Arno Pauly

Closed choice and a uniform low basis theorem

Abstract We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of hypercomputation in a uniform framework using Weihrauch reducibility. The classes of functions which are reducible to closed choice of the singleton space, the natural numbers, Cantor space and Baire space correspond to the class of computable functions, functions computable with finitely many mind changes, weakly computable functions and effectively Borel measurable functions, respectively. We also prove that all these classes correspond to classes of non-deterministically computable functions with the respective spaces as advice spaces. The class of limit computable functions can be characterized with parallelized choice of natural numbers. On top of these results we provide further insights into algebraic properties of closed choice. In particular, we prove that closed choice on Euclidean space can be considered as “locally compact choice” and it is obtained as product of closed choice on the natural numbers and on Cantor space. We also prove a Quotient Theorem for compact choice which shows that single-valued functions can be “divided” by compact choice in a certain sense. Another result is the Independent Choice Theorem, which provides a uniform proof that many choice principles are closed under composition. Finally, we also study the related class of low computable functions, which contains the class of weakly computable functions as well as the class of functions computable with finitely many mind changes. As a main result we prove a uniform version of the Low Basis Theorem that states that closed choice on Cantor space (and the Euclidean space) is low computable. We close with some related observations on the Turing jump operation and its initial topology.

On the (semi)lattices induced by continuous reducibilities

Continuous reducibilities are a proven tool in Computable Analysis, and have applications in other fields such as Constructive Mathematics or Reverse Mathematics. We study the order-theoretic properties of several variants of the two most important definitions, and especially introduce suprema for them. The suprema are shown to commutate with several characteristic numbers (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

How Incomputable is Finding Nash Equilibria

We investigate the Weihrauch-degree of several solution concepts from noncooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria, as well as various concepts of iterated strategy elimination, are dealt with. As a side result, the Weihrauch-degree of solving systems of linear inequalities is settled.

The degree structure of Weihrauch-reducibility

We answer a question by Vasco Brattka and Guido Gherardi by proving that the Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further investigate the existence of infinite infima and suprema, as well as embeddings of the Medvedev-degrees into the Weihrauch-degrees.

On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escardo, Schroder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and - closely linked to these - properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.

Non-deterministic computation and the Jayne-Rogers Theorem

We provide a simple proof of a computable analogue to the Jayne Rogers Theorem from descriptive set theory. The difficulty of the proof is delegated to a simulation result pertaining to non-deterministic type-2 machines. Thus, we demonstrate that developments in computational models can have applications in fields thought to be far removed from it.

Computation with Advice

Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice.

Infinite sequential games with real-valued payoffs

We investigate the existence of certain types of equilibria (Nash, ε-Nash, subgame perfect, ε-subgame perfect) in infinite sequential games with real-valued payoff functions depending on the class of payoff functions (continuous, upper semi-continuous, Borel) and whether the game is zero-sum. Our results hold for games with two or up to countably many players. Several of these results are corollaries of stronger results that we establish about equilibria in infinite sequential games with some weak conditions on the occurring preference relations. We also formulate an abstract equilibrium transfer result about games with compact strategy spaces and open preferences. Finally, we consider a dynamical improvement rule for infinite sequential games with continuous payoff functions.

Weihrauch Degrees of Finding Equilibria in Sequential Games

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.

On the computational content of the brouwer fixed point theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. Another main result is that connected choice is complete for dimension greater or equal to three in the sense that it is computably equivalent to Weak Kőnigs Lemma. In contrast to this, the connected choice operations in dimensions zero, one and two form a strictly increasing sequence of Weihrauch degrees, where connected choice of dimension one is known to be equivalent to the Intermediate Value Theorem. Whether connected choice of dimension two is strictly below connected choice of dimension three or equivalent to it is unknown, but we conjecture that the reduction is strict. As a side result we also prove that finding a connectedness component in a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnigs Lemma.