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Let $T$ be a theory. If $T$ eliminates $\exists^\infty$, it need not follow that $T^{eq}$ eliminates $\exists^\infty$, as shown by the example of the $p$-adics. We give a criterion to determine whether $T^{eq}$ eliminates $\exists^\infty$. Specifically, we show that $T^{eq}$ eliminates $\exists^\infty$ if and only if $\exists^\infty$ is eliminated on all interpretable sets of "unary imaginaries." This criterion can be applied in cases where a full description of $T^{eq}$ is unknown. As an application, we show that $T^{eq}$ eliminates $\exists^\infty$ when $T$ is a C-minimal expansion of ACVF., Comment: 6 pages

An interatomic potential for the ternary Ag–Cu–Sn system, an important material system related to the applications of lead-free solders, is developed on the basis of the second nearest-neighbor modified embedded-atom-method formalism. Potential parameters for the ternary and related binary systems are determined based on the recently improved unary description of pure Sn and the present improvements to the unary descriptions of pure Ag and Cu. To ensure the sufficient performance of atomistic simulations in various applications, the optimization of potential parameters is conducted based on the force-matching method that utilizes density functional theory predictions of energies and forces on various atomic configurations. We validate that the developed interatomic potential exhibits sufficient accuracy and transferability to various physical properties of pure metals, intermetallic compounds, solid solutions, and liquid solutions. The proposed interatomic potential can be straightforwardly used in future studies to investigate atomic-scale phenomena in soldering applications. Graphical abstract

Predicting the life of Ni-Cd battery for electric multiple units (EMU) can not only improve the safety and reliability of battery, but also reduce the operating costs of EMU. For this reason, a life prediction method based on linear Wiener process is proposed, which is suitable for both monotonic and non-monotonic degraded systems with accurate results. Firstly, a unary linear Wiener degradation model is established, and the parameters of the model are estimated by using the expectation-maximization algorithm (EM). With the established model, the remaining useful life (RUL) of Ni-Cd battery and its distribution are obtained. Then based on the unary Wiener process degradation model, the correlation between capacity and energy is analyzed through Copula function to build a binary linear Wiener degradation model, where its parameters are estimated using Markov Chain Monte Carlo (MCMC) method. Finally, according to the binary Wiener process model, the battery RUL and its distribution are acquired. The experimental results show that the binary linear Wiener degradation model based on capacity and energy possesses higher accuracy than the unary linear wiener process degradation model.

There are two types of stateless deterministic ordered restarting automata that both characterize the class of regular languages: the stl-det-ORWW-automaton and the stl-det-ORRWW-automaton. For the former a notion of reversibility has been introduced and studied that is very much tuned to the way in which restarting automata work. Here we suggest another, more classical, notion of reversibility for stl-det-ORRWW-automata, and we show that each regular language is accepted by such a reversible stl-det-ORRWW-automaton. We study the descriptional complexity of these automata, showing that they are exponentially more succinct than nondeterministic finite-state acceptors. We also look at the case of unary input alphabets.

We investigate finite automata whose state graphs are undirected. This means that for any transition from state p to q consuming some letter a from the input there exists a symmetric transition from state q to p consuming a letter a as well. So, the corresponding language families are subregular, and in particular in the deterministic case, subreversible. In detail, we study the operational descriptional complexity of deterministic and nondeterministic undirected finite automata. To this end, the different types of automata on alphabets with few letters are characterized. Then, the operational state complexity of the Boolean operations as well as the operations concatenation and iteration is investigated, where tight upper and lower bounds are derived for unary as well as arbitrary alphabets under the condition that the corresponding language classes are closed under the operation considered.

We introduce the notion of a Σω-bounded structure and specify a necessary and sufficient condition for a universal Σ-function to exist in a hereditarily finite superstructure over such a structure, for the class of all unary partial Σ-functions assuming values in the set ω of natural ordinals. Trees and equivalences are exemplified in hereditarily finite superstructures over which there exists no universal Σ-function for the class of all unary partial Σ-functions, but there exists a universal Σ-function for the class of all unary partial Σ-functions assuming values in the set ω of natural ordinals. We construct a tree T of height 5 such that the hereditarily finite superstructure ℍ(T) over T has no universal Σ-function for the class of all unary partial Σ-functions assuming values 0, 1 only.

