine Chaouiya. "EpiLog: A software for the logical modelling of epithelial dynamics." F1000Research 7 (March 11, 2019): 1145. http://dx.doi.org/10.12688/f1000research.15613.2.
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Cellular responses are governed by regulatory networks subject to external signals from surrounding cells and to other micro-environmental cues. The logical (Boolean or multi-valued) framework proved well suited to study such processes at the cellular level, by specifying qualitative models of involved signalling pathways and gene regulatory networks. Here, we describe and illustrate the main features of EpiLog, a computational tool that implements an extension of the logical framework to the tissue level. EpiLog defines a collection of hexagonal cells over a 2D grid, which embodies a mono-layer epithelium. Basically, it defines a cellular automaton in which cell behaviours are driven by associated logical models subject to external signals. EpiLog is freely available on the web at http://epilog-tool.org. It is implemented in Java (version ≥1.7 required) and the source code is provided at https://github.com/epilog-tool/epilog under a GNU General Public License v3.0.
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Čolić, Nevena, Pavle Milošević, Ivana Dragović, and Miljan S. Ćeranić. "IBA-VNS: A Logic-Based Machine Learning Algorithm and Its Application in Surgery." Mathematics 12, no. 7 (March 23, 2024): 950. http://dx.doi.org/10.3390/math12070950.
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The interpretability and explainability of machine learning (ML) approaches play a key role in the trustworthiness of ML models in various applications. The objective of this paper is to incorporate a logic-based reasoning in the ML model that is not only accurate but also interpretable and easily applied. More precisely, we propose a hybrid IBA-VNS approach based on interpolative Boolean algebra (IBA) and variable neighborhood search (VNS). IBA is chosen over traditional multi-valued and/or fuzzy logic techniques due to its consistency in preserving all Boolean axioms. The VNS heuristic is used for model training, i.e., determining the optimal logical aggregation function within the IBA framework for solving observed prediction problems. Obtained logic aggregation functions are easy to understand and may provide additional insight to the decision-maker. The proposed approach does not require any domain knowledge and is applicable in various domains. IBA-VNS is evaluated on several standard datasets. Further, IBA-VNS is applied to the real-world problem of predicting hospital length of stay (LOS), showing exceptional results in terms of interpretability and accuracy. In fact, the dataset is collected from the LabSerb program regarding colorectal surgeries in the period 2015–2023. The proposed approach extracted knowledge regarding the problem, i.e., the causal relations between the patient’s health condition and LOS, along with achieving an MAE of 1.144 days.
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BOTTONI, PAOLO, DANIELE GORLA, STEFANO KASANGIAN, and ANNA LABELLA. "A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes." Mathematical Structures in Computer Science 28, no. 4 (February 27, 2017): 508–32. http://dx.doi.org/10.1017/s0960129517000019.
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The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB, obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational models.
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Kucherov, Mikhail M., and Nina A. Bogulskaya. "Trilattice-Based Access Control Models: How to Secure Current Computer Network." MATEC Web of Conferences 210 (2018): 04053. http://dx.doi.org/10.1051/matecconf/201821004053.
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Designing security, from the hardware level, is essential to ensure the integrity of the intelligent cyber-physical infrastructure that is the Industrial Internet of Things (IIoT). If intelligent cyber-physical infrastructure fails to do the right things because it is insecure and vulnerable, then there will be negative social consequences [1]. Security is, in a sense, the access control to IIoT systems, which increasingly relies on the ability to compose different policies. Therefore, the advantage in any framework for compiling policies is that it is intuitive, formal, expressive, application-independent, as well as expandable to create domain-specific instances. Recently, such a scheme was proposed based on Belnap logic FOUR2 [2]. Four values of the Belnap bilattice have been interpreted as grant, deny, conflict, or unspecified with respect to access-control policy. Belnap’s four-valued logic has found a variety of applications in various fields, such as deductive database theory, distributed logic programming, and other areas. However, it turns out that the truth order in FOUR2 is a truth-and-falsity order at the same time [3]. The smallest lattice, where the orders of truth and falsity are independent of each other, which is especially important for security policy, is that of Shramko-Wansing’s SIXTEEN3. This generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering.Based onSIXTEEN3 and new Boolean predicates to control access [4], we define an expressive access-control policy language, having composition statements based on the statements of Schramko-Wansing’s logic. Natural orderings on politics are obtained by independent lifting the orders of truth and falsity of trilattice, which results in a query language in which conflict freedom analysis can be developed. The reduction of formal verification of queries to that on predicates over access requests enables to carry out policy analysis. We evaluate our approach through examples of control access model policy.
