This thesis establishes formal equational properties of multi-stage calculi and related proof techniques that support analyses of staged programs. A key promise of staging is to make programs efficient without destroying clarity, thereby reducing the likelihood of bugs. However, few publications rigorously verify that their staged programs indeed behave as intended. In fact, little is known about how staged programs can be verified, or what correctness issues staging introduces. To solve this problem, I show a reduction of the correctness of a staged program to that of an unstaged program. This reduction not only clarifies the effects of staging on program behavior but also eases verification, as unstaged programs are more susceptible to existing reasoning techniques. I also demonstrate that important single-stage reasoning techniques apply to staged programs. These techniques are useful for establishing side conditions for the reduction and for discovering or validating further reasoning principles. / NSF grant CCF-0747431

Slices are digraphs that can be composed together to form larger digraphs.In this thesis we introduce the foundations of a theory whose aim is to provide ways of defining and manipulating infinite families of combinatorial objects such as graphs, partial orders, logical equations etc. We give special attentionto objects that can be represented as sequences of slices. We have successfully applied our theory to obtain novel results in three fields: concurrency theory,combinatorics and logic. Some notable results are: Concurrency Theory: We prove that inclusion and emptiness of intersection of the causalbehavior of bounded Petri nets are decidable. These problems had been open for almost two decades. We introduce an algorithm to transitively reduce infinite familiesof DAGs. This algorithm allows us to operate with partial order languages defined via distinct formalisms, such as, Mazurkiewicztrace languages and message sequence chart languages. Combinatorics: For each constant z ∈ N, we define the notion of z-topological or-der for digraphs, and use it as a point of connection between the monadic second order logic of graphs and directed width measures, such as directed path-width and cycle-rank. Through this connection we establish the polynomial time solvability of a large numberof natural counting problems on digraphs admitting z-topological orderings. Logic: We introduce an ordered version of equational logic. We show thatthe validity problem for this logic is fixed parameter tractable withrespect to the depth of the proof DAG, and solvable in polynomial time with respect to several notions of width of the equations being proved. In this way we establish the polynomial time provability of equations that can be out of reach of techniques based on completion and heuristic search. / <p>QC 20131120</p>

We can classify several quite different calculi for automated reasoning in first-order logic as instantiation-based methods (IMs). Broadly speaking, unlike in traditional calculi such as resolution where the first-order satisfiability problem is tackled by deriving logical conclusions, IMs attempt to reduce the first-order satisfiability problem to propositional satisfiability by intelligently instantiating clauses. The Inst-Gen-Eq method is an instantiation-based calculus which is complete for first-order clause logic modulo equality. Its distinctive feature is that it combines first-order reasoning with efficient ground satisfiability checking, which is delegated in a modular way to any state-of-the-art ground solver for satisfiability modulo theories (SMT). The first-order reasoning modulo equality employs a superposition-style calculus which generates the instances needed by the ground solver to refine a model of a ground abstraction or to witness unsatisfiability. The thesis addresses the main issue in the Inst-Gen-Eq method, namely efficient extraction of instances, while providing powerful redundancy elimination techniques. To that end we introduce a novel labelled unit superposition calculus with sets, AND/OR trees and ordered binary decision diagrams (OBDDs) as labels. The different label structures permit redundancy elimination each to a different extent. We prove completeness of redundancy elimination from labels and further integrate simplification inferences based on term rewriting. All presented approaches, in particular the three labelled calculi are implemented in the iProver-Eq system and evaluated on standard benchmark problems.

