The Intersection Problem for Finite Semigroups

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

Varieties generated by unstable involution semigroups with continuum many subvarieties

This paper examines the circuit complexity of feedforward neural networks having sigmoid activation function. The starting point is the complexity class NN k defined in [18]. First two additional complexity classes N N Δ k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1772.tif/> and N N Δ , e k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1773.tif/> having less restrictive conditions (than NN k ) concerning fan-in and accuracy are defined. We then prove several relations among these three classes and well established circuit complexity classes. All the proofs are constructive as being built around binary adders. By relaxing the fan-in condition from logarithmic [18] to (almost) polynomial, the results show that substantially larger classes of sigmoid activation feedforward neural networks can be implemented in polynomial size Boolean circuits. This is done on the expense of a logarithmic factor increase in the number of layers, but having constant fan-in (of 2). The main conclusion is that there are interesting fan-in dependent depth-size tradeoffs when trying to digitally implement sigmoid activation neural networks.

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

The following theorem is proved: Suppose R is an associative ring and suppose J is the Jacobson radical of R. Suppose that for all x1, * *, xn in R, there exists a word wx... (x1, * * x), depending on x1, * *, x,,, in which at least one xi (i varies) is missing, and such that xi ... x, = w ,... .n(x1, , x). Then J is a nil ring of bounded index and R/J is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In this paper, we investigate the structure of an associative ring R with the property that, for all xl, * * * , x, in R, xl ... x = wxi.... xn(xl, * * *, Xn), where w is a word, depending on xl, * * *, xn, and where some xi (i varies) is missing in w. For such a ring R, we prove the Jacobson radical J is a nil ring of bounded index. We also show that R/J is finite. Finally, we show that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In preparation for the proofs of the main results, we first introduce DEFINITION 1. Let n be a positive integer, and let S be a semigroup. We say that S is a fln-semigroup if, for all xl, * * *, xn in S, there exists a word w .(x, *1 , xn) consisting of a product of the xi's with some x1 (j varies) missing, such that xl ... xn=wxl....xn((xl, * * *, xn). A ring R is called a fln-ring if its multiplicative semigroup is a fln-semigroup. LEMMA 1. Let S be a finite semigroup or a nilpotent semigroup. Then S is a fln-semigroup, for some n. PROOF. First, suppose S is finite, of order n, and suppose xl, Xn+1 are any elements of S. Then xi=xj for some i>j, and hence X1 ... Xn+1 = X1 ... Xi ... Xi-lXjXi+l Xn+1 = W(X1, . . , Xi-1 Xi+1 , Xn+1) Thus S is a fln+l-semigroup. Next, suppose S is nilpotent, say, Sm= (0). Presented to the Society, January 18, 1972; received by the editors August 10, 1971 and, in revised form, October 6, 1971. AMS (MOS) subject classifications (1970). Primary 16A38, 16A48; Secondary 20M10. The author was supported by a National Science Foundation Graduate Fellowship. @ American Mathematical Society 1973

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

Separations among the first order logic Ring(0,+, ∗) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN201457226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).

On the linear orderability of two classes of finite semigroups

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ϱ a congruence on S. Consider the sequence ϱ (0)⊆ϱ (1)⊆⋯⊆ϱ (n)⊆⋯ of congruences on S, where ϱ (0)=ϱ and, for an arbitrary non-negative integer n, ϱ (n+1) is defined by (a,b)∈ϱ (n+1) if and only if (xa,xb)∈ϱ (n) for all x∈S. We show that \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )\) for an arbitrary congruence ϱ on a semigroup S, where lrc(ϱ) denotes the least left reductive congruence on S containing ϱ. We focuse our attention on congruences ϱ on semigroups S for which the congruence \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is left reductive. We prove that, for a congruence ϱ on a semigroup S, \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is a left reductive congruence of S if and only if \(\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}\) is a left reductive congruence on the factor semigroup S/ϱ (here ι (S/ϱ) denotes the identity relation on S/ϱ). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S n =S n+1 is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.

We show how a positive solution to the word problem for finitely generated commutative semigroups leads to a positive solution to the conjugacy problem for a class of H N N HNN groups.

New precedence results are obtained for semigroups without zero, which describe the first row in detail and provide effective comparisons with the next few rows. These results can be used to further sharpen existing algorithms for the computation of finite commutative semigroups.

From Theorem 19.1 of [l] it follows that the lattice of varieties (equational classes) of Abelian groups is distributive.We show that the lattice of varieties of commutative semigroups is not modular by considering an example from the dual lattice of closed commutative semigroup laws.We will assume and use without comment that any given set of laws contains the commutative and associative laws.The binary operation is denoted by juxtaposition.By (a, b), where a and b are positive integers, we will mean the term xayb.Let the sets A', B', and C of laws be given by the following table.

In this paper, we describe all subsets of the lattices [Formula: see text] and [Formula: see text] of varieties (and pseudovarieties) of commutative semigroups which have the property of being a better-quasi-ordering (well-quasi-ordering). In particular, we show that the lattice [Formula: see text] of varieties of commutative nilpotent semigroups is a better-quasi-ordering, which solves a problem posed by Almeida.1.