The first definition of space complexity for P systems was based on a hypothetical real implementation by means of biochemical materials, and thus it assumes that every single object or membrane requires some constant physical space. This is equivalent to using a unary encoding to represent multiplicities for each object and membrane. A different approach can also be considered, having in mind an implementation of P systems in silico; in this case, the multiplicity of each object in each membrane can be stored using binary numbers, thus reducing the amount of needed space. In this paper, we give a formal definition for this alternative space complexity measure, we define the corresponding complexity classes and we compare such classes both with standard space complexity classes and with complexity classes defined in the framework of P systems considering the original definition of space.

In Gua, an underdocumented Tano Guang language spoken in Ghana, regressive ATR vowel harmony applies within words and non-iteratively across word boundaries. Although vowel harmony is known to cross word boundaries in some languages, little is known about the domains and extent of such harmony. We show that ATR harmony in Gua operates within phonological phrases that preferentially consist of two or three words, with binary phrases at the left edge and ternary phrases at the right edge of the utterance. Syntactic structure can exert an influence, but only with respect to subjects. In addition, we demonstrate that unary phrases are permitted, but not at the edge of the utterance. Gua is the first reported vowel harmony case that shows the same kind of phonological phrasing sensitivity as other prosodic phenomena, such as tone and duration.

Unary functions definable in o-stable ordered groups of nonvaluational type are studied. Such functions are proved to be piecewise monotone and continuous.

We present new results on the computational limitations of affine automata (AfAs). First, we show that using the endmarker does not increase the computational power of AfAs. Second, we show that the computation of bounded-error rational-valued AfAs can be simulated in logarithmic space. Third, we identify some logspace unary languages that are not recognized by algebraic-valued AfAs. Fourth, we show that using arbitrary real-valued transition matrices and state vectors does not increase the computational power of AfAs in the unbounded-error model. When focusing only the rational values, we obtain the same result also for bounded error. As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only.

For a completely distributive De Morgan algebra L, we develop a general framework of L-fuzzy rough sets. Said precisely, we introduce a pair of L-fuzzy approximation operators, called upper and lower L-fuzzifying approximation operators derived from general L-fuzzifying neighborhood systems. It is shown that the proposed approximation operators are a common extension of the L-fuzzifying approximation operators derived from L-fuzzy relations (INS 2019) and the approximation operators derived from general neighborhood systems (KBS 2014). Furthermore, we investigate the unary, serial, reflexive, transitive and symmetric conditions in general L-fuzzifying neighborhood systems, and then study the associated approximation operators from both a constructive method and an axiomatic method. Particularly, for transitivity (resp., symmetry), we give two interpretations, one is an appropriate generalization of transitivity (resp., symmetry) for L-fuzzy relations, and the other is a suitable extension of transitivity (resp., symmetry) for general neighborhood systems. In addition, for some special L-fuzzifying approximation operators, we use single axiom to characterize them, respectively. At last, the proposed approximation operators are applied in the research of incomplete information system, and a three-way decision model based on them is established. To exhibit the effectiveness of the model, a practical example is presented.

The design and principle of operation of a multi-sensor converter of binary mechanical signals into electrical signals based on a partitioned fiber-optic digital-to-analog converter with parallel structure are considered. The digital-to-analog converter is made from a set of simple and fiber-optic digital-to-analog sections adaptable to streamlined production (three- and five-bit). The advantages of the optical layout of the proposed converter on metrological and energy characteristics, in comparison with unary multi-bit converters, are substantiated. It is shown that as a result of an increase in the number of digital-to-analog sections, it is possible to increase by a large factor the information capacity of the multi-sensor converter without tightening the requirements for its manufacturing design and component base. A mathematical model of the proposed converter has been developed that reflects the features of functioning under conditions of time-sequenced conversion of the input mechanical code vectors of the separate fiber-optic sections into electrical analogs, and generating a resultant output code vector. The developed mathematical model can be used at the design phase for converters of binary mechanical signals, in order to obtain extensive and deep information on the engineering prospects of the future product, without resorting to expensive physical experiment.

