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The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the L -fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L , and lattice homomorphism from L to the interval 0,1 . In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.

The main objective of this article is to establish generalized fractional Hermite–Hadamard and related type integral inequalities forh-Godunova–Levin convexity andh-Godunova–Levin preinvexity with extended Wright generalized Bessel function acting as kernel. Moreover, Hermite–Hadamard-type and trapezoid-type inequalities for several known convexities including Godunova–Levin function, classical convex,s-Godunova–Levin function,P-function, ands-convex function are deduced as corollaries. These obtained results are analyzed in the form of generalization of fractional inequalities.

In this paper, we aim to develop formulas of spectral radius for an operator S in terms of generalized Aluthge transform, numerical radius, iterated generalized Aluthge transform, and asymptotic behavior of powers of S . These formulas generalize some of the formulas of spectral radius existing in literature. As an application, these formulas are used to obtain several characterizations of normaloid operators.

The notion of Δ -metric spaces has been proposed in this study as a generalization of b -metric spaces, extended b -metric spaces, and p -metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.

In Probabilistic Abductive Logic Programming we are given a probabilistic logic program, a set of abducible facts, and a set of constraints. Inference in probabilistic abductive logic programs aims to find a subset of the abducible facts that is compatible with the constraints and that maximizes the joint probability of the query and the constraints. In this paper, we extend the PITA reasoner with an algorithm to perform abduction on probabilistic abductive logic programs exploiting Binary Decision Diagrams. Tests on several synthetic datasets show the effectiveness of our approach.

In this paper, we study α-cosymplectic manifoldMadmitting∗-Ricci tensor. First, it is shown that a∗-Ricci semisymmetric manifoldMis∗-Ricci flat and aϕ-conformally flat manifoldMis anη-Einstein manifold. Furthermore, the∗-Weyl curvature tensorW∗onMhas been considered. Particularly, we show that a manifoldMwith vanishing∗-Weyl curvature tensor is a weakϕ-Einstein and a manifoldMfulfilling the conditionRE1,E2⋅W∗=0isη-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldMadmitting∗-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting∗-Ricci soliton and almost∗-Ricci soliton are drawn.

This paper scrutinizes the consequences of radiation and heat consumption of MHD convective flow of nanofluid on a heated stretchy plate with injection/suction and convective heating/cooling conditions. The nanofluid encompasses with C u and A g nanoparticles. We enforce the suited transformation to remodel the governing mathematical models to ODE models. The HAM (homotopy analysis method) idea is applied to derive the series solutions. The divergence of fluid velocity, temperature, skin friction coefficient, local Nusselt number, entropy generation, and Bejan number on disparate governing parameters is exhibited via graphs and tables. It is seen that the fluid velocity in both directions is subsided when elevating the magnetic field and Forchheimer number. Also, the C u nanoparticles possess hefty speed compared to A g nanoparticles because the density of A g nanoparticles is high compared to that of C u nanoparticles. The fluid temperature upturns when enlarging the heat generation and radiation parameters. The skin friction coefficients and local Nusselt number are high in A g nanoparticles than in C u nanoparticles.

A directed graph (digraph) D of order n satisfies directed cut condition D C C if there are at least S arcs from any set S ⊆ V D , S ≤ n / 2 to its complement S ¯ = V D / S . We show that a digraph D of even order n has at least 2 n − 3 arcs if D satisfies D C C .

In this paper, we define λ-approximate simulations and bisimulations for fuzzy automata over complete Heyting algebras. The value λ presents the degree of language similarity or equality between observed fuzzy automata. Algorithms for computing the greatest λ-approximate simulations and bisimulations are given. We show that λ-approximate simulations and bisimulations on a fuzzy automaton can be effectively used for factorization of fuzzy automata. We present the algorithm that splits the interval of the degrees of language similarity or equality into subintervals with same minimal corresponding factor fuzzy automata.

The new outcomes of the present paper are q -analogues ( q stands for quantum calculus) of Hermite-Hadamard type inequality, Montgomery identity, and Ostrowski type inequalities for s -convex mappings. Some new bounds of Ostrowski type functionals are obtained by using Hölder, Minkowski, and power mean inequalities via quantum calculus. Special cases of new results include existing results from the literature.

