Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Building Decision Procedures in the Calculus of Inductive Constructions

It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved

Languages Generated by Iterated Idempotencies.

The rewrite relation with parameters m and n and with the possible length limit = k or :::; k we denote by w~, =kW~· or ::;kw~ respectively. The idempotency languages generated from a starting word w by the respective operations are wDAlso other special cases of idempotency languages besides duplication have come up in different contexts. The investigations of Ito et al. about insertion and deletion, Le., operations that are also observed in DNA molecules, have established that w5 and w~ both preserve regularity.Our investigations about idempotency relations and languages start out from the case of a uniform length bound. For these relations =kW~ the conditions for confluence are characterized completely. Also the question of regularity is -k n answered for aH the languages w- D 1 are more complicated and belong to the class of context-free languages.For a generallength bound, i.e."for the relations :"::kW~, confluence does not hold so frequently. This complicatedness of the relations results also in more complicated languages, which are often non-regular, as for example the languages WWithout any length bound, idempotency relations have a very complicated structure. Over alphabets of one or two letters we still characterize the conditions for confluence. Over three or more letters, in contrast, only a few cases are solved. We determine the combinations of parameters that result in the regularity of wDIn a second chapter sorne more involved questions are solved for the special case of duplication. First we shed sorne light on the reasons why it is so difficult to determine the context-freeness ofduplication languages. We show that they fulfiH aH pumping properties and that they are very dense. Therefore aH the standard tools to prove non-context-freness do not apply here.The concept of root in Formal Language ·Theory is frequently used to describe the reduction of a word to another one, which is in sorne sense elementary.For example, there are primitive roots, periodicity roots, etc. Elementary in connection with duplication are square-free words, Le., words that do not contain any repetition. Thus we define the duplication root of w to consist of aH the square-free words, from which w can be reached via the relation w~.Besides sorne general observations we prove the decidability of the question, whether the duplication root of a language is finite.Then we devise acode, which is robust under duplication of its code words.This would keep the result of a computation from being destroyed by dupli cations in the code words. We determine the exact conditions, under which infinite such codes exist: over an alphabet of two letters they exist for a length bound of 2, over three letters already for a length bound of 1.Also we apply duplication to entire languages rather than to single words; then it is interesting to determine, whether regular and context-free languages are closed under this operation. We show that the regular languages are closed under uniformly bounded duplication, while they are not closed under duplication with a generallength bound. The context-free languages are closed under both operations.The thesis concludes with a list of open problems related with the thesis' topics

Termination of Narrowing: Automated Proofs and Modularity Properties

En 1936 Alan Turing demostro que el halting problem, esto es, el problema de decidir si un programa termina o no, es un problema indecidible para la inmensa mayoria de los lenguajes de programacion. A pesar de ello, la terminacion es un problema tan relevante que en las ultimas decadas un gran numero de tecnicas han sido desarrolladas para demostrar la terminacion de forma automatica de la maxima cantidad posible de programas. Los sistemas de reescritura de terminos proporcionan un marco teorico abstracto perfecto para el estudio de la terminacion de programas. En este marco, la evaluaci on de un t ermino consiste en la aplicacion no determinista de un conjunto de reglas de reescritura. El estrechamiento (narrowing) de terminos es una generalizacion de la reescritura que proporciona un mecanismo de razonamiento automatico. Por ejemplo, dado un conjunto de reglas que denan la suma y la multiplicacion, la reescritura permite calcular expresiones aritmeticas, mientras que el estrechamiento permite resolver ecuaciones con variables. Esta tesis constituye el primer estudio en profundidad de las propiedades de terminacion del estrechamiento. Las contribuciones son las siguientes. En primer lugar, se identican clases de sistemas en las que el estrechamiento tiene un comportamiento bueno, en el sentido de que siempre termina. Muchos metodos de razonamiento automatico, como el analisis de la semantica de lenguajes de programaci on mediante operadores de punto jo, se benefician de esta caracterizacion. En segundo lugar, se introduce un metodo automatico, basado en el marco teorico de pares de dependencia, para demostrar la terminacion del estrechamiento en un sistema particular. Nuestro metodo es, por primera vez, aplicable a cualquier clase de sistemas. En tercer lugar, se propone un nuevo metodo para estudiar la terminacion del estrechamiento desde un termino particular, permitiendo el analisis de la terminacion de lenguajes de programacion. El nuevo metodo generaliza losIborra López, J. (2010). Termination of Narrowing: Automated Proofs and Modularity Properties [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19251Palanci

QPCF: higher order languages and quantum circuits

qPCF is a paradigmatic quantum programming language that ex- tends PCF with quantum circuits and a quantum co-processor. Quantum circuits are treated as classical data that can be duplicated and manipulated in flexible ways by means of a dependent type system. The co-processor is essentially a standard QRAM device, albeit we avoid to store permanently quantum states in between two co-processor's calls. Despite its quantum features, qPCF retains the classic programming approach of PCF. We introduce qPCF syntax, typing rules, and its operational semantics. We prove fundamental properties of the system, such as Preservation and Progress Theorems. Moreover, we provide some higher-order examples of circuit encoding