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Kucherov, M., and Nina A. Bogulskaya. "Trilattice-Based Access Control Models: How to Secure Current Computer Network Mikhail." WSEAS TRANSACTIONS ON COMPUTERS 22 (August 3, 2023): 98–103. http://dx.doi.org/10.37394/23205.2023.22.12.
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Designing security, from the hardware level, is essential to ensure the integrity of the intelligent cyberphysical infrastructure that is the Industrial Internet of Things (IIoT). If intelligent cyber-physical infrastructure fails to do the right things because it is insecure and vulnerable, then there will be negative social consequences [1]. Security is, in a sense, the access control to IIoT systems, which increasingly relies on the ability to compose different policies. Therefore, the advantage in any framework for compiling policies is that it is intuitive, formal, expressive, application-independent, as well as expandable to create domain-specific instances. Recently, such a scheme was proposed based on Belnap logic FOUR2 [2]. Four values of the Belnap bilattice have been interpreted as grant, deny, conflict, or unspecified with respect to access-control policy. Belnap's four-valued logic has found a variety of applications in various fields, such as deductive database theory, distributed logic programming, and other areas. However, it turns out that the truth order in FOUR2 is a truth-and-falsity order at the same time [3]. The smallest lattice, where the orders of truth and falsity are independent of each other, which is especially important for security policy, is that of Shramko-Wansing’s SIXTEEN3. This generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering. Based on SIXTEEN3 and new Boolean predicates to control access [4], we define an expressive access-control policy language, having composition statements based on the statements of Schramko-Wansing’s logic. Natural orderings on politics are obtained by independent lifting the orders of truth and falsity of trilattice, which results in a query language in which conflict freedom analysis can be developed. The reduction of formal verification of queries to that on predicates over access requests enables to carry out policy analysis. We evaluate our approach through examples of control access model policy.
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Meaney, Christopher, Michael Escobar, Therese A. Stukel, Peter C. Austin, Sumeet Kalia, Babak Aliarzadeh, Rahim Moineddin, and Michelle Greiver. "Using ICD-9 diagnostic codes for external validation of topic models derived from primary care electronic medical record clinical text data." Health Informatics Journal 29, no. 1 (January 2023): 146045822211156. http://dx.doi.org/10.1177/14604582221115667.
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Background/Objectives: Unsupervised topic models are often used to facilitate improved understanding of large unstructured clinical text datasets. In this study we investigated how ICD-9 diagnostic codes, collected alongside clinical text data, could be used to establish concurrent-, convergent- and discriminant-validity of learned topic models. Design/Setting: Retrospective open cohort design. Data were collected from primary care clinics located in Toronto, Canada between 01/01/2017 through 12/31/2020. Methods: We fit a non-negative matrix factorization topic model, with K = 50 latent topics/themes, to our input document term matrix (DTM). We estimated the magnitude of association between each Boolean-valued ICD-9 diagnostic code and each continuous latent topical vector. We identified ICD-9 diagnostic codes most strongly associated with each latent topical vector; and qualitatively interpreted how these codes could be used for external validation of the learned topic model. Results: The DTM consisted of 382,666 documents and 2210 words/tokens. We correlated concurrently assigned ICD-9 diagnostic codes with learned topical vectors, and observed semantic agreement for a subset of latent constructs (e.g. conditions of the breast, disorders of the female genital tract, respiratory disease, viral infection, eye/ear/nose/throat conditions, conditions of the urinary system, and dermatological conditions, etc.). Conclusions: When fitting topic models to clinical text corpora, researchers can leverage contemporaneously collected electronic medical record data to investigate the external validity of fitted latent variable models.
