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The completeness of the system of eigenfunctions of the complex Schrodinger operator L c = − d 2 / d x 2 + c x 2 / 3 on the semi-axis in L 2 ( R + ) with Dirichlet boundary conditions is proved for all c: | arg ⁡ c | π / 2 + θ 0 , where π / 10 θ 0 π / 2 is determined as the only solution of a certain transcendental equation.

Action theory may be regarded as a theoretical foundation of AI, because it provides in a logically coherent way the principles of performing actions by agents. But, more importantly, action theory offers a formal ontology mainly based on set-theoretic constructs. This ontology isolates various types of actions as structured entities: atomic, sequential, compound, ordered, situational actions etc., and it is a solid and non-removable foundation of any rational activity. The paper is mainly concerned with a bunch of issues centered around the notion of performability of actions. It seems that the problem of performability of actions, though of basic importance for purely practical applications, has not been investigated in the literature in a systematic way thus far. This work, being a companion to the book as reported (Czelakowski in Freedom and enforcement in action. Elements of formal action theory, Springer 2015), elaborates the theory of performability of actions based on relational models and formal constructs borrowed from formal lingusistics. The discussion of performability of actions is encapsulated in the form of a strict logical system "Equation missing". This system is semantically defined in terms of its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since due to the nature of compound actions the system "Equation missing" is not finitary, other semantic variants of "Equation missing" are defined. The focus in on the system "Equation missing" of performability of finite compound actions. An adequate axiom system for "Equation missing" is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is highlighted. The relationship between performability of actions and dynamic logic is also discussed.

A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to be decidable both for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

In this paper we exploit the utility of the triangle symbol which has a complicated expression in terms of spider diagrams in ZX-calculus, and its role within the ZX-representation of AND-gates in particular. First, we derive spider nest identities which are of key importance to recent developments in quantum circuit optimisation and T-count reduction in particular. Then, using the same rule set, we prove a completeness theorem for quantum Boolean circuits (QBCs) whose rewriting rules can be directly used for a new method of T-count reduction. We give an algorithm based on this method and show that the results of our algorithm outperform the results of all the previous best non-probabilistic algorithms., Comment: In Proceedings QPL 2020, arXiv:2109.01534

Two new multilattice logics called submultilattice logic and indexed multilattice logic are introduced as a monosequent calculus and an indexed monosequent calculus, respectively. The submultilattice logic is regarded as a monosequent calculus version of Shramko’s original multilattice logic, which is also known as the logic of logical multilattices. The indexed multilattice logic is an extension of the submultilattice logic, and is regarded as the logic of multilattices. A completeness theorem with respect to a lattice-valued semantics is proved for the submultilattice logic, and a completeness theorem with respect to a multilattice-valued semantics is proved for the indexed multilattice logic.

This paper presents a generalization of Fine’s completeness theorem for transitive logics of finite width, and proves the Kripke completeness of transitive logics of finite “suc-eq-width”. The frame condition for each finite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points with different proper successors. The paper also presents a generalization of Rybakov’s completeness theorem for transitive logics of prefinite width, and proves the Kripke completeness of transitive logics of prefinite “suc-eq-width”. The frame condition for each prefinite suc-eq-width axiom requires, in rooted transitive frames, a finite upper bound of cardinality for antichains of points that have a finite lower bound of depth and have different proper successors. We will construct continuums of transitive logics of finite suc-eq-width but not of finite width, and continuums of those of prefinite suc-eq-width but not of prefinite width. This shows that our new completeness results cover uncountably many more logics than Fine’s theorem and Rybakov’s theorem respectively.

The present study aimed to propose the similarity triangle algebra and prove the completeness of the similarity triangle logic. The similarity was defined on triangle algebra. Similarity triangle algebra is a triangle algebra endowed with a binary operation S, which verifies specific additional properties. These properties and a class of all the similarity triangle algebras form a variety which were assessed as well. In addition, the similarity IVRL-filters (S-IVRL-filters) were investigated and introduced in the similarity triangle algebras, and the similarity triangle logic $$(\mathcal {STL})$$ was presented, which is a system of several-valued logic capturing the tautologies of similarity triangle algebras to prove the completeness theorem. According to the results, $$\mathcal {STL}$$ is a conservative extension of triangle logic ( $$\mathcal {TL}$$ ).

