Randomness and differentiability in higher dimensions

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies differentiability of computable Lipschitz functions of several variables. Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.Comment: 19 page

Regional policy spillovers : the national impact of demand-side policy in an interregional model of the UK economy

UK regional policy has been advocated as a means of reducing regional disparities and stimulating national growth. However, there is limited understanding of the interregional and national effects of such a policy. This paper uses an interregional computable general equilibrium model to identify the national impact of a policy-induced regional demand shock under alternative labour market closures. Our simulation results suggest that regional policy operating solely on the demand side has significant national impacts. Furthermore, the effects on the nontarget region are particularly sensitive to the treatment of the regional labour market

On sample complexity for computational pattern recognition

In statistical setting of the pattern recognition problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question whether such bounds exist if we restrict our attention to computable pattern recognition methods, assuming that the unknown labelling function is also computable. We find that in this case the number of examples required for a computable method to approximate the labelling function not only is not linear, but grows faster (in the VC dimension of the class) than any computable function. No time or space constraints are put on the predictors or target functions; the only resource we consider is the training examples. The task of pattern recognition is considered in conjunction with another learning problem -- data compression. An impossibility result for the task of data compression allows us to estimate the sample complexity for pattern recognition

Influence tests I: ideal composite hypothesis tests, and causal semimeasures

Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete time series is defined using generalized structural equations. Several ideal tests are introduced, and it is argued that when Halting information is transmitted, in some cases, instantaneous cause and consequence can be inferred where this is not possible classically. The approach is contrasted with Bayesian definitions of influence, where it is left open whether all Bayesian causal associations of universal semimeasures are equal within a constant. Finally the approach is also contrasted with existing engineering procedures for influence and theoretical definitions of causation.Comment: 29 pages, 3 figures, draf

The national impact of regional policy : demand-side policy simulation with labour market constraints in a two-region computable general equilibrium model

UK governments generally advocate regional policy as a means of reducing regional disparities and stimulating national growth. However, there is limited comprehension regarding the effects of regional policy on non-target economies. This paper examines the system-wide effects on the Scottish and rest of UK (RUK) economies of an increase in Scottish traded sector exports to the rest of the world. The research is carried out in an inter-regional Computable General Equilibrium framework of the Scottish and RUK economies, under alternative hypotheses regarding wage determination and inter-regional migratory behaviour. The findings suggest that regional policy can have significant national spillover effects, even when the target region is small relative to the RUK. Furthermore, the configuration of the labour market is important in determining the post-shock adjustment path of both economies. In particular, while Scottish economy results are sensitive to alternative versions of how regional labour markets function, RUK region effects prove to be even more so

A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions

Let f(1)=1, and let f(n+1)=2^{2^{f(n)}} for every positive integer n. We conjecture that if a system S \subseteq {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} \cup {x_i+1=x_k: i,k \in {1,...,n}} has only finitely many solutions in non-negative integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies x_1,...,x_n \leq f(2n). We prove: (1) the conjecture implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite, (2) the conjecture implies that the question whether or not a Diophantine equation has only finitely many rational solutions is decidable with an oracle for deciding whether or not a Diophantine equation has a rational solution, (3) the conjecture implies that the question whether or not a Diophantine equation has only finitely many integer solutions is decidable with an oracle for deciding whether or not a Diophantine equation has an integer solution, (4) the conjecture implies that if a set M \subseteq N has a finite-fold Diophantine representation, then M is computable.Comment: 13 pages, section 7 expande

Characterization theorem for the conditionally computable real functions

The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable real functions with respect to the same class of operators is a proper extension of the class of uniformly computable real functions and it computes the elementary functions of calculus on their whole domains. The definition of both classes relies on certain transformations of infinitistic names of real numbers. In the present paper, the conditional computability of real functions is characterized in the spirit of Tent and Ziegler, avoiding the use of infinitistic names

Shannon Information and Kolmogorov Complexity

We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate the basic notions of both theories: Shannon entropy versus Kolmogorov complexity, the relation of both to universal coding, Shannon mutual information versus Kolmogorov (`algorithmic') mutual information, probabilistic sufficient statistic versus algorithmic sufficient statistic (related to lossy compression in the Shannon theory versus meaningful information in the Kolmogorov theory), and rate distortion theory versus Kolmogorov's structure function. Part of the material has appeared in print before, scattered through various publications, but this is the first comprehensive systematic comparison. The last mentioned relations are new.Comment: Survey, LaTeX 54 pages, 3 figures, Submitted to IEEE Trans Information Theor