What is the Σ 3 2-circuit complexity (depth 3, bottom-fanin 2) of the 2n-bit inner product function? The complexity is known to be exponential 2 α n n for some α n = Ω ( 1 ). We show that the limiting constant α ≔ lim sup α n satisfies 0.847 . . ...

A finite automaton is called bideterministic if it is both deterministic and codeterministic – that is, if it is deterministic and its transpose is deterministic as well. The study of such automata in a weighted setting is initiated. All trim ...

Given a finite alphabet A and a quasi-metric d over A ⁎, we introduce the relation τ d , k ⊆ A ⁎ × A ⁎ such that ( x , y ) ∈ τ d , k holds whenever d ( x , y ) ≤ k. The error detection capability of variable-length codes is expressed in term of ...

We consider the McEliece cryptosystem with a binary Goppa code C ⊂ F 2 n specified by an irreducible Goppa polynomial g ( x ) ∈ F 2 m [ X ] and Goppa points ( α 1 , … , α n ) ∈ F 2 m n. Since g ( x ) together with the α i's allow for efficient ...

We show that reals z which compute complete extensions of arithmetic have the random join property: for each random x < T z there exists random y < T z such that z ≡ T x ⊕ y. The same is true for the truth-table and the weak truth-...

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several ...

Operations on regular languages that commute with any inverse alphabetic morphism are called 1-uniform transformations. In a previous work we have shown that these operations are encoded by operations on automata, called modifiers, ...

Two problems are studied. (1) How much external or internal information cost is required to compute a Boolean-valued function with an error at most 1 / 2 − ξ for small ξ > 0? A lower bound Ω ( ξ 2 ) and an upper bound O ( ξ ) are ...

Approximating the set of reachable states of a dynamical system is an algorithmic way to rigorously reason about its safety. Despite progress on efficient algorithms for affine dynamical systems, available algorithms still lack ...

We propose a new generalization of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language ...

We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the ...

We investigate the maximum number of distinct palindromic sub-words in a two-dimensional finite word. We give tight bounds for the maximum number of distinct palindromic sub-words in two-row periodic/aperiodic words. Using this, we ...

In a typical regex crossword puzzle, one is given two non-empty lists 〈 R 1 , … , R m 〉 and 〈 C 1 , … , C n 〉 of regular expressions (regexes) over some alphabet, and the challenge is to fill in an m × n grid of characters such that ...

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of read-once formulas is complete for log-space. In contrast, we observe that the problem ...

To each one-dimensional subshift X, we may associate a winning shift W ( X ) which arises from a combinatorial game played on the language of X. Previously it has been studied what properties of X does W ( X ) inherit. For example, X ...

A nearest neighbor representation of a Boolean function is a set of positive and negative prototypes in R n such that the function has value 1 on an input iff the closest prototype is positive. For k-nearest neighbor representation the ...

Recently, Scheerer [22] and Vandehey [24] showed that normality for continued fraction expansions and base-b expansions are incomparable notions. This shows that at some level, randomness for continued fractions and binary expansion ...

A DNA string is a Watson-Crick (WK-)palindrome when the complement of its reverse is equal to itself. The Watson-Crick mapping θ is an involution that is also an antimorphism. In this paper, we study a generalization of conjugates of a ...

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform ...

The CBH theorem characterises quantum theory within a C*-algebraic framework. Namely, mathematical properties of C*-algebras modelling quantum systems are equivalent to constraints that are information-theoretic in nature: (1) ...