We study an order scheduling problem with rejection, in which each order consists of multiple product types and each product type should be manufactured on a dedicated machine. The aim is to find a solution to minimize a linear sum of the maximum delivery completion time of the accepted orders and the total penalty of the rejected orders. Even if the delivery times of all orders are zero, the problem is shown to be binary $$\mathcal{NP}$$ -hard in the two-machine case and it is shown to be unary $$\mathcal{NP}$$ -hard when the number of machines is arbitrary. Three approximation algorithms are proposed and their worst-case performance ratios are analyzed. For the scenario where the number of machines is fixed, a pseudo-polynomial dynamic programming algorithm and a fully polynomial time approximation scheme are devised for it.

We consider two kinds of reducibilities on finite families of predicates on a countable set: the definability of predicates and their complements of one family via another by means of existential formulas with parameters and the same definability on isomorphism types of families. Ordered structures of degrees generated by families of unary predicates are described. It is proved that for both reducibilities, there exist continuum many minimal nonzero degrees.

A positive definite Hermitian lattice is said to be 1-universal if it represents all positive definite unary Hermitian lattices, including both free and non-free Hermitian lattices. This paper is more concerned with the representations of unary non-free Hermitian lattices by Hermitian lattices. We estimate the minimal rank $$u_m^1$$ of 1-universal Hermitian lattices and we classify all 1-universal binary and ternary Hermitian lattices over imaginary quadratic fields $$\mathbb {Q}(\sqrt{-m})$$ for all positive square-free integers m.

We investigate the scheduling problem on a single bounded parallel-batch machine where jobs belong to two non-disjoint agents (called agent A and agent B) and are of equal length but different size. Each job’s size can be arbitrarily split into two parts and processed in the consecutive batches. It is permitted to process the jobs from different agents in a common batch. We show that it is unary NP-hard for the problem of minimizing the total weighted completion time of the jobs of agent A subject to the maximum cost of the jobs of agent B being upper bounded by a given threshold. For the case of the jobs of agent A having identical weights, we study the version of Pareto problem, and give a polynomial-time algorithm to generate all Pareto optimal points and a Pareto optimal schedule corresponding to each Pareto optimal point.

Demonic composition is defined on the set of binary relations over the non-empty set X, $$Rel_X$$ , and is a variant of standard or “angelic” composition. It arises naturally in the setting of the theory of non-deterministic computer programs, and shares many of the nice features of ordinary composition (it is associative, and generalises composition of functions). When equipped with the operations of demonic composition and domain, $$Rel_X$$ is a left restriction semigroup (like $$PT_X$$ , the semigroup of partial functions on X), whereas usual composition and domain give a unary semigroup satisfying weaker laws. By viewing $$Rel_X$$ under a restricted version of its usual composition and domain as a constellation (a kind of “one-sided” category), we show how this demonic left restriction semigroup structure arises on $$Rel_X$$ , placing it in a more general context. The construction applies to any unary semigroup with a “domain-like” operation satisfying certain minimal conditions which we identify. In particular it is shown that using the construction, any Baer $$*$$ -semigroup S can be given a left restriction semigroup structure which is even an inverse semigroup if S is $$*$$ -regular. It follows that the semigroup of $$n\times n$$ matrices over the real or complex numbers is an inverse semigroup with respect to a modified notion of product that almost always agrees with the usual matrix product, and in which inverse is pseudoinverse (Moore–Penrose inverse).

We consider the competitive multi-agent scheduling problem on a single machine, where each agent’s cost function is to minimize its total weighted late work. The aim is to find the Pareto-optimal frontier, i.e., the set of all Pareto-optimal points. When the number of agents is arbitrary, the decision problem is shown to be unary $$\mathcal {NP}$$ -complete even if all jobs have the unit weights. When the number of agents is two, the decision problems are shown to be binary $$\mathcal {NP}$$ -complete for the case in which all jobs have the common due date and the case in which all jobs have the unit processing times. When the number of agents is a fixed constant, a pseudo-polynomial dynamic programming algorithm and a $$(1+\epsilon )$$ -approximate Pareto-optimal frontier are designed to solve it.

