N-ary Functions Research Articles

Monadic Expressions and Their Derivatives

There are several well-known ways to compute derivatives of regular expressions due to Brzozowski, Antimirov or Lombardy and Sakarovitch. We propose another one which abstracts the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we first introduce a new derivation technique based on the graded module monad and then show an application of this technique to generalize the parsing of expressions with capture groups and back references. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

Towards a finer classification of strongly minimal sets

Let M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with |I|>1, dcl⁎(I)=∅ (* means not in dcl of a proper subset). This implies the only definable truly n-ary functions f (f ‘depends’ on each argument), occur when n=1. We prove for Hrushovski's original construction and for the strongly minimal k-Steiner systems of Baldwin and Paolini that the symmetric definable closure, sdcl⁎(I)=∅ (Definition 2.7). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies sdcl⁎(I)≠∅. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if k=pn. The case structure depends on properties of the Hrushovski μ-function. The proofs depend on our introduction, for appropriate G⊆aut(M) (setwise or pointwise stabilizers of finite independent sets), the notion of a G-normal substructure A of M and of a G-decomposition of any finite such A. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit.

Monadic Expressions and their Derivatives

We propose another interpretation of well-known derivatives computations from regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.g. sets or linear combinations) using the notion of monad. As an example of this generalization advantage, we introduce a new derivation technique based on the graded module monad. We also extend operators defining expressions to any n-ary functions over value sets, such as classical operations (like negation or intersection for Boolean weights) or more exotic ones (like algebraic mean for rational weights). Moreover, we present how to compute a (non-necessarily finite) automaton from such an extended expression, using the Colcombet and Petrisan categorical definition of automata. These category theory concepts allow us to perform this construction in a unified way, whatever the underlying monad. Finally, to illustrate our work, we present a Haskell implementation of these notions using advanced techniques of functional programming, and we provide a web interface to manipulate concrete examples.

Some cardinality estimates for the set of correlation-immune Boolean functions

Abstract Estimates for the cardinality of the set of correlation-immune n-ary Boolean functions with fixed weight are obtained.

Pre-(quasi-)overlap functions on bounded posets

In this paper, we propose extensions of overlap functions on the bounded poset P. We generalize the concept of continuity on bounded posets for n-ary functions on P. In order to do so, we introduce the concept of conditioned continuity of n-ary functions based on directed and filtered sets in Pn. Then, we present the definition of pre-(quasi-)overlap functions by replacing the monotonicity and continuity of overlap functions on complete lattices with conditioned ones on bounded posets. Besides, we show several construction methods of pre-(quasi-)overlap functions.

Implementation complexity of Boolean functions with a small number of ones

Abstract We consider the class F n, k consisting of n-ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class F n, k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis { x & y , x ‾ } $ \{x\&y,\overline x\} $ (or in the basis { x ∨ y , x ‾ } ) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.

Directional Shift-Stable Functions

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [0,1]n, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of n=2 of directionally shift-stable aggregation and pre-aggregation functions are also given.

Semigroups of Terms, Tree Languages, Menger Algebra of n-Ary Functions and Their Embedding Theorems

The concepts of terms and tree languages are significant tools for the development of research works in both universal algebra and theoretical computer science. In this paper, we establish a strong connection between semigroups of terms and tree languages, which provides the tools for studying monomorphisms between terms and generalized hypersubstitutions. A novel concept of a seminearring of non-deterministic generalized hypersubstitutions is introduced and some interesting properties among subsets of its are provided. Furthermore, we prove that there are monomorphisms from the power diagonal semigroup of tree languages and the monoid of generalized hypersubstitutions to the power diagonal semigroup of non-deterministic generalized hypersubstitutions and the monoid of non-deterministic generalized hypersubstitutions, respectively. Finally, the representation of terms using the theory of n-ary functions is defined. We then present the Cayley’s theorem for Menger algebra of terms, which allows us to provide a concrete example via full transformation semigroups.

Menger hyperalgebras and their representations

The paper is devoted to the investigation of algebraic hyperstructures. The concept of Menger hyperalgebras, which is a canonical generalization of semihypergroups, is introduced. The emphasis of this paper is on the algebraic nature of such structure concerning subhyperalgebras, homomorphisms and quotient hyperstructures, that allows for a rich algebraic theory. Based on the theory of multiplace functions, multivalued full functions (or hyperoperation) on a general fixed arity is defined. This leads us to construct the Menger hyperalgebras of multivalued full n-ary functions. In particular, we prove that every abstract Menger hyperalgebra can be represented by multivalued full n-ary functions.

Do the Types of Seat Surface influence the pulmonary Functions during Prolonged Sitting?

