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This paper relates to system-level design of signal processing systems, which are often heterogeneous in implementation technologies and design styles. The heterogeneous approach, by combining small, specialized models of computation, achieves generality and also lends itself to automatic synthesis and formal verification. Key to the heterogeneous approach is to define interaction semantics that resolve the ambiguities when different models of computation are brought together. For this purpose, we introduce a tagged signal model as a formal framework within which the models of computation can be precisely described and unambiguously differentiated, and their interactions can be understood. In this paper, we will focus on the interaction between dataflow models, which have partially ordered events, and discrete-event models, with their notion of time that usually defines a total order of events. A variety of interaction semantics, mainly in handling the different notions of time in the two models, are explored to illustrate the subtleties involved. An implementation based on the Ptolemy system from U.C. Berkeley is described and critiqued.

Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. Some of this expressive power is captured by adding a factorisation combinator. It supports structural equality, and more generally, a large class of generic queries for updating of, and selecting from, arbitrary structures. The resulting combinatory logic is structure complete in the sense of being able to represent pattern-matching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λ-calculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combinators themselves that cannot be so represented, as they examine the internal structure of their arguments.

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

Abstract. This paper argues that basing the semantics of concurrent systems on the notions of state and state transitions is neither advisable nor necessary. The tendency to do this is deeply rooted in our notions of computation, but these roots have proved problematic in concurrent software in general, where they have led to such poor programming practice as threads. I review approaches (some of which have been around for some time) to the semantics of concurrent programs that rely on neither state nor state transitions. Specifically, these approaches rely on a broadened notion of computation consisting of interacting components. The semantics of a concurrent compositions of such components generally reduces to a fixed point problem. Two families of fixed point problems have emerged, one based on metric spaces and their generalizations, and the other based on domain theories. The purpose of this paper is to argue for these approaches over those based on transition systems, which require the notion of state. 1

was one of the founders of computability theory. His main contributions to this field were published in three papers that appeared in the span of a few years, and especially in his ground-breaking 1936–1937 paper, published when he was twenty-four years old. As indicated by its title, “On Computable Numbers, with an Application to the Entscheidungsproblem, ” Turing’s paper deals ostensibly with real numbers that are computable in the sense that their decimal expansion “can be written down by a machine. ” As he pointed out, however, the ideas carry over easily to computable functions on the integers or to computable predicates. The paper was based on work that Turing had carried out as a Cambridge graduate student, under the direction of Maxwell Newman (1897–1984). When Turing first saw a 1936 paper by Alonzo Church, he realized at once that the two of them were tackling the same problem—making computability precise— albeit from different points of view. Turing wrote to Church and then traveled to Princeton University to meet with him. The final form of the paper was

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In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the

Undecidability of various properties of first order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2-complete. The particular problem of local confluence turns out to be Π 0 2-complete for ground terms, but only Σ 0 1-complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.