Maude es un lenguaje de programación declarativo basado en la lógica de reescritura que incorpora muchas características que lo hacen muy potente. Sin embargo, a la hora de probar ciertas propiedades computacionales esto conlleva dificultades. La tarea de probar la terminación de sistemas de reesctritura es de hecho bastante dura, pero aplicada a lenguajes de programación reales se concierte en más complicada debido a estas características inherentes. Esto provoca que métodos para probar la terminación de este tipo de programas requieran técnicas específicas y un análisis cuidadoso. Varios trabajos han intentado probar terminación de (un subconjunto de) programas Maude. Sin embargo, todos ellos siguen una aproximación transformacional, donde el programa original es trasformado hasta alcanzar un sistema de reescritura capaz de ser manejado con las técnicas y herramientas de terminación existentes. En la práctica, el hecho de transformar los sistemas originales suele complicar la demostración de la terminación ya que esto introduce nuevos símbolos y reglas en el sistema. En esta tesis, llevamos a cabo el problema de probar terminación de (un subconjunto de) programas Maude mediante métodos directos. Por un lado, nos centramos en la estrategia de Maude. Maude es un lenguaje impaciente donde los argumentos de una función son evaluados siempre antes de la aplicación de la función que los usa. Esta estrategia (conocida como llamada por valor) puede provocar la no terminación si los programas no están escritos cuidadosamente. Por esta razón, Maude (en concreto) incorpora mecanismos para controlar la ejecución de programas como las anotaciones sintácticas que están asociadas a los argumentos de los símbolos. En reescritura, esta estrategia sería conocida como reescritura sensible al contexto innermost (RSCI). Por otro lado, Maude también incorpora la posibilidad de declarar atributos. / Alarcón Jiménez, B. (2011). Towards a Framework for Proving Termination of Maude Programs [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11003

The paper contains a proof of completeness of a goal-directed inference system for general E-unification with eager Variable Elimination. The proof is based on an analysis of a concept of ground, equational proof. The theory of equational proofs is developed in the first part. Solving variables in a goal is then shown to be reflected in defined transformations of an equational proof. The termination of these transformations proves termination of inferences with eager Variable Elimination.

The area of formal analysis of cryptographic protocols has been an active one since the mid 80’s. The idea is to verify communication protocols that use encryption to guarantee secrecy and that use authentication of data to ensure security. Formal methods are used in protocol analysis to provide formal proofs of security, and to uncover bugs and security flaws that in some cases had remained unknown long after the original protocol publication, such as the case of the well known Needham-Schroeder Public Key (NSPK) protocol. In this thesis we tackle problems regarding the three main pillars of protocol verification: modelling capabilities, verifiable properties, and efficiency. This thesis is devoted to investigate advanced features in the analysis of cryptographic protocols tailored to the Maude-NPA tool. This tool is a model-checker for cryptographic protocol analysis that allows for the incorporation of different equational theories and operates in the unbounded session model without the use of data or control abstraction. An important contribution of this thesis is relative to theoretical aspects of protocol verification in Maude-NPA. First, we define a forwards operational semantics, using rewriting logic as the theoretical framework and the Maude programming language as tool support. This is the first time that a forwards rewriting-based semantics is given for Maude-NPA. Second, we also study the problem that arises in cryptographic protocol analysis when it is necessary to guarantee that certain terms generated during a state exploration are in normal form with respect to the protocol equational theory. We also study techniques to extend Maude-NPA capabilities to support the verification of a wider class of protocols and security properties. First, we present a framework to specify and verify sequential protocol compositions in which one or more child protocols make use of information obtained from running a parent protocol. Second, we present a theoretical framework to specify and verify protocol indistinguishability in Maude-NPA. This kind of properties aim to verify that an attacker cannot distinguish between two versions of a protocol: for example, one using one secret and one using another, as it happens in electronic voting protocols. Finally, this thesis contributes to improve the efficiency of protocol verification in Maude-NPA. We define several techniques which drastically reduce the state space, and can often yield a finite state space, so that whether the desired security property holds or not can in fact be decided automatically, in spite of the general undecidability of such problems. / Santiago Pinazo, S. (2015). Advanced Features in Protocol Verification: Theory, Properties, and Efficiency in Maude-NPA [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48527

