Title: Model constructions for bounded arithmetic Author: Michal Garlík Abstract: We study constructions of models of bounded arithmetic theories. Us- ing basic techniques of model theory we give a new proof of Ajtai's completeness theorem for nonstandard finite structures. Working in the framework of restricted reduced powers (a generalization of the ultrapower construction) we devise two methods of constructing models of bounded arithmetic. The first one gives a new proof of Buss's witnessing theorem. Using the second method we show that the theory R1 2 is stronger than its variant strictR1 2 under a plausible computational assumption (the existence of a strong enough one-way permutation), and that the theory PV1 + Σb 1(PV ) − LLIND is stronger than PV1 + strictΣb 1(PV ) − LLIND under the same assumption. Considering relativized theories, we show that R1 2(α) is stronger than strictR1 2(α) (unconditionally). 1

The thesis formulates and proves a witnessing theorem for SPV -provable formulas in the form ∀x∃yA(x, y) where A corresponds to a polynomial time decidable relation. By SPV we understand an extension of the theory TPV (the universal theory of N in the language representing polynomial algorithms) by additional axioms ensuring the existence of a minimum of a linear ordering defined by a polynomial time decidable relation on an initial segment. As these additional axioms are not universal sentences, the theory SPV requires nontrivial use of witnessing Herbrand's and KPT theorems which have direct application only for universal theories. Based on the proven witnessing theorem, we derive a NP search problem characterizing complexity of finding y for a given x such that A(x, y). A part of the thesis is dedicated to arguments supporting the conjecture that SPV is strictly stronger than TPV . 1

The present article turns on the basis of the Communication Theory of Law as it has been developed by Professor Gregorio Robles, who has elaborated his theory being based on the conception of Law as a system of communication between the human beings. This article focuses on analyzing one of the fundamental elements concerning his Theory: the idea of conceiving the Law as a text. The article finishes with a few valuations. / El presente artículo versa sobre la fundamentación de la Teoría Comunicacional del Derecho tal y como la ha desarrollado el profesor Gregorio Robles basándose en la concepción del Derecho como un sistema de comunicación entre los seres humanos. Este artículo se centra en analizar uno de los elementos fundamentales en torno al cual pivota su Teoría: la idea de concebir el Derecho como texto. Por último, el artículo finaliza con unas valoraciones.