The original approach of Hutchinson to fractals considers the defining equation as a fixed point problem, and then applies the Banach Contraction Principle. To do this, the Blaschke Completeness Theorem is essential. Avoiding Blaschke’s result, this note presents an alternative way to fractals via the Kuratowski noncompactness measure. Moreover, our technique extends the existence part of Hutchinson’s Theorem to condensing maps instead of contractions.

This paper continues the investigation of the logic of competing theories, be they scientific, social, political etc. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired by Plo\v{s}\v{c}ica's representation of general lattices. We axiomatize the resulting many-valued, non-distributive modal logic of these structures and prove a completeness theorem. We illustrate the application of this logic through a case study in which we model competition among interacting political promises and social demands within an arena of political parties social groups., Comment: 24 pages

Fil: Castano, Diego Nicolas. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Matematica Bahia Blanca. Universidad Nacional del Sur. Departamento de Matematica. Instituto de Matematica Bahia Blanca; Argentina

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A completeness theorem with respect to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics.

Given a regular cardinal $\kappa $ such that $\kappa ^{ (or any regular $\kappa $ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the (basis for the) topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $ , when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms (geometric morphisms the inverse image of which preserves all $\kappa $ -small limits) into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory (with at most $\kappa $ many axioms), that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.

In this work, we describe well-defined dissipative boundary conditions related with a singular third-order differential equation in lim-3 case at singular point. Using the characteristic function of the corresponding dissipative operator we introduce a completeness theorem.

This paper, being a companion to the book [2] elaborates the deontology of sequential and compound actions based on relational models and formal constructs borrowed from formal linguistics. The semantic constructions presented in this paper emulate to some extent the content of [3] but are more involved. Although the present work should be regarded as a sequel of [3] it is self-contained and may be read independently. The issue of permission and obligation of actions is presented in the form of a logical system . This system is semantically defined by providing its intended models in which the role of actions of various types (atomic, sequential and compound ones) is accentuated. Since the consequence relation is not finitary, other semantically defined variants of are defined. The focus is on the finitary system in which only finite compound actions are admissible. An adequate axiom system for it is defined. The strong completeness theorem is the central result. The role of the canonical model in the proof of the completeness theorem is emphasized.

The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau&rsquo, s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

For a cardinal of the form κ = בκ, Shelah’s logic L 1 has a characterisation as the maximal logic above $${ \cup _{\lambda < \kappa }}{L_{\lambda ,\omega }}$$ satisfying a strengthening of the undefinability of well-order. Karp’s chain logic [20] L is known to satisfy the undefinability of well-order and interpolation. We prove that if κ is singular of countable cofinality, Karp’s chain logic [20] is above L 1 . Moreover, we show that if κ is a strong limit of singular cardinals of countable cofinality, the chain logic $$L_{ < \kappa , < \kappa }^c{ \cup _{\lambda < \kappa }}L_{\lambda ,\lambda }^c$$ is a maximal logic with chain models to satisfy a version of the undefinability of well-order. We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah’s [28], which asked whether for κ singular of countable cofinality there was a logic strictly between $${L_{{\kappa ^ + },\omega }}$$ and $${L_{{\kappa ^ + },{\kappa ^ + }}}$$ having interpolation. We show that modulo accepting as the upper bound a model class of Lκ, κ, Karp’s chain logic satisfies the required properties. In addition, we show that this chain logic is not κ-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic L 1 in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.

Let $\mathcal{C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathcal{C}$ is definable in first-order logic (FO) by a sentence φ if and only if $\mathcal{C}$ has a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from φ the corresponding forbidden induced subgraphs. Our results (a) and (b) show that this machinery fails on finite graphs.(a)There is a class of finite graphs that is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.(b)Even if we only consider classes $\mathcal{C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs such a characterization cannot be computed from an FO-sentence φ that defines $\mathcal{C}$ and the size of the characterization cannot be bounded by f(|φ|) for any computable function f.Besides their importance in graph theory, our results also significantly strengthen similar known theorems for arbitrary structures.