Dans cette thèse, nous visons à formaliser les effets calculatoires. En effet, les langages de programmation les plus utilisés impliquent différentes sortes d'effets de bord: changement d'état, exceptions, entrées / sorties, non-déterminisme, etc. Ils peuvent apporter facilité et flexibilité dans le processus de codage. Cependant, le problème est de prendre en compte les effets lorsque l'on veut prouver des propriétés de programmes. La principale difficulté dans ce genre de preuve de programmes est le décalage entre la syntaxe des opérations avec effets de bord et leur interprétation. Typiquement, un fragment de programme avec des arguments de type X qui retourne une valeur de type Y n'est pas interprété comme une fonction de X vers Y , à cause des effets.L'approche algébrique la plus connue pour ce problème permet une interprétation des programmes, y compris ceux comportant des effets, en utilisant des monades : l'interprétation est une fonction de X vers T (Y ) où T est une monade. Cette approche a été étendue aux théories de Lawvere et aux "gestionnaires algébriques" (algebraic handlers). Une autre approche, appelée logique décorée, fournit une sémantique équationnelle pour ces programmes. Nous spécialisons l'approche de la logique décorée pour les effets liés à l'état de la mémoire et à la gestion des exceptions en définissant la logique décorée pour les états (L_st) et la logique décorée pour les exceptions (L_exc), respectivement. Elles nous permettent de prouver des propriétés de programmes impliquant de tels effets. Ensuite, nous formalisons ces logiques en Coq et certifions les preuves associées. Ces logiques sont construites de manière à être correctes. En outre, nous introduisons une notion de complétude syntaxique relative d'une théorie dans une logique donnée par rapport à une sous-logique. Nous montrons que la théorie décorée pour les états globaux ainsi que deux théories décorées pour les exceptions sont relativement complets relativement à leur sous-logique pure. Non seulement nous pouvons utiliser le système développé pour prouver des programmes comportant des effets, mais également nous utilisons cette formalisation pour certifier les résultats de complétude obtenus. / In this thesis, we aim to formalize the effects of a computation. Indeed, most used programming languages involve different sorts of effects: state change, exceptions, input/output, non-determinism, etc. They may bring ease and flexibility to the coding process. However, the problem is to take into account the effects when proving the properties of programs. The major difficulty in such kind of reasoning is the mismatch between the syntax of operations with effects and their interpretation. Typically, a piece of program with arguments in X that returns a value in Y is not interpreted as a function from X to Y , due to the effects. The best-known algebraic approach to the problem interprets programs including effects with the use of monads: the interpretation is a function from X to T(Y) where T is a monad. This approach has been extendedto Lawvere theories and algebraic handlers. Another approach called, the decorated logic, provides a sort of equational semantics for reasoning about programs with effects. We specialize the approach of decorated logic to the state and the exceptions effects by defining the decorated logic for states (L_st) and the decorated logic for exceptions (L_exc), respectively. This enables us to prove properties of programs involving such effects. Then, we formalize these logics in Coq and certify the related proofs. These logics are built so as to be sound. In addition, we introduce a relative notion of syntactic completeness of a theory in a given logic with respect to a sublogic. We prove that the decorated theory for the global states as well as two decorated theories for exceptions are syntactically complete relatively to their pure sublogics. These proofs are certified in Coq as applications of ourgeneric frameworks.

Le lambda-calcul est un formalisation de la notion de calcul. Dans cette thèse nous nous intéresserons à certaines variantes non déterministes, et nous nous pencherons plus particulièrement sur le cas probabiliste.L'étude du lambda-calcul probabiliste n'est pas nouvelle, mais les travaux précédents considéraient le comportement probabiliste comme un effet de bord. Notre objectif est de présenter ce calcul d'une manière plus équationnelle, en intégrant le comportement probabiliste à la réduction.Tout d'abord nous définissons une sémantique opérationnelle déterministe et contextuelle pour le lambda-calcul probabiliste en appel par nom. Afin de traduire la signification de la somme nous définissons une équivalence syntaxique dans notre calcul, dont nous démontrons qu'il ne déforme pas la réduction: considérer une réduction modulo équivalence revient à considérer simplement le résultat du calcul modulo équivalence. Nous prouvons également un résultat de standardisation.Dans ce cadre nous définissons une notion de théorie équationnelle pour le lambda-calcul probabiliste. Nous étendons certaines notions usuelles, et en particulier celle de bon sens. Cette dernière se formalise facilement dans un cadre déterministe mais est bien plus complexe dans le cas probabiliste.Pour finir nous prouvons une correspondance entre l'équivalence observationnelle, l'égalité des arbres de Böhm et la théorie cohérente sensée maximale. Nous définissons une notion d'arbres de Böhm probabilistes dont nous prouvons qu'elle forme un modèle. Nous démontrons ensuite un résultat de séparabilité disant que deux termes avec des arbres de Böhm distincts ne sont pas observationnellement équivalents. / The lambda-calculus is a way to formalize the notion of computation. In this thesis we will be interested in some of these variants introducing non deterministim, and we will focus mostly on a probabilistic calculus.The probabilistic lambda-calculus has been studied for some time, but the probabilistic behaviour has always been treated as a side effect. Our purpose is to give a more equational representation of this calculus, by handling the probabilities inside the reduction rather than as a side effect.To begin with we give a deterministic and contextual operational semantics for the call-by-name probabilistic lambda-calculus. To express the probabilistic behaviour of the sum we introduce a syntactic equivalence in our calculus, and we show it has little consequence on the calculus: reducing modulo equivalence amount to reducing and then looking at the result modulo equivalence. We also prove a standardization theorem.Then using this operational semantics we define a notion of equational theories for the probabilistic lambda-calculus. We extend some usual notions to this setting, and in particular the sensibility of a theory. This notion is quite simple in a deterministic setting but becomes more complicated when we have a probabilistic computation.Finally we prove a generalization of the equality between the observational equivalence, the Böhm tree equality and the maximal coherent sensible lambda-theory. We give a notion of probabilistic Böhm trees, and prove that this forms a model of the probabilistic lambda-calculus. Then we prove a separability result stating that two terms with different Böhm trees are separable, i.e. are not observationally equivalent.