Large-scale complex network data poses significant challenges for the analytic ideas and tools to monitor and analyze complex networks. As classical structure descriptors, average path length ( $L$ ), global and local network efficiency ( $E_{glob}$ and $E_{loc}$ ), general and loop clustering coefficient ( $C_{3}$ and $D$ ), play significant roles in complex network analysis. In this paper, we make use of resistive distance instead of the shortest path distance to suggest resistive distance descriptors, i.e., $L_{r}$ , $\Omega _{glob}$ and $\Omega _{loc}$ , $\Omega _{3}$ and $D_{r}$ . We investigate all the resistive descriptors on classical WS, BA and ER models and find some interesting phenomenons. On one hand, $L_{r}$ and $\Omega _{3}$ (resp., $\Omega _{glob}$ and $\Omega _{loc}$ ) can be used to characterize the features of small-word networks. On the other hand, $L_{r}$ , $\Omega _{3}$ , and $\Omega _{glob}$ can be utilized to measure network invulnerability. To access the effectiveness of the resistive distance descriptors, we conduct extensive numerical simulations on synthetic and real networks. In comparison with the baselines, the proposed metrics show competitive performance on classical network models and are more efficient for networks with small and medium size.

A $(m,n)$ sonar sequence is an $m\times n$ array with exactly one dot in each column and where all lines connecting two dots in the array are distinct as vectors. These arrays are known to have many applications such as sonar and radar detection and these are studied as a particular case of Golomb rectangles or two-dimensional Sidon sets. The main open problem for sonar sequences is: for fixed $m$ , find the largest $n$ for which there is an $(m,n)$ sonar sequence, these sequences are called the best sonar sequences. The extended sonar sequences are generalizations of sonar sequences where each column has at most one dot, the motivation to study these arrays are the best results obtained when applied to radar and sonar detection. In this paper, we give the best sonar sequences with $m\leq 100$ obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions and we present new constructions of extended sonar sequences that use Sidon sets.

In this paper, we present a Model-Free Stochastic Inverse Optimal Control (IOC) algorithm for the discrete-time infinite-horizon stochastic linear quadratic regulator (LQR). Our proposed algorithm exploits the richness of the available system trajectories to recover the control gain $K$ and cost function parameters $(Q,R)$ in a low (space, sample, and computational) complexity manner. By leveraging insights on the stochastic LQR, we guarantee well-posedness of the Model-Free Stochastic IOC LQR via satisfaction of the Certainty Equivalence optimality conditions. The exact solution of the control gain $K$ is recovered via a deterministic, low complexity Least Squares approach. Using $K$ , we solve a completely model-free non-iterative SemiDefinite Programming (SDP) problem to obtain a unique (up to a scalar ambiguity) $(Q,R)$ , in which optimality and feasibility are jointly ensured. Via derivation of the sample complexity bounds, we show that the non-asymptotic performance of the Model-Free Stochastic IOC LQR can be characterized by the signal-to-noise (SNR) ratio of the finite set of system state and input signals. We present a model-based version of the algorithm for the special case where $(A,B)$ is available, and we, further, provide the extension to the Stochastic Model-Free IOC linear quadratic tracking (LQT) case.

Let $G$ be an additive group of order $v$ . A $k$ -element subset $D$ of $G$ is called a $(v, k, \lambda, t)$ -almost difference set if the expressions $g-h$ , for $g$ and $h$ in $D$ , represent $t$ of the non-identity elements in $G$ exactly $\lambda $ times and every other non-identity element $\lambda + 1$ times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers $A$ is called a Golomb ruler if the difference between two distinct elements of $A$ are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.

