We continue the work on mathematical modeling of hardware and software started in [11]. The main objective of this paper is the definition of a program. We start with the concept of partial product, i.e. the set of all partial functions f from I to ⋃ i∈I Ai, fulfilling the condition f .i ∈ Ai for i ∈ dom f . The computation and the result of a computation are defined in usual way. A finite part...

The articles [16], [19], [20], [2], [21], [4], [9], [3], [1], [11], [15], [5], [17], [18], [10], [7], [8], [6], [13], [22], [12], and [14] provide the notation and terminology for this paper. We use the following convention: n is a natural number, x, X, X1 are sets, and s, r, p are real numbers. Let S, T be 1-sorted structures. A partial function from S to T is a partial function from the carri...

We consider multi-dimensional variational integrals F [u] := Ω f (·, u, Du) dx where the integrand f is a strictly convex function of its last argument. We give an elementary proof for the partial C 1,α-regularity of minimizers of F. Our approach is based on the method of A-harmonic approximation, avoids the use of Gehring's lemma, and establishes partial regularity with the optimal Hölder expo...

Let $$\sigma _{1}, \dots , \sigma _{k}$$ be the elementary symmetric functions of complex variables $$x_{1}, x_{k}$$ . We say that $$F \in {\mathbb {C}}[\sigma _{k}]$$ is a trace function if their exists $$f {C}}[z]$$ such $$F(\sigma _{k}) = \sum _{j=1}^{k} f(x_{j})$$ for all {C}}^{k}$$ give an explicit finite family second order differential operators in Weyl algebra $$W_{2}:= _{k}]\langle \fr...

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity fbs(f). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show unbounded separations between fbs(f) and other adversary bo...

A class F of partial recursive functions is called recursively enumerable if there exists an r.e. set J ⊆ N such that F = {φi | i ∈ J}. We prove that every r.e. class F of partial recursive functions with infinite domains must have a recursive witness array, i.e. there is a computable array of finite sets X = [Xn]n∈ω such that (i) for every f ∈ F one has f(n) ∈ Xn for infinitely many n and (ii)...

According to Larsen [15, p. 13], a function F from a nonvoid subset DF of a commutative semigroup A into A is called a partial multiplier on A if F (D) · E = F (E) ·D for all D, E ∈ DF . Note that if in particular DF is a subsemigroup of A and F ( D · E ) = F (D) · E for all D , E ∈ DF , then F is a partial multiplier on A . A partial multiplier F on A is called total if DF = A . Clearly, the i...

The purpose of this note is to observe a generalization of the concept “computable in. . . ” to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f ] is constructed such that in A[f ], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley’s 2-categ...

We describe a method that automatically induces LFG f{structures from treebank tree representations , given a set of f{structure annotation principles that deene partial, modular c{ to f{structure correspondences in a linguistically informed, principle-based way. This work extends the approach of van Genabith, Sadler and Way (1999a,b,c) where f{ structure annotation of treebanks is driven by ma...