Tesis por compendio / [ES] La herramienta criptográfica Maude-NPA es un verificador de modelos especializado para protocolos de seguridad criptográficos que tienen en cuenta las propiedades algebraicas de un sistema criptográfico. En la literatura, las propiedades criptográficas adicionales han descubierto debilidades de los protocolos de seguridad y, en otros casos, son parte de los supuestos de seguridad del protocolo para funcionar correctamente. Maude-NPA tiene una base teórica en la rewriting logic, la unificación ecuacional y el narrowing para realizar una búsqueda hacia atrás desde un patrón de estado inseguro para determinar si es alcanzable o no. Maude-NPA se puede utilizar para razonar sobre una amplia gama de propiedades criptográficas, incluida la cancelación del cifrado y descifrado, la exponenciación de Diffie-Hellman, el exclusive-or y algunas aproximaciones del cifrado homomórfico. En esta tesis consideramos nuevas propiedades criptográficas, ya sea como parte de protocolos de seguridad o para descubrir nuevos ataques. También hemos modelado diferentes familias de protocolos de seguridad, incluidos los Distance Bounding Protocols or Multi-party key agreement protocolos. Y hemos desarrollado nuevas técnicas de modelado para reducir el coste del análisis en protocolos con tiempo y espacio. Esta tesis contribuye de varias maneras al área de análisis de protocolos criptográficos y muchas de las contribuciones de esta tesis pueden ser útiles para otras herramientas de análisis criptográfico. / [CA] L'eina criptografica Maude-NPA es un verificador de models especialitzats per a protocols de seguretat criptogràfics que tenen en compte les propietats algebraiques d'un sistema criptogràfic. A la literatura, les propietats criptogràfiques addicionals han descobert debilitats dels protocols de seguretat i, en altres casos, formen part dels supòsits de seguretat del protocol per funcionar correctament. Maude-NPA te' una base teòrica a la rewriting lògic, la unificació' equacional i narrowing per realitzar una cerca cap enrere des d'un patró' d'estat insegur per determinar si es accessible o no. Maude-NPA es pot utilitzar per raonar sobre una amplia gamma de propietats criptogràfiques, inclosa la cancel·lació' del xifratge i desxifrat, l'exponenciacio' de Diffie-Hellman, el exclusive-or i algunes aproximacions del xifratge homomòrfic. En aquesta tesi, considerem noves propietats criptogràfiques, ja sigui com a part de protocols de seguretat o per descobrir nous atacs. Tambe' hem modelat diferents famílies de protocols de seguretat, inclosos els Distance Bounding Protocols o Multi-party key agreement protocols. I hem desenvolupat noves tècniques de modelització' de protocols per reduir el cost de l'analisi en protocols amb temps i espai. Aquesta tesi contribueix de diverses maneres a l’àrea de l’anàlisi de protocols criptogràfics i moltes de les contribucions d’aquesta tesi poden ser útils per a altres eines d’anàlisi criptogràfic. / [EN] The Maude-NPA crypto tool is a specialized model checker for cryptographic security protocols that take into account the algebraic properties of the cryptosystem. In the literature, additional crypto properties have uncovered weaknesses of security protocols and, in other cases, they are part of the protocol security assumptions in order to function properly. Maude-NPA has a theoretical basis on rewriting logic, equational unification, and narrowing to perform a backwards search from an insecure state pattern to determine whether or not it is reachable. Maude-NPA can be used to reason about a wide range of cryptographic properties, including cancellation of encryption and decryption, Diffie-Hellman exponentiation, exclusive-or, and some approximations of homomorphic encryption. In this thesis, we consider new cryptographic properties, either as part of security protocols or to discover new attacks. We have also modeled different families of security protocols, including Distance Bounding Protocols or Multi-party key agreement protocols. And we have developed new protocol modeling techniques to reduce the time and space analysis effort. This thesis contributes in several ways to the area of cryptographic protocol analysis and many of the contributions of this thesis can be useful for other crypto analysis tools. / This thesis would not have been possible without the funding of a set of research projects. The main contributions and derivative works of this thesis have been made in the context of the following projects: - Ministry of Economy and Business of Spain : Project LoBaSS Effective Solutions Based on Logic, Scientific Research under award number TIN2015-69175-C4-1-R, this project was focused on using powerful logic-based technologies to analyze safety-critical systems. - Air Force Office of Scientific Research of United States of America : Project Advanced symbolic methods for the cryptographic protocol analyzer Maude-NPA Scientific Research under award number FA9550-17-1-0286 - State Investigation Agency of Spain : Project FREETech: Formal Reasoning for Enabling and Emerging Technologies Scientific I+D-i Research under award number RTI2018-094403-B-C32 / Aparicio Sánchez, D. (2022). Modeling and Analysis of Advanced Cryptographic Primitives and Security Protocols in Maude-NPA [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/190915 / Compendio

