Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $\omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

Natural Factors of the Medvedev Lattice Capturing IPC

Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that for every non-trivial factor of the Medvedev lattice its theory is contained in Jankov's logic, the deductive closure of IPC plus the weak law of the excluded middle. This answers a question by Sorbi and Terwijn

The universality of polynomial time Turing equivalence

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees

On the Relation Between Representations and Computability

Computability and decidability are intimately linked problems which have interested computer scientists and mathematicians for a long time, especially during the last century. Work performed by Turing, Church, Godel, Post, Kleene and other authors considered the questions "What is computable?" and "What is an algorithm?". Very important results with plenty of implica- tions were obtained, such as the halting theorem [12], the several solutions to the Entscheidungsproblem [12, 5], the Church-Turing thesis [12] or Godel's incompleteness theorem. Further work was performed on topics which as of today have remained purely theoretical but which have o ered us a great understanding of computability and related questions. Some of this work in- cludes the one related to degrees of recursive unsolvability [1] [7] and Rice's theorem [11]. Several formalisms were described and compared, some of the most im- portant ones being Turing machines and -calculus. These formalisms were mathematical constructions which allowed the study of the concept of com- putation or calculation and all of its related questions. We have found that an often ignored detail and, as we show, important aspect of computability is related to representation. In particular, we show that the computability of an abstract problem can only be considered once a choice of representation has been made. We inquire to what extent this is essential and what e ects it may have and in what manner. We o er a wide discussion on its implications, a formalisation of these considerations and some important results deriving from these formalisations. In particular, the main result of the work is a proof that computably enumerable repre- sentations cannot be strictly stronger or weaker than other representations. We also discuss the Church-Turing thesis with particular interest, inquiring about its deep meaning and the actual facts and false assumptions related to it. Furthermore, we consider the relationship between representation and the so-called representation degrees and the degrees of recursive unsolvability de- rived from the concept of oracle machine. We show that these two concepts o er parallel hierarchies which are very similar in their construction but quite di erent in their essential meaning and properties.La computabilidad y la decidibilidad son problemas estrechamente relacionados que han interesado ampliamente a informáticos y matemáticos, especialmente a lo largo del ultimo siglo. Los trabajos realizados por Turing, Church, Godel, Post, Kleene y otros autores se planteaban las preguntas "Qué es computable?" y "Qué es un algoritmo?". Se lograron muchos resultados importantes con multitud de implicaciones, como el teorema de la parada [12], la solución al Entscheidungsproblem [12, 5], la hipótesis de Church-Turing [12] o el teorema de incompletidud de Godel. Gran cantidad del trabajo posterior se realizó en relación a otros temas que han permanecido hasta hoy en el campo de la teoría pero que nos han permitido entender en mayor medida la computabilidad y problemas relacionados. Por ejemplo, el relacionado con los grados de indecibilidad [1] [7] y el teorema de Rice [11]. Varios formalismos fueron descritos y comparados, algunos de los más importantes son las máquinas de Turing y el cálculo lambda. Estos formalismos constituían construcciones matemáticas que permitían el estudio del concepto de computación o cálculo y todas las preguntas relacionadas. Un aspecto comúnmente ignorado y relevante de la computabilidad está relacionado con la representación. En particular, percatamos que la com- putabilidad de un problema abstracto sólo puede ser considerada una vez se ha producido una elección de representación. Nos preguntamos hasta qué punto esto es esencial y qué efectos puede tener y de qué manera. Ofrecemos una amplia discusión sobre sus implicaciones, una formalización de estas consideraciones y algunos resultados importantes derivados de las mismas. En particular, el resultado principal del trabajo es una demostración de que las representaciones computacionalmente enumerables no pueden ser más fuertes o más débiles que otras. Realizamos una discusión especialmente enfrascada en relación a la tesis de Church-Turing, su significado más profundo y los hechos y falacias que giran en torno a ella. Además, consideramos la relación existente entre la representación y los llamados grados de representación, y los grados de indecibilidad derivados del concepto de máquina oráculo. Demostramos que estos dos conceptos ofrecen jerarquías paralelas con una construcción muy similar pero notablemente distintas en su significado esencial y sus propiedades