Purpose: The purpose of this study was to identify the effects of the types of seat surface (static or dynamic seat surface) on the pulmo nary functions during prolonged sitting. Methods: Thirty-four participants (20 males and 14 females) were recruited, and distributed randomly into dynamic prolonged sitting (DPS, n=17) and static prolonged sitting (SPS, n=17) groups. The DPS group was seated on a chair with a dynamic air cushion, and the SPS group was seated on a chair without a dynamic air cushion. The pulmonary function was assessed before sitting, and after partici pants had been seated for one hour. The pulmonary function [forced vital capacity (FVC), forced expiratory volume in 1 second (FEV1), and Peak expiratory flow (PEF)] was measured using a spirometer. Results: Statistical analyses revealed significant differences in the time x group interactions of FVC, FEV1, PEF, and FEV1/FVC. The DPS group were significantly different in FVC, FEV1, PEF, and FEV1/FVC after prolonged sitting for one hour, compared to the SPS group (p<0.05). Conclusion: These findings suggest that dynamic sitting can prevent a decrease in the physiological function, such as pulmonary func tions, rather than static sitting during prolonged sitting.

Quasiuniversal Boolean Automaton with Four Constant States

The problem of realization of Boolean functions by initial Boolean automata with constant states and n inputs is considered. Initial Boolean automaton with constant states and n inputs is an initial automaton with output such that in all states the output functions are n-ary constant Boolean functions 0 or 1. An example of an initial Boolean automaton with the minimum number of constant states and n inputs realizing the maximum possible number of n-ary Boolean fonctions, where n ≥ 3, is constructed.

Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure

The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

Lower Bounds of Complexity for Polarized Polynomials over Finite Fields

We obtain an efficient lower bound of complexity for n-ary functions over a finite field of arbitrary order in the class of polarized polynomials. The complexity of a function is defined as the minimal possible number of nonzero terms in a polarized polynomial realizing the function.

Some Characterizations of Lattice OWA Operators

Ordered Weighted Averaging (OWA) operators are a family of aggregation functions for data fusion. If the data are real numbers, then OWA operators can be characterized either as a special kind of discrete Choquet integral or simply as an arithmetic mean of the given values previously ordered. This paper analyzes the possible generalizations of these characterizations when OWA operators are defined on a complete lattice. In addition, the set of all n-ary OWA operators is studied as a sublattice of the lattice of all the n-ary aggregation functions defined on a distributive lattice.

Lower bound for the complexity of five-valued polarized polynomials

Abstract The paper is devoted to the complexity of representation of q-valued functions by polarized polynomials and by matrix Kronecker forms of certain type. The complexity of a function is the minimal possible number of nonzero coefficients of a polynomial or a Kronecker form representing the function. It is known that for polynomial representation and representation by Kronecker forms of a certain type the maximal values of complexity in the class of all q-valued n-ary functions coincide. We establish the lower bound of these maximal values for five-valued functions.

Estimates for the number of Boolean functions realized by an initial Boolean automaton with three constant states

The problem of realization of Boolean functions by initial Boolean automata with constant states and n inputs is considered. Such automata are those whose output function coincides with one of n-ary constant Boolean functions 0 or 1 in all states. The exact value of the maximum number of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with three constant states and n inputs is obtained.

Maximal number of Boolean functions realized by an initial Boolean automaton with two constant states

The problem of realization of Boolean functions by initial Boolean automata with two constant states and n inputs is considered. An initial Boolean automaton with two constant states and n inputs is an initial automaton with output such that in all states the output functions are n-ary constant Boolean functions 0 or 1. The maximum cardinality of set of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with two constant states and n inputs is obtained.

Efficient Binary Linear Programming Formulations for Boolean Functions

A very useful tool when designing linear programs for optimization problems is the formulation of logical operations by linear programming constraints. We give efficient linear programming formulation of important n-ary boolean functions f(x_1,\ldots,x_n)=x_{n+1} such as conjunction, disjunction, equivalence, and implication using n+1 boolean variables x1,...,x_{n+1}. For the case that the value f(x1, ...,xn) is not needed for further computations, we even give more compact formulation. Our formulations show that every binary boolean function f(x1,x2)=x3 can be realized by the only three boolean variables x1,x2,x3 and at most four linear programming constraints.

Superpositions of continuous functions defined on a Baire space

We consider uniformly continuous functions on a Baire space and introduce the notion of a continuity modulus of a function. We formulate a condition on the growth of the continuity modulus ? guaranteeing that superpositions of n-ary functions with continuity modulus ? do not exhaust all (n + 1)-ary functions with continuity modulus ? for any n. Moreover, negating this property leads to the inverse effect.

Polynomial-Space Decidable Membership Problems for Recurrent Systems over Sets of Natural Numbers

A finite recurrent system over sets of natural numbers of dimension n is a pair composed of n n-ary functions over sets of natural numbers and an n-tuple of singleton sets of natural numbers. Every function is applied to the entries of the tuple and computes a set of natural numbers, that may also be empty. The results are composed into another tuple, and the process is started anew. Thus, a finite recurrent system defines an infinite sequence of n-tuples containing sets of natural numbers. The last entry of a generated n-tuple is called the output of a step, and the union of the output sets of all steps is the set defined by the finite recurrent system. Membership problems ask whether a given number is in a specified output set or in some output set. We study membership problems for special finite recurrent systems, whose functions are built from the set operations union, intersection and complementation and the arithmetical operations addition and multiplication. Sum and product of two sets of natural numbers are defined elementwise. We restrict the set of operations from which functions are built and determine the impact on the complexity of the membership problems. We focus on PSPACE-decidable membership problems and show completeness results for the complexity classes NL, NP and PSPACE.