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Coniglio, Marcelo E., Aldo Figallo-Orellano, and Ana Claudia Golzio. "Non-deterministic algebraization of logics by swap structures1." Logic Journal of the IGPL 28, no. 5 (November 29, 2018): 1021–59. http://dx.doi.org/10.1093/jigpal/jzy072.
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Abstract Multialgebras (or hyperalgebras or non-deterministic algebras) have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency (or LFIs) that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics (as independently introduced by M. Fidel and D. Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI (which is closely connected with Kalman’s functor), suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics.
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Ozawa, Masanao. "Transfer principle in quantum set theory." Journal of Symbolic Logic 72, no. 2 (June 2007): 625–48. http://dx.doi.org/10.2178/jsl/1185803627.
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AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.
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Monro, G. P. "Logic, sheaves, and factorization systems." Journal of Symbolic Logic 58, no. 3 (September 1993): 872–93. http://dx.doi.org/10.2307/2275101.
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In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.
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Marino, P. "JOHN L. BELL. Set Theory: Boolean-Valued Models and Independence Proofs. Oxford: Clarendon Press, 2005. Oxford Logic Guides, No. 47. Pp. xxii + 191. ISBN 0-19-856852-5, 987-0-19-856852-0 (pbk)." Philosophia Mathematica 14, no. 3 (January 13, 2006): 392–94. http://dx.doi.org/10.1093/philmat/nkl006.
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Sachaniuk-Kavets’ka, N., O. Prozor, V. Khomyuk, and R. Shevchuk. "MATHEMATICAL DESCRIPTION OF THE DIFFERENTIATION OPERATION IN THE LOGICAL-TIME ENVIRONMENT." Information technology and computer engineering 57, no. 2 (2023): 93–98. http://dx.doi.org/10.31649/1999-9941-2023-57-2-93-98.
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Effective and timely processing of information is one of the most important problems of creating means of its processing at the level of human perception and thinking. Analytical processing of digital signals in a graphical or purely digital representation is somewhat limited and does not take into account the dynamics of signals and messages. The classical apparatus of logic is insufficient to describe the dynamics of system behavior over time. Therefore, it is important to develop models of so-called Boolean differential calculus, as this approach is based on the general concept of changing the logical variable, which will lead to a universal, in terms of dynamics, system of concepts and operations. To facilitate the preprocessing of dynamic digital variables and signals, the logic-time function of multivalued logic can be used. The purpose of this article is a mathematical representation of the differentiation of logic-time functions of multi-valued logic in the index form of the record using the simulation of its scheme. The paper shows the expediency of the idea of replacing an arbitrary digital signal (variable) that changes over time with a logic-time function, which allows to facilitate the preliminary analytical processing of digital signals and variables using the properties of such functions. The article presents a new mathematical apparatus for describing logic-time functions of multivalued logic and individual operations on them using modeling of known implementation schemes. The paper considers the peculiarities of the derivative of the multivalued logical-time function (LMF), as one of the most used and basic operations used in the study of signals and images. Its characteristics and features are shown for functions presented in index form. The general expression of the derivative of the n order is obtained and it is shown that different logic-time functions can have the same derivative. The concepts of the left and right derivative of the LMF are introduced and the relationship between them is shown. The properties of the k derivative of the BLMF were considered and the expressions for the derivative conjunction and disjunction were obtained. A possible structural scheme of the differentiator is presented, which opens up the possibility of hardware processing of multi-valued LMFs.
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Holliday, Wesley. "Possibility Frames and Forcing for Modal Logic." Australasian Journal of Logic 22, no. 2 (January 20, 2025): 44–288. https://doi.org/10.26686/ajl.v22i2.5680.
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This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability/correspondence theory, and completeness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with complete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allowing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of filter-descriptive possibility frames with p-morphisms. Applying our duality theory to definability theory, we prove analogues for possibility semantics of theorems of Goldblatt [1974] and Goldblatt and Thomason [1975] characterizing modally definable classes of frames. In addition, we discuss analogues for possibility semantics of first-order correspondence results in the style of Lemmon and Scott [1977], Sahlqvist [1975], and van Benthem [1976a]. Finally, applying our duality theory to completeness theory, we show that there are continuum many normal modal logics that can be characterized by full possibility frames but not by Kripke frames, that all Sahlqvist logics can be characterized by full possibility frames that contain no worlds, and that all normal modal logics can be characterized by filter-descriptive possibility frames.
