Sets and Supersets

Peer reviewed

Sets and Supersets

Peer reviewed

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

Formal Verification of Temporal Properties for Parametrized Concurrent Programs and Concurrent Data Structures

Los tipos de datos concurrentes son implementaciones concurrentes de las abstracciones de datos clásicas, con la diferencia de que han sido específicamente diseñados para aprovechar el gran paralelismo disponible en las modernas arquitecturas multiprocesador y multinúcleo. La correcta manipulación de los tipos de datos concurrentes resulta esencial para demostrar la completa corrección de los sistemas de software que los utilizan. Una de las mayores dificultades a la hora de diseñar y verificar tipos de datos concurrentes surge de la necesidad de tener que razonar acerca de un número arbitrario de procesos que invocan estos tipos de datos de manera concurrente. Esto requiere considerar sistemas parametrizados. En este trabajo estudiamos la verificación formal de propiedades temporales de sistemas concurrentes parametrizados, poniendo especial énfasis en programas que manipulan estructuras de datos concurrentes. La principal dificultad a la hora de razonar acerca de sistemas concurrentes parametrizados proviene de la interacción entre el gran nivel de concurrencia que éstos poseen y la necesidad de razonar al mismo tiempo acerca de la memoria dinámica. La verificación de sistemas parametrizados resulta en sí un problema desafiante debido a que requiere razonar acerca de estructuras de datos complejas que son accedidas y modificadas por un numero ilimitado de procesos que manipulan de manera simultánea el contenido de la memoria dinámica empleando métodos de sincronización poco estructurados. En este trabajo, presentamos un marco formal basado en métodos deductivos capaz de ocuparse de la verificación de propiedades de safety y liveness de sistemas concurrentes parametrizados que manejan estructuras de datos complejas. Nuestro marco formal incluye reglas de prueba y técnicas especialmente adaptadas para sistemas parametrizados, las cuales trabajan en colaboración con procedimientos de decisión especialmente diseñados para analizar complejas estructuras de datos concurrentes. Un aspecto novedoso de nuestro marco formal es que efectúa una clara diferenciación entre el análisis del flujo de control del programa y el análisis de los datos que se manejan. El flujo de control del programa se analiza utilizando reglas de prueba y técnicas de verificación deductivas especialmente diseñadas para lidiar con sistemas parametrizados. Comenzando a partir de un programa concurrente y la especificación de una propiedad temporal, nuestras técnicas deductivas son capaces de generar un conjunto finito de condiciones de verificación cuya validez implican la satisfacción de dicha especificación temporal por parte de cualquier sistema, sin importar el número de procesos que formen parte del sistema. Las condiciones de verificación generadas se corresponden con los datos manipulados. Estudiamos el diseño de procedimientos de decisión especializados capaces de lidiar con estas condiciones de verificación de manera completamente automática. Investigamos teorías decidibles capaces de describir propiedades de tipos de datos complejos que manipulan punteros, tales como implementaciones imperativas de pilas, colas, listas y skiplists. Para cada una de estas teorías presentamos un procedimiento de decisión y una implementación práctica construida sobre SMT solvers. Estos procedimientos de decisión son finalmente utilizados para verificar de manera automática las condiciones de verificación generadas por nuestras técnicas de verificación parametrizada. Para concluir, demostramos como utilizando nuestro marco formal es posible probar no solo propiedades de safety sino además de liveness en algunas versiones de protocolos de exclusión mutua y programas que manipulan estructuras de datos concurrentes. El enfoque que presentamos en este trabajo resulta ser muy general y puede ser aplicado para verificar un amplio rango de tipos de datos concurrentes similares. Abstract Concurrent data types are concurrent implementations of classical data abstractions, specifically designed to exploit the great deal of parallelism available in modern multiprocessor and multi-core architectures. The correct manipulation of concurrent data types is essential for the overall correctness of the software system built using them. A major difficulty in designing and verifying concurrent data types arises by the need to reason about any number of threads invoking the data type simultaneously, which requires considering parametrized systems. In this work we study the formal verification of temporal properties of parametrized concurrent systems, with a special focus on programs that manipulate concurrent data structures. The main difficulty to reason about concurrent parametrized systems comes from the combination of their inherently high concurrency and the manipulation of dynamic memory. This parametrized verification problem is very challenging, because it requires to reason about complex concurrent data structures being accessed and modified by threads which simultaneously manipulate the heap using unstructured synchronization methods. In this work, we present a formal framework based on deductive methods which is capable of dealing with the verification of safety and liveness properties of concurrent parametrized systems that manipulate complex data structures. Our framework includes special proof rules and techniques adapted for parametrized systems which work in collaboration with specialized decision procedures for complex data structures. A novel aspect of our framework is that it cleanly differentiates the analysis of the program control flow from the analysis of the data being manipulated. The program control flow is analyzed using deductive proof rules and verification techniques specifically designed for coping with parametrized systems. Starting from a concurrent program and a temporal specification, our techniques generate a finite collection of verification conditions whose validity entails the satisfaction of the temporal specification by any client system, in spite of the number of threads. The verification conditions correspond to the data manipulation. We study the design of specialized decision procedures to deal with these verification conditions fully automatically. We investigate decidable theories capable of describing rich properties of complex pointer based data types such as stacks, queues, lists and skiplists. For each of these theories we present a decision procedure, and its practical implementation on top of existing SMT solvers. These decision procedures are ultimately used for automatically verifying the verification conditions generated by our specialized parametrized verification techniques. Finally, we show how using our framework it is possible to prove not only safety but also liveness properties of concurrent versions of some mutual exclusion protocols and programs that manipulate concurrent data structures. The approach we present in this work is very general, and can be applied to verify a wide range of similar concurrent data types.