Sea ice concentration algorithms using brightness temperatures ($T_{B}$) from satellite microwave radiometers are used to compute sea ice concentration ($c_{\text{ice}}$), sea ice extent, and generate sea ice climate data records. Therefore, it is important to minimize the sensitivity of $c_{\text{ice}}$ estimates to geophysical noise caused by snow/sea ice thermal microwave emission signature variations, and presence of WV and clouds in the atmosphere and/or near-surface winds. In this study, we investigate the effect of geophysical noise leading to systematic $c_{\text{ice}}$ biases and affecting $c_{\text{ice}}$ standard deviations (STD) using simulated top of the atmosphere $T_{B}$s over open water and 100% sea ice. We consider three case studies for the Arctic and the Antarctic and eight different $c_{\text{ice}}$ algorithms, representing different families of algorithms based on the selection of channels and methodologies. Our simulations show that, over open water and low $c_{\text{ice}}$, algorithms using gradients between V-polarized 19-GHz and 37-GHz $T_{B}$s show the lowest sensitivity to the geophysical noise, while the algorithms exclusively using near-90-GHz channels have by far the highest sensitivity. Over sea ice, the atmosphere plays a much smaller role than over open water, and the $c_{\text{ice}}$ STD for all algorithms is smaller than over open water. The hybrid and low-frequency (6 GHz) algorithms have the lowest sensitivity to noise over sea ice, while the polarization type of algorithms has the highest noise levels.

$(\alpha,\beta)$ -Pythagorean fuzzy set is a very efficient way of dealing with uncertainty. In this article, we have introduced the notions of $(\alpha,\beta)$ - Pythagorean fuzzy subring and $(\alpha,\beta)$ - Pythagorean fuzzy ideal of a ring. Further, we have briefly described various results related to it. Also, we have discussed the level subring of an $(\alpha,\beta)$ - Pythagorean fuzzy subring. Moreover, we have studied the direct product and ring homomorphism of $(\alpha,\beta)$ - Pythagorean fuzzy subrings.

In this paper, a class of Clifford-valued neutral-type recurrent neural networks with $D$ operator is explored. By using non-decomposition method and the Banach fixed point theorem, we obtain several sufficient conditions for the existence of anti-periodic solutions for Clifford-valued neutral-type recurrent neural networks with $D$ operator. By using the proof by contradiction and inequality techniques, we obtain the global exponential synchronization of anti-periodic solutions for Clifford-valued neutral-type recurrent neural networks with $D$ operator. Finally, we give one example to illustrate the feasibility and effectiveness of main results.

A category theory constitutes a convenient conceptual apparatus to organize the worlds of mathematical entities. The concept of fuzzy natural transformation as an abstract mapping on functors is one of the essential concepts of this theory. If we admit a piece of non-commutativity in its definition diagrams, then the ‘upward’ and the ‘downward’ diagram parts generate different result sets. In this way, we can introduce the concept of fuzzy natural transformation. We deal with the multi-fuzzy natural transformation if such a transformation is based on a multi-diagram. This paper aims to describe the multi-fuzzy natural transformation situation when the symmetric difference of the result is finite. It allows us to organize the whole spectrum of the result sets in a unique quotient algebra $\mathcal {P}^{comp}(\omega)/\mathrm {fin}$ . Different algebraic properties of this structure will be explored, and a piece of classical encoding theory will be reconstructed in the environment determined by this algebra. In particular, the concepts of the $k$ -multi similarity, the abstract $k$ -multi similarity, the $k$ -similarity balls, and the abstract $k$ -similarity balls are introduced as some generalization of the idea of Hamming’s distances and Hamming’s balls. Finally, it is shown how to automate some verification processes in this context using an R-based programming environment.

The physical and structural properties of molecular structure or graph such as boiling point, melting point, surface tension, or solubility are studied using topological index (TI). Topological index is a mathematical formula that can be applied to any graph which models some molecular structures. The various operations play an important role in graph theory such as joining, union, intersection, products, and subdivision. In this paper, we computed the bounds for general Randic coindex of F -sum graphs such as ( S -sum, R -sum, Q -sum, and T -sum) in the form of their factor graphs. At the end, results are illustrated by numerical table for the particular F -sum graphs.

In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field ξ . Furthermore, if the potential vector field ξ of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of η -Ricci-Yamabe soliton on an imperfect fluid GRW spacetime with a harmonic potential function ψ . Finally, we examine necessary and sufficient conditions for a 1 -form η , which is the g -dual of the vector field ξ on imperfect fluid GRW spacetime to be a solution of the Schrödinger-Ricci equation.