This paper describes a unified global constraint to model scheduling problems with unary resources, i.e., each resource can process only a single activity at a time. In addition, the constraint enforces sequence-dependent transition times between activities. It often happens that activities are grouped into families with zero transition times within a family. Moreover, some of the activities might be optional from the resource viewpoint (typically in the case of alternative resources). The global constraint unifies reasoning with both optional activities and families of activities. The scalable filtering algorithms we discuss keep a low time complexity of $$\mathcal {O}(n \cdot \log (n) \cdot \log (f))$$ , where n is the number of tasks on the resource and f is the number of families. This results from the fact that we extend the $$\varTheta $$ -tree data structure used for the Unary Resource constraint without transition times. Our experiments demonstrate that our global constraint strengthens the pruning of domains as compared with existing approaches, leading to important speedups. Moreover, our low time complexity allows maintaining a small overhead, even for large instances. These conclusions are particularly true when optional activities are present in the problem.

The notion of neighborhood systems is abstracted from the geometric notion of “near,” and it is primitive in the theory of topological spaces. Now, the notion of neighborhood systems has been extensively applied in the study of rough set. The notion of fuzzifying neighborhood systems is a fuzzification of the notion of neighborhood systems, and it is initially in the theory of fuzzifying topological spaces. Said briefly, each element x and each subset A of a universe are associated with a number in the unit interval, interpreted as the degree of A being a neighborhood of x. In this paper, a model of rough sets derived from fuzzifying neighborhood systems is developed. It is shown that this model unifies many well-known rough sets such as binary relation-based rough sets, covering-based rough sets and neighborhood system-based rough sets into one framework. The new rough sets are studied from two approaches: the constructive and axiomatic approaches. Furthermore, when the fuzzifying neighborhood system is serial, reflexive, unary, transitive, symmetric and Euclidean, then the corresponding rough sets are discussed and characterized, respectively. At last, the reduction theory of this rough set model is established.

The aim of this work is to introduce a novel approach for the universal description of the thermodynamic functions of pure substances on the basis of artificial neural networks. The proposed approximation method is able to describe the thermodynamic functions ($$C_{p}(T), S(T), H(T)-H(T_{\mathrm{ref}}), G(T)$$ C p ( T ) , S ( T ) , H ( T ) - H ( T ref ) , G ( T ) ) of the different phases of unary material systems in a wide temperature range (between 0 and 6000 K). Phase transition temperatures and the respective enthalpies of transformation, which are computationally determined by the minimization of the Gibbs free energy, are also approximated. This is achieved by using artificial neural networks as models for the thermodynamic functions of the individual phases and by expressing the thermodynamic quantities in terms of the free network parameters. The resulting expressions are then optimized with machine learning algorithms on the basis of measurement data. A physical basis for the resulting approximation is given by the use of, among others, Planck–Einstein functions as activation function of the neurons of the network. This article provides a description of the method and as an example of a specific application the approximation of the thermodynamic functions of the different phases of pure iron. The article focuses on the problem of the representation of thermodynamic data and their practical application.

Approaches to anaphora generally seek to explain the potential for a DP to covary with a pronoun in terms of a combination of factors, such as (i) the inherent semantics of the antecedent DP (i.e., whether it is indefinite, quantificational, referential), (ii) its scope properties, and (iii) its structural position. A case in point is Reinhart’s classic condition on bound anaphora, paraphrasable as A DP can antecede a pronoun pro only if the DP c-commands pro at S-structure, supplemented with some extra machinery to allow indefinites to covary with pronouns beyond their c-command domains. In the present paper, I explore a different take. I propose that anaphora is governed not by DPs and their properties; it is governed by predicates (i.e., in the unary case, objects of type ) and their properties. To use a metaphor from dynamic semantics: discourse referents can only be ‘activated’ by predicates, never by DPs (Dynamic Predication Principle). This conceptually simple assumption is shown to have far-reaching consequences. For one, it yields a new take on weak crossover, arguably worthy of consideration. Moreover, it leads to a further general “restatement of the anaphora question”, in Reinhart’s (Linguist Philos 6: 47–88, 1983) words.

