Distilling Abstract Machines (Long Version)

It is well-known that many environment-based abstract machines can be seen as strategies in lambda calculi with explicit substitutions (ES). Recently, graphical syntaxes and linear logic led to the linear substitution calculus (LSC), a new approach to ES that is halfway between big-step calculi and traditional calculi with ES. This paper studies the relationship between the LSC and environment-based abstract machines. While traditional calculi with ES simulate abstract machines, the LSC rather distills them: some transitions are simulated while others vanish, as they map to a notion of structural congruence. The distillation process unveils that abstract machines in fact implement weak linear head reduction, a notion of evaluation having a central role in the theory of linear logic. We show that such a pattern applies uniformly in call-by-name, call-by-value, and call-by-need, catching many machines in the literature. We start by distilling the KAM, the CEK, and the ZINC, and then provide simplified versions of the SECD, the lazy KAM, and Sestoft's machine. Along the way we also introduce some new machines with global environments. Moreover, we show that distillation preserves the time complexity of the executions, i.e. the LSC is a complexity-preserving abstraction of abstract machines.Comment: 63 page

State machines for large scale computer software and systems

A method for specifying the behavior and architecture of discrete state systems such as digital electronic devices and software using deterministic state machines and automata products. The state machines are represented by sequence maps $f:A^*\to X$ where $f(s)=x$ indicates that the output of the system is $x$ in the state reached by following the sequence of events $s$ from the initial state. Examples provided include counters, networks, reliable message delivery, real-time analysis of gates and latches, and producer/consumer. Techniques for defining, parameterizing, characterizing abstract properties, and connecting sequence functions are developed. Sequence functions are shown to represent (possibly non-finite) Moore type state machines and general products of state machines. The method draws on state machine theory, automata products, and recursive functions and is ordinary working mathematics, not involving formal methods or any foundational or meta-mathematical techniques. Systems in which there are levels of components that may operate in parallel or concurrently are specified in terms of function composition

Criteria for homotopic maps to be so along monotone homotopies

The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.Comment: 7 pages, 5 figures, partially presented at GETCO 2006. title change; strengthened Cor. 3.3. -> Prop. 3.7, Prop. 3.2 -> Lem. 3.2; corrected def of category of continuous lattices in sec. 2; added 5 figures, 8 eg's, Def. 3.4, Lemmas 2.8, 3.5, refs [1],[4],[5]; rewording throughout; conclusion and abstract rewritte

On the topological aspects of the theory of represented spaces

Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escard\'o, Schr\"oder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.Comment: Earlier versions were titled "Compactness and separation for represented spaces" and "A new introduction to the theory of represented spaces