First-order Programs Research Articles

Optimization of functional programs by grammar thinning

We describe a new technique for optimizing first-order functional programs. Programs are represented as graph grammars, and optimization proceeds by counterexample: when a graph generated by the grammar is found to contain an unnecessary computation, the optimizer attempts to reformulates the grammar so that it never again generates any graph that contains that counterexample. This kind of program reformulation corresponds to an interesting problem on context-free grammars. Our reformulation technique is derived from an (approximate) solution to this CFG problem. An optimizer called Thinner is the proof of concept for this technique. Thinner is a fully automatic, source-to-source optimizer for a Lisp-like language of purely functional, first-order programs. Thinner rediscovers a wide variety of common compiler optimizations. It also finds other more exotic transformations, including the well-known Fibonacci reformulation and the Knuth-Morris-Pratt optimization.

Safe fusion of functional expressions II: Further improvements

AbstractLarge functional programs are often constructed by decomposing each big task into smaller tasks which can be performed by simpler functions. This hierarchical style of developing programs has been found to improve programmers' productivity because smaller functions are easier to construct and reuse. However, programs written in this way tend to be less efficient. Unnecessary intermediate data structures may be created. More function invocations may be required.To reduce such performance penalties, Phil Wadler proposed a transformation algorithm, called deforestation, which could automatically fuse certain composed expressions together to eliminate intermediate tree-like data structures. However, his technique is currently safe (terminates with no loss of efficiency) for only a subset of first-order expressions.This paper will generalise the deforestation technique to make it safe for all first-order and higher-order functional programs. Our generalisation is explained using a model for safe fusion which views each function as a producer and its parameters as consumers. Through this model, syntactic program properties are proposed to classify producers and consumers as either safe or unsafe. This classification is used to identify sub-terms that can be safely fused/eliminated. We present the generalised transformation algorithm, illustrate it with examples and provide a termination proof for the transformation algorithm of first-order programs. This paper also contains a suite of additional techniques to further improve the basic safe fusion method. These improvements could be viewed as enhancements to compensate for some inadequacies of the syntactic analyses used.

Equivalence-preserving first-order unfold/fold transformation systems

Two unfold/fold transformation systems for first-order programs, one basic and the other extended, are presented. The systems comprise an unfolding rule, a folding rule and a replacement rule. They are intended to work with a first-order theory Δ specifying the meaning of primitives, on top of which new relations are built by programs. They preserve the provability relationship Δ ∪ Γ ⊬ G between a call-consistent program Γ and a goal formula G such that Γ is strict with respect to G. They also preserve the logical consequence relationship in three-valued logic.

Protected completions of first-order general logic programs

Minker and Perlis [15] have made the important observation that in certain circumstances, it might be desirable to prevent the inference of ⌝A when A is in the finite failure set of a logic program P. In this paper, we investigate the model-theoretic aspects of their proposal and develop a Fitting-style [5] declarative semantics for protected completions of general logic programs (containing function symbols). This extends the Minker-Perlis proposal which applies to function-free pure logic programs. In addition, an operational semantics is proposed and it is proven to be sound for existentially quantified positive queries and negative ground queries to general, canonical protected logic programs. Completeness issues are investigated and completeness is proved for positive existential queries and negative ground queries for the following classes of programs: (1) function-free general protected logic programs (the Minker-Perlis operational semantics apply to function-free pure protected logic programs), (2) pure protected logic programs (with function symbols) and (3) protected general logic programs that do not contain any internal variables (though they may contain function symbols).

Computers and the design of ion beam optical systems

Advances in microcomputers have made it possible to maintain a library of advanced ion optical programs which can be used on inexpensive computer hardware, which are suitable for the design of a variety of ion beam systems including ion implanters, giving excellent results. This paper describes in outline the steps typically involved in designing a complete ion beam system for materials modification applications. Two computer programs are described which, although based largely on algorithms which have been in use for many years, make possible detailed beam optical calculations using microcomputers, specifically the IBM PC. OPTICIAN is an interactive first-order program for tracing beam envelopes through complex optical systems. SORCERY is a versatile program for solving Laplace's and Poisson's equations by finite difference methods using successive over-relaxation. Ion and electron trajectories can be traced through these potential fields, and plots of beam emittance obtained.

The First-Order Approach to Principal-Agent Problems

The first-order approach to principal-agent problems involves relaxing the constraint that the agent choose an action which is utility maximizing to require instead only that the agent choose an action at which his utility is at a stationary point. Although more mathematically tractable, this approach is generally invalid. This paper identifies sufficient conditions-the monotone likelihood ratio condition and convexity of the distribution function condition-for the first-order approach to be valid. The Pareto-optimal wage contract is shown to be nondecreasing in output under these same conditions. MIRRLEES [5] WAS THE FIRST to point out that the standard method for analyzing the principal-agent problem is not generally correct. This method, the so-called first-order approach, involves weakening the constraint that the agent choose a utility-maximizing action to require instead only that the agent choose an action at which his utility is at a stationary point. The resulting problem is more mathematically tractable. However, as Mirrlees [5] has shown, necessary conditions for a contract to solve the first-order program are not generally even necessary conditions for the valid program. Therefore qualitative propositions about the nature of the Pareto-optimal contract derived from the first-order approach are not in general valid. This has motivated researchers to try to identify classes of cases where the first-order approach is valid.

Unbounded program memory adds to the expressive power of first-order programming logic

It is proved that no logic of programs with unbounded memory is reducible to a bounded memory programming logic. This is achieved by carefully analyzing firstorder definability in the algebra of finite binary trees.