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

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

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 $\mathcal{D}_{h}$

The weakness of being cohesive, thin or free in reverse mathematics

Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey's theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to stable Ramsey's theorem for pairs, revealing the combinatorial nature of this non-reducibility and prove that whenever $k$ is greater than $\ell$, stable Ramsey's theorem for $n$-tuples and $k$ colors is not computably reducible to Ramsey's theorem for $n$-tuples and $\ell$ colors. In this sense, Ramsey's theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey's theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page

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