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

Collusion between Algorithms: a literature review and limits to enforcement

Algorithms play an increasingly important role in economic activity, as they become faster and smarter. Together with the increasing use of ever larger data sets, they may lead to significant changes in the way markets work. These developments have raised concerns not only over the right to privacy and consumers’ autonomy, but also on competition. Infringements of antitrust laws involving the use of algorithms have occurred in the past. However, current concerns are of a different nature as they relate to the role algorithms can play as facilitators of collusive behavior in repeated games, and the role increasingly sophisticated algorithms can play as autonomous implementers of firms’ strategies, as they learn to collude without any explicit instructions provided by human agents. In particular, it is recognized that the use of ‘learning algorithms’ can facilitate tacit collusion and lead to an increased blurring of borders between tacit and explicit collusion. Several authors who have addressed the possibilities for achieving tacit collusion equilibrium outcomes by algorithms interacting autonomously, have also considered some form of ex-ante assessment and regulation over the type of algorithms used by firms. By using well-known results in the theory of computation, I show that such option faces serious challenges to its effectiveness due to undecidability results. Ex-post assessment may be constrained as well. Notwithstanding several challenges faced by current software testing methodologies, competition law enforcement and policy have much to gain from an interdisciplinary collaboration with computer science and mathematics

Controlling algorithmic collusion : short review of the literature, undecidability, and alternative approaches

Algorithms have played an increasingly important role in economic activity, as they becoming faster and smarter. Together with the increasing use of ever larger data sets, they may lead to significant changes in the way markets work. These developments have been raising concerns not only over the rights to privacy and consumers’ autonomy, but also on competition. Infringements of antitrust laws involving the use of algorithms have occurred in the past. However, current concerns are of a different nature as they relate to the role algorithms can play as facilitators of collusive behavior in repeated games, and the role increasingly sophisticated algorithms can play as autonomous implementers of pricing strategies, learning to collude without any explicit instructions provided by human agents. In particular, it is recognized that the use of ‘learning algorithms’ can facilitate tacit collusion and lead to an increased blurring of borders between tacit and explicit collusion. Several authors who have addressed the possibilities for achieving tacit collusion equilibrium outcomes by algorithms interacting autonomously, have also considered some form of ex-ante assessment and regulation over the type of algorithms used by firms. By using well-known results in the theory of computation, I show that such option faces serious challenges to its effectiveness due to undecidability results. Ex-post assessment may be constrained as well. Notwithstanding several challenges face by current software testing methodologies, competition law enforcement and policy have much to gain from an interdisciplinary collaboration with computer science and mathematics.info:eu-repo/semantics/publishedVersio