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Wu, Jiachao. "A Boolean model for conflict-freeness in argumentation frameworks." AIMS Mathematics 8, no. 2 (2023): 3913–19. http://dx.doi.org/10.3934/math.2023195.
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<abstract><p>The Boolean models of argumentation semantics have been established in various ways. These models commonly translate the conditions of extension-based semantics into some constraints of the models. The goal of this work is to explore a simple method to build Boolean models for argumentation. In this paper, the attack relation is treated as an operator, and its value is calculated by the values of its target and source arguments. By examining the values of the attacks, a Boolean model of conflict-free sets is introduced. This novel method simplifies the existing ways by eliminating the various constraints. The conflict-free sets can be calculated by simply checking the values of the attacks.</p></abstract>
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Hug, Daniel, Günter Last, and Wolfgang Weil. "Generalized contact distributions of inhomogeneous Boolean models." Advances in Applied Probability 34, no. 1 (March 2002): 21–47. http://dx.doi.org/10.1239/aap/1019160948.
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The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.
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Hug, Daniel, Günter Last, and Wolfgang Weil. "Generalized contact distributions of inhomogeneous Boolean models." Advances in Applied Probability 34, no. 01 (March 2002): 21–47. http://dx.doi.org/10.1017/s000186780001137x.
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The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝ d and a gauge body B ⊂ ℝ d , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.
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Kara, Ahmet, Dan Olteanu, and Dan Suciu. "From Shapley Value to Model Counting and Back." Proceedings of the ACM on Management of Data 2, no. 2 (May 10, 2024): 1–23. http://dx.doi.org/10.1145/3651142.
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In this paper we investigate the problem of quantifying the contribution of each variable to the satisfying assignments of a Boolean function based on the Shapley value. Our main result is a polynomial-time equivalence between computing Shapley values and model counting for any class of Boolean functions that are closed under substitutions of variables with disjunctions of fresh variables. This result settles an open problem raised in prior work, which sought to connect the Shapley value computation to probabilistic query evaluation. We show two applications of our result. First, the Shapley values can be computed in polynomial time over deterministic and decomposable circuits, since they are closed under OR-substitutions. Second, there is a polynomial-time equivalence between computing the Shapley value for the tuples contributing to the answer of a Boolean conjunctive query and counting the models in the lineage of the query. This equivalence allows us to immediately recover the dichotomy for Shapley value computation in case of self-join-free Boolean conjunctive queries; in particular, the hardness for non-hierarchical queries can now be shown using a simple reduction from the \#P-hard problem of model counting for lineage in positive bipartite disjunctive normal form.
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Mazo, Raúl, Camille Salinesi, Daniel Diaz, Olfa Djebbi, and Alberto Lora-Michiels. "Constraints." International Journal of Information System Modeling and Design 3, no. 2 (April 2012): 33–68. http://dx.doi.org/10.4018/jismd.2012040102.
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Drawing from an analogy between features based Product Line (PL) models and Constraint Programming (CP), this paper explores the use of CP in the Domain Engineering and Application Engineering activities that are put in motion in a Product Line Engineering strategy. Specifying a PL as a constraint program instead of a feature model carries out two important qualities of CP: expressiveness and direct automation. On the one hand, variables in CP can take values over boolean, integer, real or even complex domains and not only boolean values as in most PL languages such as the Feature-Oriented Domain Analysis (FODA). Specifying boolean, arithmetic, symbolic and reified constraint, provides a power of expression that spans beyond that provided by the boolean dependencies in FODA models. On the other hand, PL models expressed as constraint programs can directly be executed and analyzed by off-the-shelf solvers. This paper explores the issues of (a) how to specify a PL model using CP, including in the presence of multi-model representation, (b) how to verify PL specifications, (c) how to specify configuration requirements, and (d) how to support the product configuration activity. Tests performed on a benchmark of 50 PL models show that the approach is efficient and scales up easily to very large and complex PL specifications.
