Theoretical Computer Science
Using simple systems with a notion of discrete deterministic evolution over time, we study discrete causality via tools from theoretical computer science and logic. We consider the set of all representations (i.e. partial descriptions) of such systems, from algebraic, domaintheoretic, and categorical viewpoints. The order theory introduced, is based in the notion of comparing high-level and lowlevel descriptions of the same system. This is shown to give a Complete Partial Order where the
... osure of each element is a locale. This partial order has a very close connection to a categorical construction known as the 'particle-style' trace, via analogues of domain-theoretic equations. Thus, the trace may be thought of as the computation of suprema in this partial order. As a sample application, we show how to construct algebraic models of space-bounded Turing machines in these terms, and derive compositionality from the abstract properties of the trace. Abstract Machine as a directed graph. As an example application, we study algebraic models of space-bounded Turing machines in these terms. Although algebraic models have previously been derived by a brute-force analysis  , the framework of CPOs and the particle-style trace, demonstrates why both the particle-style trace, and the corresponding Geometry of Interaction construction, appear in this seemingly unrelated context. The abstract properties of the trace also give compositionality of these algebraic models for free, using the same categorical structures commonly used to model confluence in logical systems. Definitions The basic definitions below are taken from  . However, the terminology and notation used has changed somewhat, for clarity. Definition 1 (Abstract Machines, Configurations, the Primitive Evolution ). An Abstract Machine, or A.M., consists of a set X of configurations, together with a partial function P : X → X called the primitive evolution. We will often treat P as a (special type of) relation -in particular, the transitive closure of P plays a vital rôle in this paper. Abstract Machines may be drawn as directed graphs, as shown in Fig. 1 , although not all directed graphs specify an Abstract Machine. We use such diagrams throughout, to illustrate key concepts. As the definition of an Abstract Machine is very broad, a wide range of computational examples are available: Examples. (1) (Deterministic) Turing machines -a configuration is a specification of the tape contents, together with the position and label of the machine head. The primitive evolution is given by the transition rule for the T.M. (2) Assorted variants on deterministic state machines (finite state/two-way/Mealy automata, space/time bounded T.M.s, &c.). These can be given (as in  ) as special cases of (1). (3) Cellular automata -a configuration is a specification of the contents of every cell, and the primitive evolution is given by the neighbourhood rule. (4) Universal register machines -a configuration is the contents of all registers, and the primitive evolution is immediate. (5) von Neumann computers -a configuration is a specification of the memory contents, and the primitive evolution is given by the fetch-execute cycle. We will use this as the canonical example of an Abstract Machine, in order to motivate various general definitions. (6) A quantum computer executing Grover's algorithm. A configuration is a specification of the contents of the quantum registers, and the primitive evolution is the 'inversion about the mean' unitary map. 2 (7) The linear combinators of  . This follows from the intuition of the trace on partial functions (see Section 7) as 'a particle moving through a labelled network' , and an exact specification of the configurations and primitive evolution is an interesting, non-trivial exercise. Although the above are all computational examples, any system that evolves in deterministic discrete steps may be considered as an Abstract Machine. Sometimes, as in (5) and (6) above, the distinction between computational and physical examples is somewhat blurred & some authors (notably  ) claim that such a distinction is entirely arbitrary. An interesting non-example is that of the pure untyped lambda calculus. It is tempting to think of lambda-terms as configurations, and β-reduction as the primitive evolution. However, arbitrary λ-terms can be reduced in many different ways, so we do not have a deterministic partial function to act as the primitive evolution. Given a fixed deterministic reduction strategy, λ-calculus may be treated as an Abstract Machine, but this clearly misses much of the interesting structure of β-reduction. Tools to study Abstract Machines The above definition of a set, together with a partial function acting on it, is of course not original. This paper differs in the questions we ask, and hence the tools used, in the study of these structures. We now make a few basic definitions: 2 There are, as may be expected, various subtleties in this example. It is considered further in Section 10.