The family of cumulative paired ϕ-entropies offers a wide variety of ordinal dispersion measures, covering many well-known dispersion measures as a special case. After a comprehensive analysis of this family of entropies, we consider the corresponding sample versions and derive their asymptotic distributions for stationary ordinal time series data. Based on an investigation of their asymptotic bias, we propose a family of signed serial dependence measures, which can be understood as weighted types of Cohen’s κ, with the weights being related to the actual choice of ϕ. Again, the asymptotic distribution of the corresponding sample κϕ is derived and applied to test for serial dependence in ordinal time series. Using numerical computations and simulations, the practical relevance of the dispersion and dependence measures is investigated. We conclude with an environmental data example, where the novel ϕ-entropy-related measures are applied to an ordinal time series on the daily level of air quality.

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this paper, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking., To appear in Theory and Practice of Logic Programming (TPLP)

Formal reasoning about finite sets and cardinality is an important tool for many applications, including software verification, where very often one needs to reason about the size of a given data structure and not only about what its elements are. The Constraint Logic Programming tool {log} provides a decision procedure for deciding the satisfiability of formulas involving very general forms of finite sets, without cardinality. In this paper we adapt and integrate a decision procedure for a theory of finite sets with cardinality into {log}. The proposed solver is proved to be a decision procedure for its formulas. Besides, the new CLP instance is implemented as part of the {log} tool. In turn, the implementation uses Howe and King's Prolog SAT solver and Prolog's CLP(Q) library, as an integer linear programming solver. The empirical evaluation of this implementation based on +250 real verification conditions shows that it can be useful in practice., Comment: Under consideration in Theory and Practice of Logic Programming (TPLP)

We introduce the concepts of Cohen positive strongly p -summing and positive p -dominated m-homogeneous polynomials. The version of Pietsch’s domination theorem for the first class among other results and a Bu-type theorem is proved, as well as some inclusions with other known spaces. Moreover, we present a characterization of these classes in tensor terms.

In this article, we derive four theorems concerning the fractional integral image for the product of the q -analogue of general class of polynomials with the q -analogue of the I -functions. To illustrate our main results, we use q -fractional integrals of Erdélyi–Kober type and generalized Weyl type fractional operators. The study concludes with a variety of results that can be obtained by using the relationship between the Erdélyi–Kober type and the Riemann–Liouville q -fractional integrals, as well as the relationship between the generalized Weyl type and the Weyl type q -fractional integrals.

We advocate the use of de Bruijn's universal abstraction $\lambda^\infty$ for the quantification of schematic variables in the predicative setting and we present a typed $\lambda$-calculus featuring the quantifier $\lambda^\infty$ accompanied by other practically useful constructions like explicit substitutions and expected type annotations. The calculus stands just on two notions, i.e., bound rt-reduction and parametric validity, and has the expressive power of $\lambda\rightarrow$. Thus, while not aiming at being a logical framework by itself, it does enjoy many desired invariants of logical frameworks including confluence of reduction, strong normalization, preservation of type by reduction, decidability, correctness of types and uniqueness of types up to conversion. This calculus belongs to the $\lambda\delta$ family of formal systems, which borrow some features from the pure type systems and some from the languages of the Automath tradition, but stand outside both families. In particular, the calculus includes and evolves two earlier systems of this family. Moreover, a machine-checked specification of its theory is available., Comment: 37 pages, minor revision

In this paper, we define α -admissible and α - ϕ -fuzzy cone contraction in fuzzy cone metric space to prove some fixed point theorems. Some related sequences with contraction mappings have been discussed. Ultimately, our theoretical results have been utilized to show the existence of the solution to a nonlinear integral equation. This application is also illustrative of how fuzzy metric spaces can be used in other integral type operators.

In this paper, we study the sharp bounds for the generalized m -linear n -dimensional p -adic Hardy-Littlewood-Pólya operator on central and noncentral p -adic Morrey spaces with power weight. As an application, the sharp bounds for the p -adic Hardy-Littlewood-Pólya operator on corresponding Morrey spaces are obtained. These results generalize substantially some well-known results.

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p -adic variable exponent Lebesgue spaces.

