Higher-Order Termination: from Kruskal to Computability

Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal. In the early days of verification, computer scientists used similar ideas, interpreting the arguments of a program call by a natural number, such as their size. Showing the size of the arguments to decrease for each recursive call gives a termination proof of the program, which is however rather weak since it can only yield quite small ordinals. In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied the strong normalization property for that expression. The predicate being defined by induction on types, or even as a fixpoint, this method could yield much larger ordinals. Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi..

The play's the thing

For very understandable reasons phenomenological approaches predominate in the field of sensory urbanism. This paper does not seek to add to that particular discourse. Rather it takes Rorty’s postmodernized Pragmatism as its starting point and develops a position on the role of multi-modal design representation in the design process as a means of admitting many voices and managing multidisciplinary collaboration. This paper will interrogate some of the concepts underpinning the Sensory Urbanism project to help define the scope of interest in multi-modal representations. It will then explore a range of techniques and approaches developed by artists and designers during the past fifty years or so and comment on how they might inform the question of multi-modal representation. In conclusion I will argue that we should develop a heterogeneous tool kit that adopts, adapts and re-invents existing methods because this will better serve our purposes during the exploratory phase(s) of any design project that deals with complexity

Computer simulation of diffusion processes in tilt spatio-periodic potentials

Нещодавно було показано, що в істотно нерівноважних системах коефіцієнт дифузії може вести себе немонотонно з температурою. Одним із прикладів таких систем з аномальною температурної залежністю є рух броунівських часток в просторово-періодичних структурах. Метою статті було дослідження зміни температурної залежності дифузії в недодемпфованих системах з низьким коефіцієнтом тертя. В роботі методами комп'ютерного моделювання вивчено зміна коефіцієнта дифузії частинок в широкому діапазоні температур в нахилених просторово-періодичних потенціалах для різних значень коефіцієнта тертя. Показано, що дифузія досягає максимуму при певній величині зовнішньої сили. Її значення залежить від величини коефіцієнта тертя. Показано, що на відміну від звичайної залежності Аррениуса, в разі нахиленого періодичного потенціалу, максимальний коефіцієнт дифузії зростає, а не зменшується з пониженням температури експоненціальним чином. Встановлено, що така залежність характерна для всіх недодемпфованих систем. Показано, що для просторово-періодичних структур існує обмежена ділянка сил, в якому спостерігається зростання коефіцієнта дифузії зі зменшенням температури. Це область так званої температурно-аномальної дифузії (ТАД). Визначено ширина і положення області ТАД в залежності від коефіцієнта тертя γ і параметрів системи. Показано, що зі зменшенням γ, ширина області ТАД зменшується пропорційно γ. При цьому коефіцієнт дифузії в області ТАД, навпаки зростає ~γ. Отримані дані про температурно-аномальної дифузії мають важливе значення для різних областей фізики і техніки та відкривають перспективи створення новітніх технологій управління процесами дифузії.It was recently shown that in essentially nonequilibrium systems, the diffusion coefficient can behave nonmonotonically with temperature. One example of such systems with anomalous temperature dependence is the motion of Brownian particles in spatially periodic structures. The aim of the article was to study the change in the temperature dependence of diffusion in underdamped systems with a low coefficient of friction. In this paper, computer simulation methods are used to study the change in the diffusion coefficient of particles in a wide range of temperatures in oblique spatially periodic potentials for different values of the friction coefficient. It is shown that diffusion reaches a maximum at a certain external force. Its value depends on the coefficient of friction. It is shown that, in contrast to the usual Arrhenius dependence, in the case of an inclined periodic potential, the maximum diffusion coefficient increases while temperature is decreasing exponentially. It is established that such a dependence is common to all underdamped systems. It is shown that for spatially periodic structures there is a limited portion of forces in which an increase in the diffusion coefficient while decreasing temperature is observed. This is the area of the so-called temperature-anomalous diffusion (TAD). The width and position of the TAD region are determined depending on the friction coefficient γ and the system parameters. It has been shown that a decrease in γ, width TAD region decreases proportionally γ. In this case, the diffusion coefficient in the TAD region, on the contrary, increases ~γ. The data obtained on the temperature and the anomalous diffusion are important for various fields of physics and engineering, and opens new prospects for a diffusion process control technology

A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not terminating modulo E. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into a terminating part and a possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system

A Completion Method to Decide Reachability in Rewrite Systems

International audienceThe Knuth-Bendix method takes in argument a finite set of equations and rewrite rules and, when it succeeds, returns an algorithm to decide if a term is equivalent to another modulo these equations and rules. In this paper, we design a similar method that takes in argument a finite set of rewrite rules and, when it succeeds, returns an algorithm to decide not equivalence but reachability modulo these rules, that is if a term reduces to another. As an application, we give new proofs of the decidability of reachability in finite ground rewrite systems and in pushdown systems