Completely regular semigroups with the unary operation of inversion within their maximal subgroups form a variety under inclusion denoted by $$\mathcal {C}\mathcal {R}$$. The lattice of its subvarieties is denoted by $$\mathcal {L}(\mathcal {C}\mathcal {R})$$. Kernel, trace, left trace and right trace relations on $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ induce operators which can be used to produce networks. For the pairs (kernel, trace)- and (left trace, right trace)-networks we establish strong properties. We consider examples of local- and core-relations networks in some special cases, as well as $$\mathbf {B}^\wedge \mathbf {B}^\vee $$-networks.

In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

This research develops a new mathematical modeling method by combining industrial big data and process mechanism analysis under the framework of generalized additive models (GAM) to generate a practical model with generalization and precision. Specifically, the proposed modeling method includes the following steps. Firstly, the influence factors are screened using mechanism knowledge and data-mining methods. Secondly, the unary GAM without interactions including cleaning the data, building the sub-models, and verifying the sub-models. Subsequently, the interactions between the various factors are explored, and the binary GAM with interactions is constructed. The relationships among the sub-models are analyzed, and the integrated model is built. Finally, based on the proposed modeling method, two prediction models of mechanical property and deformation resistance for hot-rolled strips are established. Industrial actual data verification demonstrates that the new models have good prediction precision, and the mean absolute percentage errors of tensile strength, yield strength and deformation resistance are 2.54%, 3.34% and 6.53%, respectively. And experimental results suggest that the proposed method offers a new approach to industrial process modeling.

We revisit the classical single-machine scheduling problem to minimize total tardiness with deadlines. The problem is binary NP-hard even without the deadline restrictions. It was reported early in Koulamas and Kyparisis (Eur J Oper Res 133:447–453, 2001) that the exact complexity (unary NP-hardness or pseudo-polynomial-time solvability) of the problem is still open. We show that this problem is unary NP-hard.

In this paper, we present correspondence analysis for the well-known four-valued logic First Degree Entailment (FDE). Correspondence analysis is Kooi and Tamminga’s technique for finding adequate natural deduction systems for all the truth-functional unary and binary extensions of an arbitrary functionally incomplete many-valued logic. In particular, Kooi and Tamminga initially presented correspondence analysis for Asenjo–Priest’s three-valued the Logic of Paradox. Generally speaking, a tabular functionally incomplete many-valued logic is added with an arbitrary unary or binary connective $$ \circ $$ following the truth-table definition of $$ \circ $$ . As a result, a tremendous amount of logics obtains a sound and complete natural deduction system in one go. In this paper, we generalize its application proposed by Kooi and Tamminga for the unary extensions of FDE and obtain correspondence analysis for the binary extensions of the logic in question. On the other hand, we use a proof-searching algorithm to have been successfully applied to classical and a variety of non-classical logics and present a finite, sound, and complete proof-searching algorithm for natural deduction systems for the binary extensions of FDE obtained via correspondence analysis. In the end, a comparative study of this method and the method presented by Avron and his collaborators is given.

Demonic composition $$*$$ is an associative operation on binary relations, and demonic refinement $${\sqsubseteq }$$ is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation $$\circ $$ given by $$s\circ t=D(s)*t$$ . We show that the class of relation algebras of signature $$\{\, \sqsubseteq , D, *\, \}$$ , or equivalently $$\{\, \subseteq , \circ , *\, \}$$ , has no finite axiomatisation. A large number of other non-finite axiomatisability consequences of this result are also given, along with some further negative results for related signatures. On the positive side, a finite set of axioms is obtained for relation algebras with signature $$\{\, \sqsubseteq , \circ , *\, \}$$ , hence also for $$\{\, \subseteq , \circ , *\, \}$$ .