We report on an inversion tool for a class of oriented conditional constructor term rewriting systems. Four well-behaved rule inverters ranging from trivial to full, partial and semi-inverters are included. Conditional term rewriting systems are theoretically well founded and can model functional and non-functional rewrite relations. We illustrate the inversion by experiments with full and partial inversions of the Ackermann function. The case study demonstrates, among others, that polyvariant inversion and input-output set propagation can reduce the search space of the generated inverse systems., Comment: In Proceedings VPT 2021, arXiv:2109.02001

We introduce a class of neighbourhood frames for graded modal logic embedding Kripke frames into neighbourhood frames. This class of neighbourhood frames is shown to be first-order definable but not modally definable. We also obtain a new definition of graded bisimulation with respect to Kripke frames by modifying the definition of monotonic bisimulation., Comment: removed journal template

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

Applicative bisimilarity is a coinductive characterisation of observational equivalence in call-by-name lambda-calculus, introduced by Abramsky in 1990. Howe (1989) gave a direct proof that it is a congruence. In previous work with Borthelle (2020), we abstract over this result by proposing a categorical framework for specifying operational semantics, in which we prove that (an abstract analogue of) applicative bisimilarity is automatically a congruence. However, the framework presents a few infelicities: (1) it requires a non-trivial refinement of the standard approach of Fiore, Plotkin, and Turi (1999) based on monoid algebras for specifying syntax with variable binding; (2) it relies on so-called prebisimulations instead of the more standard notion of bisimulation by lifting; (3) one of the axioms, called weak compositionality, feels ad hoc; (4) the proofs involve directed unions of relations, leading to quite a few painful inductions. In this paper, we rectify all of these deficiencies. In particular, a notable novelty is that the so-called Howe closure is defined as an initial monoid algebra in a category of spans. Finally, the familiality/cellularity axiom of the previous framework is now viewed as a mere sufficient condition for the main hypothesis, preservation of functional bisimulations.

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces.

This paper introduces a novel approach to computing logic programming semantics. First, a propositional Herbrand base is represented in a vector space and if-then rules in a program are encoded in a matrix. Then the least fixpoint of a definite logic program is computed by matrix-vector products with a non-linear operation. Second, disjunctive logic programs are represented in third-order tensors and their minimal models are computed by algebraic manipulation of tensors. Third, normal logic programs are represented by matrices and third-order tensors, and their stable models are computed. The result of this paper exploits a new connection between linear algebraic computation and symbolic computation, which has the potential to realize logical inference in huge scale of knowledge bases.

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that PRS n spacetimes are Ricci flat in trivial, A , and B subspaces, whereas perfect fluid in subspaces I , I ⊕ A , and I ⊕ B , and have zero scalar curvature in subspace A ⊕ B . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

We report on a dynamic range enhanced optical frequency domain reflectometry distributed backscattering interrogator based on dual-loop composite optical phase-locked loop (OPLL). Exploiting simultaneously an acousto-optic frequency shifter based an external modulation loop and a piezo based direct modulation loop, the proposed composite OPLL allows offering a larger loop bandwidth and gain, permitting a more efficient coherence enhancement as well as sweep linearization. A high fidelity frequency sweep of ${\rm{\sim}} \text{8.2 GHz}$ at ${\text{164 GHz/s}}$ sweep rate is generated with a peak-to-peak frequency error as low as ${\rm{\sim}} \text{120 kHz}$. It leads to a dynamic range enhancement of more than ${\text{3 dB}}$ for the measured power loss compared to the case when only piezo loop is applied. This corresponds to ${\rm{\sim}} \text{15 km}$ extension for the measurement range of Rayleigh backscattering without any spatial resolution penalties. Fourier transform-limited spatial resolution has been demonstrated at a range window more than about $\text{28}$ times of the intrinsic coherence length of the adopted fiber laser. The proposed method provides a straightforward optimization of the real-time sweep control and is expected to be a useful tool in industrial and commercial applications.

Nominal syntax is a generalisation of first-order syntax that includes names, a notion of name binding and an elegant axiomatisation of alpha-equivalence, based on nominal set theory. However, it does not take into account non-capturing atom substitution, which is not a primitive notion in nominal syntax. We consider an extension of nominal syntax with non-capturing atom substitutions and show that matching is decidable and finitary but unification is undecidable in general. The proof of undecidability of unification is obtained by reducing Hilbert's tenth problem to unification of extended nominal terms. We provide a general matching algorithm and characterise a class of problems for which matching is unitary, giving rise to expressive and efficient rewriting systems.

