1.1 Informal introduction to Π1 relations Let 2N denote Cantor space. • A relation B ⊆ N × (2N)r is Π1 if it is obtained from an arithmetical relation by a universal quantification over sets. • If k = 1, r = 0 we have a Π1 set ⊆ N. • If k = 0, r = 1 we have a Π1 class ⊆ 2N. • The relation B is∆1 if both B and its complement are Π1. There is an equivalent representation of Π1 relations where the...

An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr’s Theorem. In the new setting, while K-trivial sets exist that are not hyper-arithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z i...

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are Rn, Cn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2N, the Baire space NN, the infinite symmetric group S∞...

Proof. As the edges in GC represent causal relations, a path of length 0 (no edge) is not considered a causal relation, and existence of a directed path from a variable back to itself would contravene the causal DAG assumption, hence: irreflexive and acyclic (or, more accurate, asymmetric). Transitivity follows immediately, by concatenation, from the sequence 〈X, .., Y, .., Z〉, in which each no...

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L0 are two logics and ì is an asymptotic measure on finite structures, then L a.e. L0 ...