In this paper, we study the ordered $$*$$ -semihypergroups in terms of fuzzy subsets in detail and define a unary operation $$\star $$ on the set of all the fuzzy subsets of an ordered $$*$$ -semihypergroup. To begin with, we define and study the fuzzy hyperideals of an ordered $$*$$ -semihypergroup. In particular, we investigate the properties of fuzzy hyperideals generated by ordered fuzzy points of an ordered $$*$$ -semihypergroup. Furthermore, we introduce the concepts of prime, weakly prime and semiprime fuzzy hyperideals of ordered $$*$$ -semihypergroups. Especially, the relationships among these three types of fuzzy hyperideals are established. In the sequel, we give some characterizations of intra-regular ordered $$*$$ -semihypergroups and semisimple ordered $$*$$ -semihypergroups in terms of fuzzy hyperideals. Especially, we prove that an ordered $$*$$ -semihypergroup S is semisimple if and only if every fuzzy hyperideal of S can be expressed as the intersection of all weakly prime fuzzy hyperideals of S containing it.

International audience; We study a variant of the Multi-Agent Path Finding (MAPF) problem in which the group of agents are required to stay connected with a supervising base station throughout the execution. In addition, we consider the problem of covering an area with the same connectivity constraint. We show that both problems are PSPACE-complete on directed and undirected topological graphs while checking the existence of a bounded plan is NP-complete when the bound is given in unary (and PSPACE-hard when the encoding is in binary). Moreover, we identify a realistic class of topo-logical graphs on which the decision problem falls in NLOGSPACE although the bounded versions remain NP-complete for unary encoding.

We introduce a general representation of unary hyperintensional modalities and study various hyperintensional modal logics based on the representation. It is shown that the major approaches to hyperintensionality known from the literature, that is state-based, syntactic and structuralist approaches, all correspond to special cases of the general framework. Completeness results pertaining to our hyperintensional modal logics are established.

We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $\Gamma$. We prove that whenever $\Gamma$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $\Gamma$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $\Gamma$., Comment: manuscript to be published in Theory of Computing Systems

Every Berman’s variety $$\mathbb {K}_p^q$$ which is the subvariety of Ockham algebras defined by the equation $${\sim ^{2p+q}}a = {\sim ^q}a$$ ($$p\ge 1$$ and $$q\ge 0$$) determines a finitary substitution invariant consequence relation $$\vdash _p^q$$. A sequent system $$\mathsf {S}_p^q$$ is introduced as an axiomatization of the consequence relation $$\vdash _p^q$$. The system $$\mathsf {S}_p^q$$ is characterized by a single finite frame $$\mathfrak {F}_p^q$$ under the frame semantics given for the formal language. By the duality between frames and algebras, $$\mathsf {S}_p^q$$ can be viewed as a $$4^{2p+q}$$-valued logic as it is characterized by a distributive lattice of $$4^{2p+q}$$ elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system $$\mathsf {G}_p^q$$ is established for $$\vdash _p^q$$. The Craig interpolation property of $$\vdash _p^q$$ is shown proof-theoretically utilizing $$\mathsf {G}_p^q$$.

A canonical (propositional) Gentzen-type system is a system in which every rule has the subformula property, it introduces exactly one occurrence of a connective, and it imposes no restrictions on the contexts of its applications. A larger class of Gentzen-type systems which is also extensively in use is that of quasi-canonical systems. In such systems a special role is given to a unary connective $$\lnot $$ of the language (usually, but not necessarily, interpreted as negation). Accordingly, each application of a logical rule in such systems introduces either a formula of the form $$\diamond (\psi _1,\ldots ,\psi _n)$$ , or of the form $$\lnot \diamond (\psi _1,\ldots ,\psi _n)$$ , and all the active formulas of its premises belong to the set $$\{\psi _1,\ldots ,\psi _n,\lnot \psi _1,\ldots ,\lnot \psi _n\}$$ . In this paper we investigate the whole class of quasi-canonical systems. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system is coherent iff it is analytic iff it has a characteristic non-deterministic matrix (Nmatrix) which is based on some subset of the set of the four truth-values which are used in the famous Dunn–Belnap four-valued (deterministic) matrix for information processing. In addition, we determine when a quasi-canonical system admits (strong) cut-elimination.