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Arenas, Marcelo, Pablo Barceló, Leopoldo Bertossi, and Mikaël Monet. "The Tractability of SHAP-Score-Based Explanations for Classification over Deterministic and Decomposable Boolean Circuits." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 8 (May 18, 2021): 6670–78. http://dx.doi.org/10.1609/aaai.v35i8.16825.
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Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).
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Baumgartner, James E. "J. L. Bell. Boolean-valued models and independence proofs in set theory. Second edition of XLVI 165. Oxford logic guides, no. 12. Clarendon Press, Oxford University Press, Oxford and New York1985, xx + 165 pp. - Dana Scott. Foreword. A revised reprint of XLVI 165. Therein, pp. vii–xiii." Journal of Symbolic Logic 51, no. 4 (December 1986): 1076–77. http://dx.doi.org/10.2307/2273926.
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Hug, Daniel, and Wolfgang Weil. "Determination of Boolean models by densities of mixed volumes." Advances in Applied Probability 51, no. 01 (March 2019): 116–35. http://dx.doi.org/10.1017/apr.2019.5.
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AbstractIn Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.
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Resconi, Germano. "Boolean Models Guide Intentionally Continuous Information and Computation Inside the Brain." Oriental journal of computer science and technology 12, Issue 3 (September 13, 2019): 90–98. http://dx.doi.org/10.13005/ojcst12.03.03.
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In 1943 Machculloch and Pitts create the formal neuron where many input signals are linearly composed with different weights on the neuron soma. When the soma electrical signal goes over a specific threshold an output is produced. The main topic in this model is that the response is the same response as in a Boolean function used a lot for the digital computer. Logic functions can be simplified with the formal neuron. But there is the big problem for which not all logic functions, as XOR , cannot be designed in the formal neuron. After a long time the back propagation and many other neural models overcame the big problem in some cases but not in all cases creating a lot of uncertainty. The model proposed does not consider the formal neuron but the natural network controlled by a set of differential equations for neural channels that model the current and voltage on the neuron surface. The steady state of the probabilities is the activation state continuous function whose maximum and minimum are the values of the Boolean function associated with the activation time of spikes of the neuron. With this method the activation function can be designed when the Boolean functions are known. Moreover the neuron differential equation can be designed in order to realize the wanted Boolean function in the neuron itself. The activation function theory permits to compute the neural parameters in agreement with the intention.
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Noskov, Sergeу I., and Aleksandra R. Chekalova. "CONSTRUCTION OF A HOMOGENEOUS NESTED PIECEWISE LINEAR REGRESSION WITH LAGGING VARIABLES." Vestnik Chuvashskogo universiteta, no. 4 (December 25, 2024): 75–83. https://doi.org/10.47026/1810-1909-2024-4-75-83.
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When constructing regression models of objects of any nature, it is often necessary to use nonlinear approximating constructions, including piecewise linear ones, while the process under study can have a pronounced dynamic nature, therefore, lagging (lag) variables can be used as regressors. The research purpose is to develop an algorithmic method of identifying the parameters of a homogeneous nested piecewise linear regression with lagging variables. Materials and methods. To achieve the goal, the methods of reducing the problems of estimating the parameters of nested piecewise linear models to linear Boolean programming problems proposed earlier by one of the authors were used. The least absolute values method, known in regression analysis, was also applied. The volume of oil production in the Russian Federation was adopted as the modeling object using statistical initial data in 2013–2022. Research results. An algorithmic method of constructing a homogeneous nested piecewise linear regression with lagging variables has been developed, which is reduced to solving a linear Boolean programming problem. It has been applied to construct a model of the possible volume of oil production in the Russian Federation. In this case, data on the volume of capital investments of Russian vertically integrated oil companies and on the commissioning of new wells have been used as independent variables. Conclusions. The developed method of constructing a homogeneous nested piecewise linear regression with lagging variables using the least absolute values method is reduced to a linear Boolean programming problem. Such models allow us to identify limiting values of dependent variables, taking into account possible delays in influence at external and internal levels.
