David B. Benson

Life in the game of Go

The Oriental game of Go contains a unique method by which pieces, called stones, are captured and made safe from capture. A group of stones safe from capture is called safe, unconditionally alive, or similar terms. Life or its lack can be determined by lookahead through the game tree, at some expense. We present a graph-theoretic static analysis of the board arrangement which determines unconditional life or its lack, together with proofs of its equivalency to look ahead. An algorithm for the static evaluation is given and we argue that it is the preferable method for computer Go play. These results constitute the first realistic theorems in the theory of Go.

The basic algebraic structures in categories of derivations

The basic algebraic structures within the categories of derivations determined by rewriting systems are presented. The similarity congruence relation in categories of derivations is given in three versions. The syntax category is formed by taking derivations modulo similarity. This category is a free strict monoidal category, a simple form of a 2-category. The syntax category is central to the study of rewriting systems, morphisms in the category generalizing the notion of “derivation tree,” so a detailed development is given. Griffiths interchange operators on derivations form a 2-category over a category of derivations. Representability of a similarity class is defined and shown to imply the existence of group of operators on the class, induced by interchanges. Uniform representability of rewriting systems is defined and shown to imply that the set of left divisors of each derivation in the syntax category is a distributive lattice.

Functional behavior of nondeterministic and concurrent programs

The functions behavior of a deterministic program segment is a function f:D→D, where D is some set of states for the computation. This notion of functional behavior can be extended to nondeterministic and concurrent programs using techniques from linear algebra. In particular, the functional behavior of a nondeterministic program segment is a linear transformation f:A→A, where A is a free semiring module. Other notions from linear algebra carry over into this setting. For example, weakest preconditions and predicate transformers correspond to well-studied concepts in linear algebra. Using multilinear algebra, programs with tuples of inputs and outputs can be handled. For nondeterministic concurrent programs, the functional behavior is a linear transformation f:A→A, where A is a free semiring algebra. In this case, f may also be an algebra morphism, which indicates that the program involves no interprocess communication. Finally, a model of syntax for programs is studied whose semantics is given using linear algebra. It is shown that in this model, free interpretations (essentially Herbrand universes) do not generally exist.

Bisimulation of automata

Abstract We view CCS terms as defining nondeterministic automata. An algebraic representation of automata is given, and categories of automata and simulations between them are defined. The crucial feature is the consideration of only the pure simulations which carry the pure (actual, determined) states of the domain automation to the pure states of the codomain automaton. The pure epimorphisms between the automata partition the category into bisimulation equivalence classes. There is a unique canonical representative for each bisimulation equivalence class. These results hold for weak bisimulation and hence for strong bisimulation. Essentially the same results are obtained with regard to rooted bisimulation equivalence classes of automata with start states.

Algebraic solutions to recursion schemes

Abstract The main problem in recursive scheme theory is determining how to solve a scheme and express its solution. Up to now this was always achieved by adding restrictive hypotheses either on the schemes themselves, or on the domains where they take their values, e.g., assuming the domains have a metric or an order structure and are complete with respect to this structure, or are iterative. Here we develop a strictly algebraic theory of recursion schemes with second-order substitutions. As it is strictly algebraic, the theory applies not only to all recursion schemes on trees, but also to recursion schemes on arbitrary algebras presented in the usual way by generators and relations. In particular, this gives a semantics for nondeterminism and for process algebras.

Fixed points in free process algebras: part II

The left simple polynomials over free process algebras are either semilinear or nonlinear. The semilinear left simple polynomials are subdivided further into the classes of δ-semilinear, τ-semilinear, {τ, δ}-semilinear and A-semilinear, depending upon the right multipliers of the variable x. The linear polynomials considered in Part I are a special case of τ-semilinear polynomials. For δ-semilinear polynomials p(x), p5(x) = p6(x). For the special case of {τ, δ}- semilinear polynomials in which the right multipliers lie in P(∅), p6(x) = p7(x). The solutions to x = p(x) for A-semilinear polynomials are completely characterized. As is demonstrated, the remaining semilinear cases remain unsettled. Necessary conditions for the solutions to x = p(x) for nonlinear left simple polynomials are given. In addition, for a certain subclass of the nonlinear polynomials, sufficient conditions are given. This paper depends upon Part I (Benson and Tiuryn, 1989).

Parameter passing in nondeterministic recursive programs

Abstract Call by value and call by name have some subtleties when used in a nondeterministic programming language. A common formalism is used to establish the denotational semantics of recursive programs called by name and called by value. Ashcroft and Hennessy introduced the idea of differentiating between selecting arguments from a set of arguments at the point of invocation and selecting arguments during the run of the procedure. This distinction is shown to be independent of the evaluation according to value or name, giving rise to four possible parameter passing methods, all of which have a suitable least fixed point semantics.

Semantic preserving translations

Let X1, X2 be derivation systems (freex-categories) generated by context free grammars. Let X0 be a translation category withx-functorsfi:X0→Xi,i=1, 2. Let T be an Ω*-theory, a generalization of algebraic theories. LetIi:Xi→T be algebraic interpretations of the derivations systems, giving the semantics of derivation systems. The translation category X0 is shown to preserve the common semantics through the translation if there is a natural transformation from the functorf2ºI2 to the functorf1ºI1. This is used to show that certain elementary conditions on well-behaved generalized2 sequential machine maps (g2sm maps) result in semantics preservation by the g2sm maps.

An abstract machine theory for formal language parsers

SummaryThe usual data necessary for any abstract machine theory is given in categorical terminology. In these terms, an abstract machine theory for formal language parsers is developed, exposing the essential nature of any left-to-right parsing scheme. A weak classification of all parsers for a given language is developed and the usual notions of initial machine, reachable machine and minimal machine apply. Minimality is an extremely weak notion in this theory, although it is equivalent to a simple form of immediate error detection for parsers. Remarks on the construction of parsing procedures are given.

Functional Behaviour of Nondeterministic Programs

The functional behavior of a deterministic program is a function f:D→D, where D is some set of states for the computation. This notion of functional behaviors can be extended to nondeterministic programs using techniques from linear algebra. In particular, the functional behavior of a nondeterministic program is a linear transformation f:A→A, where A is a free semiring module. Other notions from linear algebra carry over into this setting. For example, weakest preconditions and predicate transformers correspond to well-studied concepts in linear algebra. Finally, we consider multiple-input and multiple-output programs. The functional behavior of a nondeterministic program with multiple inputs and outputs is a linear transformation f:⊗ m A→⊗ n A, where ⊗ x A is an iterated tensor product of the semiring module A. This is in contrast to the deterministic case, where such a program is a function f:D m →D n , using the Cartesian products D m and D n .