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many elements. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this proof, we provide a Coq tactic to automatically prove the countability of concrete inductive types. We also developed Coq libraries for countability and for the syntax of inductive definitions, which have value on their own., Comment: In Proceedings LFMTP 2021, arXiv:2107.07376

The algebraic structures have many applications in coding theory, cryptography, and security networks. In this paper, the notion of hybrid subalgebras ofBCH-algebras is introduced and related properties are investigated. Moreover, some characterizations of hybrid subalgebras ofBCH-algebras are given. Furthermore, we state and prove some theorems in hybrid subalgebras ofBCH-algebras. The homomorphic images and inverse images of fuzzyBCH-subalgebras are studied and discussed.

Fuzzy multiset is an extension of fuzzy set in multiset framework. In this paper, we review the concept of fuzzy multisets and study the notions of metric and norm on fuzzy multiset. Some results on metric and norm are established in fuzzy multiset context.

Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone, with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

In this paper, we introduce extended b -gauge spaces and the extended family of generalized extended pseudo- b -distances. Moreover, we define the sequential completeness and construct the Caristi-type G -contractions in the framework of extended b -gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.

Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.

ion logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as predicate logic plus operators and variable binding. We argue that abstraction logic is the best foundational logic possible because it maximises both simplicity and practical expressivity. This argument is supported by the observation that abstraction logic has simpler terms and a simpler notion of proof than all other general logics. At the same time, abstraction logic can formalise both intuitionistic and classical abstraction logic, and is sound and complete for these logics and all other logics extending deduction logic with equality.

In this paper, we study two problems: determining action model equivalence and minimizing the event space of an action model under certain structural relationships. The Kripke model equivalence is perfectly caught by the structural relationship called bisimulation. In this paper, we propose the generalized action emulation perfectly catching the action model equivalence. Previous structural relationships sufficient for the action model equivalence, i.e. the bisimulation, the propositional action emulation, the action emulation, and the action emulation of canonical action models, can be described by various restricted versions of the generalized action emulation. We summarize four critical properties of the atom set over preconditions, and prove that any formula set satisfying these properties can be used to restrict the generalized action emulation to determine the action model equivalence by an iteration algorithm. We also construct a new formula set with these four properties, which is generally more efficient than the atom set. The technique of the partition refinement has been used to minimize the world space of a Kripke model under the bisimulation. Applying the partition refinement to action models allows one to minimize their event spaces under the bisimulation. The propositional action emulation is weaker than bismulation but still sufficient for the action model equivalence. We prove that it is PSPACE-complete to minimize the event space of an action model under the propositional action emulation, and provide a PSPACE algorithm for it. Finally, we prove that minimize the event space under the action model equivalence is PSPACE-hard, and propose a computable method based on the canonical formulas of modal logics to solve this problem., 30 pages, 0 figure

interpretation, Hoare logic, and incorrectness (or reverse Hoare) logic are powerful techniques for static analysis of computer programs. All of them have been successfully extended to the quantum setting, but largely developed in parallel. In this paper, we examine the relationship between these techniques in the context of verifying quantum while-programs, where the abstract domain and the set of assertions for quantum states are well-structured. In particular, we show that any complete quantum abstract interpretation induces a quantum Hoare logic and a quantum incorrectness logic, both of which are sound and relatively complete. Unlike the logics proposed in the literature, the induced logic systems are in a forward manner, making them more useful in certain applications. Conversely, any sound and relatively complete quantum Hoare logic or quantum incorrectness logic induces a complete quantum abstract interpretation. As an application, we are able to show the non-existence of any sound and relatively complete quantum Hoare logic or incorrectness logic if tuples of local subspaces are taken as assertions., 26 pages

A novel model of reversible computing, the $\aleph$-calculus, is introduced. It is declarative, reversible-Turing complete, and has a local term-rewriting semantics. Unlike previously demonstrated reversible term-rewriting systems, it does not require the accumulation of history data. Terms in the $\aleph$-calculus, in combination with the program definitions, encapsulate all program state. An interpreter was also written., 7 pages, 1 listing, 14th Conference on Reversible Computation