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Bibilo, P. N. "Synthesis of Modular Multipliers." Programmnaya Ingeneria 14, no. 8 (August 14, 2023): 377–87. http://dx.doi.org/10.17587/prin.14.377-387.
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The results of experiments on the circuit implementation of modular multipliers in the design library of ASIC (Application-Specific Integrated Circuits) and FPGA (Field-Programmable Gate Array) are presented. The initial descriptions of modular multiplier projects were given by systems of not fully defined (partial) Boolean functions and algorithmic VHDL descriptions. Logical optimization was carried out in the class of disjunctive normal forms (DNF) and representations of Boolean function systems by BDD (Binary Decision Diagrams). The synthesized circuits were evaluated by area and time delay. It is established that the use of partial Boolean function models and preliminary logical BDD optimization allows one to improve the parameters of synthesized ASIC and FPGA blocks for small module values, however, the best solutions for large module values can be obtained using algorithmic VHDL descriptions of modular multipliers. In the synthesis of modular multiplier circuits as part of an FPGA and the use of ISE and Vivado design systems it is advisable to use synthesized VHDL operations (a*b) mod p.
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K, Alice, and Surya K. "An instance of Satisfiability Problem learnt with Instance based, Decision trees, Naive Bayes." Knowledge Transactions on Applied Machine Learning 01, no. 04 (September 20, 2023): 21–30. http://dx.doi.org/10.59567/ktaml.v1.04.03.
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Though it is not recommended for mining the data sets produced by any closed-form solution, our approach is very much aiming at the standard satisfiability (SAT) or assignment problem in terms of searching for the existence of learning models and their performance using machine learning algorithms. Here we claim that we do not apply the analytical or theoretical method in the sense it is not based on conventional or well-established minimization rules or circuit synthesis in mind but the application of computational codes for prediction for the given Boolean circuits. The main objectives are to get classification on mapping Boolean literal values to some predefined functional output and to get maximum accuracy of such classifiers. Here we have enumerated the results up to 20 bits majority operator as an instance for constructing the learning models based on machine learning algorithms. The similarity and significance of the results are shown and thus it may open the possibility for extensibility and generalizability to other types of Boolean circuits. The maximum average performance is found to be 80.72% of accuracy and 74.81% of precision based on the selected set of classifiers.
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Kryvonosov, Valery, and Serhii Vasylenko. "MATHEMATICAL MODELS FOR DIAGNOSTICS OF A BOLTED CONDUCTIVE JOINT UNDER CONDITIONS OF CHANGING MODE PARAMETERS." Bulletin of the National Technical University "KhPI". Series: Energy: Reliability and Energy Efficiency, no. 1 (2) (July 2, 2021): 45–51. http://dx.doi.org/10.20998/2224-0349.2021.01.07.
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The reliability of the transmission and distribution of electricity in ports, ships and energy enterprises depends on the condition of the bolted conductive joints. Operational practice shows that 2 % of cases of emergency shutdowns of technological lines and fires on electrical equipment occur due to the weakening of bolted conductive joints. The main reasons for the bolted joint weakening are dynamic changes in operating current loads and changes in climatic parameters. It is possible to increase the reliability of bolted conductive joints using modern diagnostic methods that allow identifying the appearance of bolted joint weakening. The existing methods for diagnosing the state of bolted joints, which are based on monitoring the temperature of the bolted joint, are analysed. It has been established that the use of such methods to identify the initial moment of the joint weakening is impractical, since they react to the emergency state of the bolted joint, that is, to the critical temperature value. To identify the initial moment of the bolted joint weakening, it is proposed to simultaneously monitor the values of the load current and the parameters of the ambient air, which makes it possible to increase the reliability of detecting pre-emergency situations. To diagnose the initial moment of the bolted joint weakening, it is proposed to use the estimated Boolean functions. Arguments of Boolean variables are obtained for a stationary and continuous process of changing current and ambient temperatures, a stationary and deterministic process as well as a random and deterministic process. The minimized Boolean functions are given for the normal operation of the bolted joint, when the initial moment of weakening appears and when the emergency mode occurs.