We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain $A$, a codomain $B$ and a "filter" $T$ on finite approximations of functions from $A$ to $B$, a generalised form GDC$_{A,B,T}$ of the axiom of dependent choice and dually a generalised bar induction principle GBI$_{A,B,T}$ such that: GDC$_{A,B,T}$ intuitionistically captures the strength of$\bullet$ the general axiom of choice expressed as $\forall a\exists\beta R(a, b) \Rightarrow\exists\alpha\forall a R(\alpha,(a \alpha (a)))$ when $T$ is a filter that derives point-wise from a relation $R$ on $A x B$ without introducing further constraints,$\bullet$ the Boolean Prime Filter Theorem / Ultrafilter Theorem if $B$ is the two-element set $\mathbb{B}$ (for a constructive definition of prime filter),$\bullet$ the axiom of dependent choice if $A = \mathbb{N}$,$\bullet$ Weak K{\"o}nig's Lemma if $A = \mathbb{N}$ and $B = \mathbb{B}$ (up to weak classical reasoning): GBI$_{A,B,T}$ intuitionistically captures the strength of$\bullet$ G{\"o}del's completeness theorem in the form validity implies provability for entailment relations if $B = \mathbb{B}$,$\bullet$ bar induction when $A = \mathbb{N}$,$\bullet$ the Weak Fan Theorem when $A = \mathbb{N}$ and $B = \mathbb{B}$.Contrastingly, even though GDC$_{A,B,T}$ and GBI$_{A,B,T}$ smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when $A$ is $\mathbb{B}^\mathbb{N}$ and $B$ is $\mathbb{N}$., Comment: LICS 2021 - 36th Annual Symposium on Logic in Computer Science, Jun 2021, Rome / Virtual, Italy

We ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic.

The condition of $$\sigma $$-completeness related to orthomodular lattices places an important role in the study of quantum probability theory. In the framework of algebras with infinitary operations, an equational theory for the category of $$\sigma $$-complete orthomodular lattices is given. In this structure, we study the congruences theory and directly irreducible algebras establishing an equational completeness theorem. Finally, a Hilbert style calculus related to $$\sigma $$-complete orthomodular lattices is introduced and a completeness theorem is obtained.

Shramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic ...

Reynolds' separation logical system for pointer program verification is investigated. This paper proves its completeness theorem that states that every true asserted program is provable in the logical system. In order to prove the completeness, this paper shows the expressiveness theorem that states the weakest precondition of every program and every assertion can be expressed by some assertion. This paper also introduces an extension of the assertion language with inductive definitions and proves the soundness theorem, the expressiveness theorem, and the completeness theorem.

After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if T is a complete recursive L -theory in continuous logic, and v ( φ ) is the truth value of the L -sentence φ in models of T, then v ( φ ) is a recursive real uniformly recursive in φ. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex ⁎-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures.

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

In order to solve the parallel algorithm of Petri net system with concurrent function, so as to achieve the parallel control and simulation operation of this system, this paper proposes the function partition completeness theory and algorithms of Petri net parallelization, thereby providing the theoretical support for the realization of Petri parallel algorithms. Firstly, according to the concurrent characteristics of Petri net model, we analyze the parallelism of Petri net system; then, by giving the solving process of place invariants and the function partitioning of Petri net, we propose the function partitioning conditions and determination theorem of Petri net parallelization, and conduct its theoretical proof and practical verification. On this basis, we conduct the theoretical study and analysis on the situation that Petri net system has several kinds of parallel function partitioning, propose the completeness theorem of parallelism function partitioning in Petri net system, and verify it. Finally, we give the algorithms, application examples and simulation experiment results of parallel function partitioning of Petri net systems based on place invariant. The theoretical proof and experimental results show that the function partitioning conditions and completeness theory of Petri net parallelization based on place invariant are correct, and the parallel algorithms under such theoretical basis are also correct and effective.

Starting from certain metalogical results (the completeness theorem, the soundness theorem, and Lindenbaum-Scott theorem), I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate.

In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs LP 0 . Moreover, we prove a version of the uniform arithmetical completeness theorem of LP 0 .

The article compares two different approaches of incorporating probability into coalition logics. One is based on the semantics of games with stochastic transitions and the other on games with the stochastic failures. The work gives an example of a non-trivial property of coalition power for the first approach and a complete axiomatization for the second approach. It turns out that the logical properties of the coalition power modality under the second approach depend on whether the modal language allows the empty coalition. The main technical results for the games with stochastic failures are a strong completeness theorem for the logical system without the empty coalition and an incompleteness theorem which shows that there is no strongly complete logical system in the language with the empty coalition.