Recurrent neural networks (RNNs) are now widely used on sequence generation tasks due to their ability to learn long-range dependencies and to generate sequences of arbitrary length. However, their left-to-right generation procedure only allows a limited control from a potential user which makes them unsuitable for interactive and creative usages such as interactive music generation. This article introduces a novel architecture called anticipation-RNN which possesses the assets of the RNN-based generative models while allowing to enforce user-defined unary constraints. We demonstrate its efficiency on the task of generating melodies satisfying unary constraints in the style of the soprano parts of the J.S. Bach chorale harmonizations. Sampling using the anticipation-RNN is of the same order of complexity than sampling from the traditional RNN model. This fast and interactive generation of musical sequences opens ways to devise real-time systems that could be used for creative purposes.

The multi-covering rough sets (MCRSs) are a popular aspect of rough sets. It is easy to see that classical rough sets, covering rough sets (CRSs) and multi-granulation rough sets (MGRSs) are all the special cases of the MCRSs. Recently, the algebraic theory of these rough set models mentioned above have been researched in detail. However, the algebraic theory of MCRSs has not been studied until now. It is necessary for researchers to explore the algebraic theory of MCRSs. In this paper, we focus on the operation and algebraic theories of two types of MCRS models. First, the properties of the two types of multi-covering set approximations are discussed. Especially, the properties of multi-covering approximation operators based on the unary coverings are deeply researched. Second, the operation properties with respect to intersection and union of MCRSs are researched. Meanwhile, to compute the intersection and union of MCRSs, several algorithms are constructed. Finally, on the basis of the operation properties of MCRSs, many meaningful algebraic properties of MCRSs are deeply studied.

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP (which is built in the $$ \{\vee ,\wedge ,\lnot \} $$ -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete analogue of LP, strong Kleene logic $$ \mathbf K_3 $$ . In this paper, we generalize these results for the negative fragments of LP and $$ \mathbf K_3 $$ , respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or $$ \mathbf K_3 $$ , but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of $$ \mathbf K_3 $$ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic $$ \mathbf G_3 $$ and its dual $$ \mathbf DG_3 $$ (which have different interpretations of negation than $$ \mathbf K_3 $$ and LP).

In connection to developing the third generation of Calphad databases a new thermodynamic description is presented for unary carbon. Models used in this work have more physical basis and are valid down to 0 K. The anisotropy in graphite, caused by weak Van der Waals inter-plane forces makes it impossible to fit the heat capacity data by a single Einstein temperature for modelling the harmonic vibration of the atoms. By using multiple Einstein temperatures this problem is solved and a good agreement with the experimental data at both low and other temperatures is achieved. Diamond is modeled using multiple Einstein temperatures due to its specific heat capacities at very low temperatures too, and the two-state model is used for modelling the liquid phase.

Inclusion dependencies form one of the most fundamental classes of integrity constraints. Their importance in classical data management is reinforced by modern applications like data profiling, data cleaning, entity resolution, and schema matching. Their discovery in an unknown dataset is at the core of any data-analysis effort. Therefore, several research approaches have focused on their efficient discovery in a given, static dataset. However, none of these approaches are appropriate for application on dynamic datasets. In these cases, discovery techniques should be able to efficiently update the inclusion dependencies after an update in the dataset, without reprocessing the entire dataset. We present the first approach for incrementally updating the unary inclusion dependencies. In particular, our approach is based on the concept of attribute clustering, from which the unary inclusion dependencies are efficiently derivable. We incrementally update the clusters after each update of the dataset. An update of the clusters does not need access to the dataset because of special data structures designed to efficiently support the updating process. We performed an exhaustive analysis of our approach by applying it to large datasets with several hundred attributes and more than 116.2 million tuples. The results showed that the incremental discovery significantly reduces the runtime needed by the static discovery. This reduction in the runtime is up to 99.9996% for both the insertion and the deletion.

