Santiago Figueira

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We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jumptraceable if for each order(More)
Dickson's Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and model-checking, constraint solving, logic, etc. While Dickson's Lemma is well-known, most computer scientists are not aware of the complexity upper bounds that are entailed by(More)
Consider a Martin-Löf random ∆2 set Z. We give lower bounds for the number of changes of Zs n for computable approximations of Z. We show that each nonempty Π 1 class has a low member Z with a computable approximation that changes only o(2) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced(More)
We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic HL(↓), for example, can be thought of as a particular case of a memory logic where(More)
We define a program size complexity function H∞ as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in {0, 1} relative to the H∞ complexity. We prove that the classes of Martin-Löf random sequences and H∞-random(More)
Well quasi-orders (wqo’s) are an important mathematical tool for proving termination of many algorithms. Under some assumptions upper bounds for the computational complexity of such algorithms can be extracted by analyzing the length of controlled bad sequences. We develop a new, self-contained study of the length of bad sequences over the product ordering(More)
We study the length functions of controlled bad sequences over some well quasi-orders (wqo’s) and classify them in the Fast Growing Hierarchy. We develop a new and selfcontained study of the length of bad sequences over the disjoint product in N (Dickson’s Lemma), which leads to recently discovered upper bounds but through a simpler argument. We also give a(More)
The present work investigates several questions from a recent survey of Miller and Nies related to Chaitin’s Ω numbers and their dependence on the underlying universal machine. It is shown that there are universal machines for which ΩU is just ∑ x 2 1−H(x). For such a universal machine there exists a co-r.e. set X such that ΩU [X] = ∑ p:U(p)↓∈X 2 −|p| is(More)
In an unpublished manuscript Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing’s ideas as(More)