We uncover a close relationship between combinatorial and syntactic proofs for first-order logic (without equality). Whereas syntactic proofs are formalized in a deductive proof system based on inference rules, a combinatorial proof is a syntax-free presentation of a proof that is independent from any set of inference rules. We show that the two proof representations are related via a deep inference decomposition theorem that establishes a new kind of normal form for syntactic proofs. This yields (a) a simple proof of soundness and completeness for first-order combinatorial proofs, and (b) a full completeness theorem: every combinatorial proof is the image of a syntactic proof., To be published in LICS 2021. This is the author version of the paper with full proofs in the appendix

Quantum graphs consisting of a ring with two semi-infinite edges attached to the same point of the ring is considered. We deal with the Rashba spin-orbit Hamiltonian on the graph. A theorem concerning to completeness of the resonance states on the ring is proved. Due to use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrodinger, Dirac and Landau quantum graphs.

Krein–de Branges spectral theory provides a correspondence between canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We revisit this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. Our result can be viewed as a spectral version of the classical Szegő theorem in the theory of polynomials orthogonal on the unit circle. It extends Krein–Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

The notion of argumentation and the one of belief stand in a problematic relation to one another. On the one hand, argumentation is crucial for belief formation: as the outcome of a process of arguing, an agent might come to (justifiably) believe that something is the case. On the other hand, beliefs are an input for argument evaluation: arguments with believed premisses are to be considered as strictly stronger by the agent to arguments whose premisses are not believed. An awareness epistemic logic that captures qualified versions of both principles was recently proposed in the literature. This paper extends that logic in three different directions. First, we try to improve its conceptual grounds, by depicting its philosophical foundations, critically discussing some of its design choices and exploring further possibilities. Second, we provide a (heretofore missing) completeness theorem for the basic fragment of the logic. Third, we study, using techniques from dynamic epistemic logic, how different forms of information change can be captured in the framework., Comment: In Proceedings TARK 2021, arXiv:2106.10886. The research activity of Antonio Yuste-Ginel has been partially funded by the predoctoral grant no. MECD-FPU 2016/04113 of Ministerio de Universidades (Spain)

In this chapter we consider a popular version of sequent calculus called G3. In sections 3.2 and 3.3 we focus on different strategies of proving admissibility of cut and some auxiliary results. Section 3.4 is devoted to additional topics connected with different ways of interpretation of sequents. As a by-product of these considerations we will provide a Post-style completeness proof via conjunctive normal forms in the setting of G3. In section 3.5 a general survey of different types of SC, including nostandard and generalised systems, is sketched. Summary of kinds of rules and their properties is provided for better characterization of standard SCs. In the next section we provide three different proofs of the interpolation theorem, all constructive and based on the cut elimination theorem. The remaining sections contain the presentation of tautology elimination rule, which is equivalent to cut, and a proof of a refined version of the strong completeness theorem for nonstandard SC with elimination rules.

A convex (or concave) real-valued function, f , on a dual Banach space P is continuous for the Mackey topology m (P∗, P ) if (and only if) it is Mackey continuous on bounded subsets of P∗ . Equivalence of Mackey continuity to sequential Mackey continuity follows when P is strongly weakly compactly generated, e.g., when P = L1(T ), where T is a set that carries a sigma-finite measure σ. This result of Delbaen, Orihuela and Owari extends their earlier work on the case that P∗ is either L∞ (T ) or a dual Orlicz space. An earlier result of this kind is recalled also: it derives Mackey continuity from bounded Mackey continuity for a nondecreasing concave function, F , that is defined and finite only on the nonnegative cone L∞+. Applied to a linear f , the Delbaen-Orihuela-Owari result shows that the convex bounded Mackey topology is identical to the Mackey topology, i.e., cbm (P∗, P ) = m (P∗, P ); here, this is shown to follow also from Grothendieck’s Completeness Theorem. As for the bounded Mackey topology, bm (P∗, P ), it is conjectured here not to be a vector topology, or equivalently to be strictly stronger than m (P∗, P ), except when P is reflexive.