Formal Definition of Artificial Intelligence

* This publication is partially supported by the KT-DigiCult-Bg project.

Formal Definition of the Concept “Infos”

The concept INFOS is very important for understanding the information phenomena. Because of this, it is basic for the General Information Theory. The more precise formal definition of this concept is given in the paper.

A Metaontology for Medical Diagnostics of Acute Diseases. Part 2. A Formal Description of Cause-and-effect Relations

This article is the continuation of the formal description of the metaontology for medical diagnostics in the language of applied logic. It contains a description of interrelations between terms of knowledge and reality in the form of ontological agreements.

A Metaontology for Medical Diagnostics of Acute Diseases. Part 3. A Formal Description of the Causes of Signs’ Values and of Diseases

This article is the final part of the formal description of the metaontology for medical diagnostics in the language of applied logic. It contains a description of the causes of signs’ values and of the causes of diseases.

The ADHD debate and the philosophy of truth

There is ongoing and wide-ranging dispute over the proliferation of childhood behaviour disorders. In particular, the veracity of the category Attention Deficit Hyperactivity Disorder (ADHD), has been the subject of considerable scepticism. With no end to the debate in sight, it will be argued here that the problem might effectively be approached, not by addressing the specific features of ADHD itself, but rather by a philosophical analysis of one of the terms around which this entire problem revolves: that is, the notion of truth. If we state: “It is true that ADHD is a real disorder”, what exactly do we mean? Do we mean that it is an objective fact of nature? Do we mean that it fits seamlessly with other sets of ideas and explanations? Or do we simply mean that it works as an idea in a practical sense? This paper will examine the relationship between some of the dominant models of truth, and the assertions made by those in the field of ADHD. Specifically, the paper will contrast the claim that ADHD is a real disorder, with the claim that ADHD is a product of social governance. The intention is, first, to place some significant qualifications upon the validity of the truth-claims made by ADHD advocates, and second, to re-emphasise the potential and promise of philosophical investigation in providing productive new ways of thinking about some obstinate and seemingly intractable educational problems.