L’objectif principal de cette thèse est l’étude et la formalisation de nouvelles constructions permettant d’augmenter l’expressivité du filtrage et des langages à base de règles en général. Ceci est motivé par le développement de Tom, un système qui enrichit les langages impératifs comme Java et C avec des constructions de haut niveau comme le filtrage et les stratégies. Une première extension que l’on propose est la notion d’anti-patterns, i.e. des motifs qui peuvent contenir des symboles de complément. Nous définissons de manière formelle la sémantique des anti-patterns dans le cas syntaxique et modulo une théorie équationnelle arbitraire. Puis nous étendons la notion classique de filtrage entre les motifs et les termes clos au filtrage entre les anti-patterns et les termes clos (anti-filtrage). Ensuite, nous proposons plusieurs extensions aux constructions de filtrage fournies par Tom. La condition pour l’application d’une règle devient une conjonction ou disjonction de contraintes de filtrage et d’anti-filtrage ainsi que d’autres types de conditions. Les techniques classiques de compilation du filtrage ne sont pas bien adaptées à ces conditions complexes. On propose donc une nouvelle méthode de compilation basée sur des systèmes de réécriture contrôlés par des stratégies. Nous avons complètement réécrit le compilateur de Tom en utilisant cette technique. Tous ces éléments rassemblés constituent un environnement pour décrire et implémenter des transformations de manière élégante et concise. Pour promouvoir son utilisation dans des projets à grand échelle, on développe une technique pour extraire automatiquement des informations structurelles à partir d’une hiérarchie de classes Java. Cela permet l’intégration du filtrage offert par Tom dans n’importe quelle application Java / The main objective of this thesis is the study of new constructs and formalisms that increase the expressivity of pattern matching and rule based languages in general. This is motivated by the development of Tom, a system that adds high level constructs such as pattern matching and strategies to languages like Java and C. A first extension that we propose is the notion of anti-patterns, i.e. patterns that may contain complement symbols. We define formally the semantics of anti-patterns both in the syntactic case and modulo an arbitrary equational theory. We then extend the classical notion of matching between patterns and ground terms to matching between anti-patterns and ground terms. We further propose several extensions to the matching constructs provided by Tom. Consequently, the condition for the application of a rule becomes a combination of matching and anti-matching constraints together with other types of conditions. Classical compilation techniques for pattern matching are not very well suited for these complex conditions. Therefore we propose a new compilation method based on rewrite systems controlled by strategies, which provides a high level of modularity. Tom’s compiler has been rewritten from scratch using this technique. All this constitutes a software environment for expressing transformations in a clear and concise way. To promote its use in large scale applications, we propose an approach for extracting automatically structural information from arbitrary Java hierarchies. This allows a seamless integration of Tom’s pattern matching facilities in any application