Currently, we are dealing with a very limited set of activation functions such as Sigmoid, ReLu, Leaky ReLu among others. These activation functions used in the existing digital brain neural network systems are chosen using assumption with the help of “trial and error” approach. However, they do not ethically and appropriately establish any relationship with the referenced AI datasets. Jamilu (2019) proposed that a digital brain should have at least 2000 to 100 billion distinct activation functions implying distinct artificial neurons satisfies Jameel’s criterion(s) for it to normally mimic the human brain. The objectives of this paper are to propose a theorem called “Digital Brain Completeness Theorem”, “superintelligent digital brain neural network systems” and why it is tremendously important to have an extremely huge distinct activation functions implying distinct artificial neurons in a digital brain just like in the case of its counterpart for it to function rationally.

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally missestests, which are needed to model standard programming constructs such as conditionals and$$\mathsf {while}$$while-loops. It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO). Our main contribution is a completeness theorem for CKAO. Our result resorts on a more general study of CKA “with hypotheses”, of which CKAO turns out to be an instance: this analysis is of independent interest, as it can be applied to extensions of CKA other than CKAO.

Formal mathematics is getting increasing attention in mathematics and computer science. In particular, the formalization of calculus has important applications in engineering design and analysis. In this paper, we present a formal proof of some fundamental theorems of continuous functions on closed intervals based on the Coq proof assistant. In this formalization, we build a real number system referring to Landau’s Foundations of Analysis. Then we complete the formalization of the basic definitions of interval, function, and limit and formally prove the theorems including completeness theorem, intermediate value theorem, uniform continuity theorem and others in Coq. The proof process is normalized, rigorous and reliable.

By Gӧdel’s Completeness Theorem 5.5 we know that every consistent theory T has a model, and if T has an infinite model, then it also has arbitrarily large models.

Quantum graph with the Landau operator (Schrodinger operator with a magnetic field) at the edges is considered. The Kirchhoff condition is assumed at the internal vertices. We derive conditions for the graph structure ensuring the completeness of the resonance states on finite subgraphs obtained by cutting all infinite leads of the initial graph. Due to the use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrodinger quantum graph.

We present a probabilistic logic for reasoning about degrees of confirmation. We provide a sound and strongly complete axiomatization for the logic. We show that the problem of deciding satisfiability is in PSPACE.

It follows from Godel's Completeness Theorem that a theory is consistent if and only if it has a model. In particular, the consistency of Peano Arithmetic follows from \(\mathbb{N} \models \mathbf{PA}\).

In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax–Phillips scattering theory and Sz.-Nagy–Foias characteristic function theory we prove a completeness theorem.

We prove a relative n-completeness theorem asserting that a complex space Y that fibers over a complex space X of dimension less than n is n-complete provided that Y admits a continuous exhaustion function that is strictly plurisubharmonic along fibers. We apply it to a particular case related to Skoda’s conjecture.

Girard introduced phase semantics as a complete set-theoretic semantics of linear logic, and Okada modified phase-semantic completeness proofs to obtain normal-form theorems. On the basis of these works, Okada and Takemura reformulated Girard’s phase semantics so that it became phase semantics for proof-terms, i.e., lambda-terms. They formulated phase semantics for proof-terms of Laird’s dual affine/intuitionistic lambda-calculus and proved the normal-form theorem for Laird’s calculus via a completeness theorem. Their semantics was obtained by an application of computability predicates. In this paper, we first formulate phase semantics for proof-terms of second-order intuitionistic propositional logic by modifying Tait-Girard’s saturated sets method. Next, we prove the completeness theorem with respect to this semantics, which implies a strong normalization theorem.

We study a modal extension of the Continuous First-Order Logic of Ben Yaacov and Pedersen (J Symb Logic 75(1):168–190, 2010). We provide a set of axioms for such an extension. Deduction rules are just Modus Ponens and Necessitation. We prove that our system is sound with respect to a Kripke semantics and, building on Ben Yaacov and Pedersen (2010), that it satisfies a number of properties similar to those of first-order predicate logic. Then, by means of a canonical model construction, we get that every consistent set of formulas is satisfiable. From the latter result we derive an Approximated Strong Completeness Theorem, in the vein of Continuous Logic, and a Compactness Theorem.