Joseph S. Miller

The undecidability of iterated modal relativization

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ11–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 Σ01–complete. These results go via reduction to problems concerning domino systems.

Decidability and Complexity Results for Timed Automata and Semi-linear Hybrid Automata

We define a new class of hybrid automata for which reachability is decidable--a proper superclass of the initialized rectangular hybrid automata--by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show that reachability is undecidable for these algebraic timed automata by simulating two-counter Minsky machines. Modifying the construction to apply to parametric timed automata, we reprove the undecidability of the emptiness problem, and then distinguish the dense and discrete-time cases with a new result. The algorithmic complexity-- both classical and parametric--of one-clock parametric timed automata is also examined. We finish with a table of computability-theoretic complexity results, including that the existence of a Zeno run is Σ11 -complete for semi-linear hybrid automata; it is too complex to be expressed in first-order arithmetic.

RELATIVIZING CHAITIN'S HALTING PROBABILITY

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that and are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

The K-Degrees, Low for K Degrees,and Weakly Low for K Sets

We call A weakly low for K if there is a c such that KA(σ) ≥ K(σ)−c for infinitely many σ; in other words, there are infinitely many strings that A does not help compress. We prove that A is weakly low for K iff Chaitin’s Ω is A-random. This has consequences in the K-degrees and the low for K (i.e., low for random) degrees. Furthermore, we prove that the initial segment prefix-free complexity of 2-random reals is infinitely often maximal. This had previously been proved for plain Kolmogorov complexity.

Randomness and differentiability

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function [0,1]->R is differentiable at z. (2) We prove that a real number z in [0,1] is weakly 2-random if and only if each almost everywhere differentiable computable function [0,1]->R is differentiable at z. (3) Recasting in classical language results dating from 1975 of the constructivist Demuth, we show that a real z is ML random if and only if every computable function of bounded variation is differentiable at z, and similarly for absolutely continuous functions. We also use our analytic methods to show that computable randomness of a real is base invariant, and to derive other preservation results for randomness notions.

Randomness for non-computable measures

Different approaches have been taken to defining randomness for non-computable probability measures. We will explain the approach of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas. We will show that these approaches are fundamentally equivalent. Having clarified what it means to be random for a non-computable probability measure, we turn our attention to Levin’s neutral measures, for which all sequences are random. We show that every PA degree computes a neutral measure. We also show that a neutral measure has no least Turing degree representation and explain why the framework of the continuous degrees (a substructure of the enumeration degrees studied by Miller) can be used to determine the computational complexity of neutral measures. This allows us to show that the Turing ideals below neutral measures are exactly the Scott ideals. Since X ∈ 2 is an atom of a neutral measure μ if and only if it is computable from (every representation of) μ, we have a complete understanding of the possible sets of atoms of a neutral measure. One simple consequence is that every neutral measure has a Martin-Lof random atom. 1. Defining randomness Let X be an element of Cantor space and μ a Borel probability measure on Cantor space. What should it mean for X to be random with respect to μ? In the case that μ is the Lebesgue measure, then the theory of μrandomness is well developed (for recent treatises on the subject the reader is referred to Downey and Hirschfeldt, and Nies [2, 13]). In fact if μ is a computable measure, then early work of Levin showed that μ-randomness can be seen as essentially a variant on randomness for Lebesgue measure [10]. This leaves the question of how to define randomness if μ is non-computable. We will show that the two approaches that have previously been used to define μ-randomness, for non-computable μ, are equivalent. Later, in Theorem 4.12, we will provide another characterization of μ-randomness using the enumeration degrees. Last compilation: September 8, 2011 Last time the following date was changed: November 22, 2010. 2010 Mathematics Subject Classification. Primary 03D32; Secondary 68Q30, 03D30. The second author was supported by the National Science Foundation under grants DMS-0945187 and DMS-0946325, the latter being part of a Focused Research Group in Algorithmic Randomness.

Two notes on subshifts

We prove two unrelated results about subshifts. First, we give a condition on the lengths of forbidden words that is sufficient to guarantee that the corresponding subshift is nonempty. The condition implies that, for example, any sequence of binary words of lengths 5, 6, 7, . . . is avoidable. As another application, we derive a result of Durand, Levin and Shen [2, 3] that there are infinite sequences such that every substring has high Kolmogorov complexity. In particular, for any d < 1, there is a b ∈ N and an infinite binary sequence X such that if τ is a substring of X, then τ has Kolmogorov complexity greater than d |τ | − b. The second result says that from the standpoint of computability theory, any behavior possible from an arbitrary effectively closed subset of nN (i.e., a Π1 class) is exhibited by an effectively closed subshift. In technical terms, every Π1 Medvedev degree contains a Π 0 1 subshift. This answers a question of Simpson [10].

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

We study computably enumerable equivalence relations (ceers), under the reducibility R ≤ S if there exists a computable function f such that x R y if and only if f(x) S f(y), for every x, y. We show that the degrees of ceers under the equivalence relation generated by ≤ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if R′ ≤ R, where R′ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are Σ3-complete (the former answering an open question of Gao and Gerdes).

Cupping with random sets

We prove that a set is K-trivial if and only if it is not weakly ML-cuppable. Further, we show that a set below zero jump is K-trivial if and only if it is not ML-cuppable. These results settle a question of Ku\v{c}era, who introduced both cuppability notions.

The Denjoy alternative for computable functions

The Denjoy-Young-Saks Theorem from classical analysis states that for an arbitrary real-valued function f, the Denjoy alternative holds outside a null set. This means that for almost every real x, either the derivative of f exists at x, or the derivative fails to exist in the worst possible way: the slopes of f around x take arbitrarily large positive values and arbitrarily large negative values. Demuth studied effective versions of this theorem, in particular the effective version when the function f is Markov computable. He then looked at the set DA of reals x such that any Markov computable function satisfies the Denjoy alternative at x. He introduced a notion of algorithmic randomness (now known as Demuth randomness) which he proved to be sufficient for a point to belong to DA. In this paper, we show that DA is in fact strictly contained in the set of Demuth random points. We also obtain interesting side results about effective versions of Lebesgues density theorem.