Recently, Hillert and Selleby proposed a simple method for expression of the lattice stability or Gibbs energy of formation that does not violate the third law of thermodynamics. This method describes the derivation of the Gibbs energy function from high temperatures down to 0 K by interpolation, instead of extrapolation from room temperature to 0 K. In the present work, their original method is discussed in terms of determination of the characteristic parameter values. Keeping the essential interpolation character of their method, a generalized approach is presented for expressing the lattice stability through parameter optimizations. This approach retains the zero point entropy of substances and is in line with the development of the third generation CALPHAD databases. Using the Al unary system as a case study, the lattice stabilities of the hcp and bcc phases are investigated. The respective Einstein temperatures are also evaluated. At high temperatures, the present descriptions reproduce the lattice stabilities suggested by SGTE for the existing second generation of databases, with a reasonable accuracy. More importantly, information from ab initio calculations (total energy at 0 K) is also used for this optimization and the present method results in a physically sounder description of thermodynamic properties at lower temperatures down to 0 K. The present approach provides a simple and flexible way to estimate the lattice stabilities, with potential applicability for the Gibbs energy of formation of stoichiometric compounds and the excess energy of solution phases, in accordance with the third law of thermodynamics.

Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: (a) chords inferentially related to subchords, (b) pairs of chords in the space of major–minor tonality and (c) arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments.

In order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized ?ukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations.

Green’s relations are a fundamental tool in the structure theory of semigroups. They can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes of Green’s relations then correspond to the strongly connected components. We study the complexity of Green’s relations in semigroups generated by transformations on a finite set. We show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of strongly connected components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for a constant size alphabet is rather involved. We also investigate the special cases of unary and binary alphabets. All these results are extended to deterministic finite automata and their syntactic semigroups.

We study the class of binary coded versions of unary languages that can be accepted by alternating machines with loglog n space. We show that there exists a binary PSpace-complete language $\mathcal {L}$ such that the unary coded version of $\mathcal {L}$ is in ASpace(loglog n). Consequently, the standard translation between unary languages accepted with loglog n space and binary languages accepted with log n space works for alternating machines if and only ifP = PSpace. In general, if a binary language is accepted deterministically in 2n⋅nO(1) time and, simultaneously, in nO(1) space—which covers many PSpace-complete problems—then its unary coded version is accepted by an alternating Turing machine using an initially delimited worktape of size loglog n. This unexpected power follows from the fact that, with an auxiliary worktape of size O(loglog n) on a unary input 1n, an alternating machine can simulate a stack with log n bits, representing the contents of the stack by its input head position. The standard push/pop operations on the stack are implemented by moving the head along the input.

We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product.

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the $$\overline{\mathcal R}_E$$ -class of every $$x\in S$$ contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of $$\overline{\mathcal R}_E$$ and $$\overline{\mathcal L}_E$$ . The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup $$T_X$$ can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.

A numerical algorithm that can quantitatively solve transcendental equation sets with multiple independent variables in the complex number domain is presented in this paper. A strict mathematical proof of solving the unary transcendental equation in the real domain is given in detail to show the process of the algorithm, which can be generalized to the equations with two or more variables. The complex variable concerned in this method is treated as two independent variables in real domain that represent the real part and the imaginary part of the variable, respectively. The criterion for choosing the dimension of scanning elements is given to overcome the difficulty in solving the general multivariate equation. Finally, the algorithm is applied in the problem of plotting 3-D dispersion curves in complex wave number and real frequency domains. Three examples are used to illustrate the correctness and effectiveness of the algorithm.

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theories of typed monoids and of rational series are used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in NC1. Affine Parikh automata, where each transition applies an affine transformation on the registers, are also considered. Relying on these characterizations, the landscape of relationships and closure properties of the classes at hand is completed, in particular over unary languages.

Fourier inversions of operations identify those with certain symmetries. A general method with invertible matrices over semirings sets up many different inversive schemes, such as disjunctive normal form, algebraic normal form and classical Fourier analysis. First, unary functions are inverted, and then, with tensor products, the same is done for operations of more than one argument. Classical Fourier analysis is applied to inverting operations on a set of k elements by replacing it by the set of $${k^{\rm th}}$$ roots of unity. The Fourier transform of an operation preserving rotations of the roots has many coefficients that are zero, in fact, at most $${1/k}$$ are non-zero. A related result holds for operations preserving reflections of the roots.

In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner (affirmative/negative, extensional/intensional). First-order FDE is also briefly revisited, and we conclude by listing some open problems for future research.