We show that the usual positional representations of a real number are either random, in the sense of Martin-Lof, for all bases or not so for any base. Thus, randomness is an invariant of number representations. All our proofs are constructive.

Randomness as an Invariant for Number Representations

Type-two Theory of Effectivity (TTE) provides a general framework for Computable Analysis. To refine it to polynomial-time computability while keeping as much generality as possible, Kawamura and Cook recently proposed a modification to TTE using machines that have random access to an oracle and run in time depending on the “size” of the oracle. They defined type-two analogues of P, NP, PSPACE and applied them to real functions and operators. We further refine their model and study computation below P: type-two analogues of the classes L, NC, and P-completeness under log-space reductions. The basic idea is to use second-order polynomials as resource bounds, as Kawamura and Cook did, but we need to make some nontrivial (yet natural, as we will argue) choices when formulating small classes in order to make them well-behaved. Most notably: we use a modification of the constant stack model of Aehlig, Cook and Nguyen for query tapes in order to allow sufficient oracle accesses without interfering with space bounds; representations need to be chosen carefully, as computational equivalence between them is now finer; uniformity of circuits must be defined with varying sizes of oracles taken into account. As prototypical applications, we recast several facts (some in a stronger form than was known) about the complexity of numerical problems into our framework.

Small Complexity Classes for Computable Analysis

Infinite linear systems and one counter languages

Lethal Autonomous Weapons promise to revolutionize warfare -- and raise a multitude of ethical and legal questions. It has thus been suggested to program values and principles of conduct (such as the Geneva Conventions) into the machines' control, thereby rendering them both physically and morally superior to human combatants. 
We employ mathematical logic and theoretical computer science to explore fundamental limitations to the moral behaviour of intelligent machines in a series of "Gedankenexperiments": Refining and sharpening variants of the Trolley Problem leads us to construct an (admittedly artificial but) fully deterministic situation where a robot is presented with two choices: one morally clearly preferable over the other -- yet, based on the undecidability of the Halting problem, it provably cannot decide algorithmically which one. Our considerations have surprising implications to the question of responsibility and liability for an autonomous system's actions and lead to specific technical recommendations.

Logical Limitations to Machine Ethics with Consequences to Lethal Autonomous Weapons.

Promises are a standard way to formalize partial algorithms; and advice quantifies nonuniformity. For decision problems, the latter is captured in common complexity classes such as \(\mathcal{P}/\operatorname{poly}\), that is, with advice growing in size with that of the input. We advertise constant-size advice and explore its theoretical impact on the complexity of classification problems – a natural generalization of promise problems – and on real functions and operators. Specifically we exhibit problems that, without any advice, are decidable/computable but of high complexity while, with (each increase in the permitted size of) advice, (gradually) drop down to polynomial-time.

Real Benefit of Promises and Advice

Promise problems have been introduced in 1985 by S. Even e.a. as a generalization of decision problems. Using a very general approach we study solvability and unsolvability conditions for promise problems of set and language families. We show, that cores of unsolvability are completely determined by partitions of cohesive sets. We prove the existence of cores in unsolvable promise problems assuming certain closure properties for the given set family. Connections to immune sets and complexity cores are presented. Furthermore, results about cohesiveness with respect to the language families from the Chomsky hierarchy are given.

/pdf/cohesiveness-in-promise-problems-1v8nf9ik7s.pdf

Cohesiveness in promise problems

Number representations and registers

We exhibit a new class of grammars with the help of weightfunctions. They are characterized by decreasing the weight during the derivation process. A decision algorithm for the emptiness problem is developed. This class contains non-contextfree grammars. The corresponding language class is identical to the class of ultralinear languages.

/pdf/weightreducing-grammars-and-ultralinear-languages-3kqr676tw8.pdf

Ulrike Brandt

Papers

Real Benefit of Promises and Advice

Cohesiveness in promise problems

Number representations and registers

Weightreducing grammars and ultralinear languages

On Testing Output Faults in the McCluskey Fault Model.