Combinatory Terms Research Articles

Combinators as presheaves

Abstract Computer scientists, logicians and other researchers have recently paid renewed attention to the model of computation based on the logic of combinators. We develop a new formalization of the syntax of combinators employing the ‘generic figures’ approach to the categories of presheaves that provides an intuitive and easily interpreted diagrammatic syntax for combinators, one that eschews the need to label proper combinators with variable names. Furthermore, we show that this formal framework provides the possibility of representing choices of operand from among given alternatives at each level of a combinatory term, allowing for a more general type of combinatory expression, which we call a multi-combinator.

A Historical Account of My Early Research Interests

This paper presents a brief account of some of the my early research interests. This historical account starts from my laurea thesis on Signal Theory and my master thesis on Computation Theory. It recalls some results in Combinatory Logic and Term Rewriting Systems. Some other results concern Program Transformation, Parallel Computation, Theory of Concurrency, and Proof of Program Properties. My early research activity has been mainly done in cooperation with Andrzej Skowron, Anna Labella, and Maurizio Proietti.

Normal-order reduction grammars

AbstractWe present an algorithm which, for given n, generates an unambiguous regular tree grammar defining the set of combinatory logic terms, over the set {S, K} of primitive combinators, requiring exactly n normal-order reduction steps to normalize. As a consequence of Curry and Feys's standardization theorem, our reduction grammars form a complete syntactic characterization of normalizing combinatory logic terms. Using them, we provide a recursive method of constructing ordinary generating functions counting the number of SK-combinators reducing in n normal-order reduction steps. Finally, we investigate the size of generated grammars giving a primitive recursive upper bound.

Algebra and Geometry of Sets in Boolean Space

In the present paper, geometry of the Boolean space Bn in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in Bn and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.

Three syntactic theories for combinatory graph reduction

We present a purely syntactic theory of graph reduction for the canonical combinators S, K, and I, where graph vertices are represented with evaluation contexts and let expressions. We express this first syntactic theory as a storeless reduction semantics of combinatory terms. We then factor out the introduction of let expressions to denote as many graph vertices as possible upfront instead of on demand . The factored terms can be interpreted as term graphs in the sense of Barendregt et al. We express this second syntactic theory, which we prove equivalent to the first, as a storeless reduction semantics of combinatory term graphs. We then recast let bindings as bindings in a global store, thus shifting, in Strachey's words, from denotable entities to storable entities. The store-based terms can still be interpreted as term graphs. We express this third syntactic theory, which we prove equivalent to the second, as a store-based reduction semantics of combinatory term graphs. We then refocus this store-based reduction semantics into a store-based abstract machine. The architecture of this store-based abstract machine coincides with that of Turner's original reduction machine. The three syntactic theories presented here therefore properly account for combinatory graph reduction As We Know It. These three syntactic theories scale to handling the Y combinator. This article therefore illustrates the scientific consensus of theoreticians and implementors about graph reduction: it is the same combinatory elephant.

Using Inhabitation in Bounded Combinatory Logic with Intersection Types for Composition Synthesis

We describe ongoing work on a framework for automatic composition synthesis from a repository of software components. This work is based on combinatory logic with intersection types. The idea is that components are modeled as typed combinators, and an algorithm for inhabitation {\textemdash} is there a combinatory term e with type tau relative to an environment Gamma? {\textemdash} can be used to synthesize compositions. Here, Gamma represents the repository in the form of typed combinators, tau specifies the synthesis goal, and e is the synthesized program. We illustrate our approach by examples, including an application to synthesis from GUI-components.

How to assign ordinal numbers to combinatory terms with polymorphic types

The article investigates a system of polymorphically typed combinatory logic which is equivalent to Godel's T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ? 0-recursive function. The ordinal assignments used to obtain this result are also used to prove that the system under consideration is indeed equivalent to Godel's T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style.

Algorithms for modeling anatomic and target volumes in image‐guided neurosurgery and radiotherapy

The use of image‐guidance in surgery and radiotherapy has significantly improved patient outcome in neurosurgery and radiotherapy treatments. This work developed volume definition and verification techniques for image‐guided applications, using a number of algorithms ranging from image processing to visualization. Stereoscopic visualization, volumetric tumor model overlaid on an ultrasound image, and visualization of the treatment geometry were experimented with on a neurosurgical workstation. Visualization and volume definition tools were developed for radiotherapy treatment planning system. The magnetic resonance inhomogeneity correction developed in this work, possibly the first published data‐driven method with wide applicability, automatically mitigates the radio‐frequency (RF) field inhomogeneity artifact present in magnetic resonance images. Correcting the RF inhomogeneity improves the accuracy of the generated volumetric models. Various techniques to improve region growing are also presented. The simplex search method and combinatory similarity terms were used to improve the similarity function with a low additional computational cost and high yield in region correctness. Moreover, the effects of different priority queue implementations were studied. A fast algorithm for calculating high‐quality digitally reconstructed radiographs has been developed and shown to better meet typical radiotherapy needs than the two alternative algorithms. A novel visualization method, beam's light view, is presented. It uses texture mapping for projecting the fluence of a radiation field on an arbitrary surface. This work describes several improved algorithms for image processing, segmentation, and visualization used in image‐guided treatment systems. These algorithms increase the accuracy of image‐guidance, which can further improve the applicability and efficiency of image‐guided treatments.

Expedited Broda-Damas bracket abstraction

AbstractA bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schönfinkel [4] algorithm plus two mappings which convert the Schönfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n − 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered.

Compact bracket abstraction in combinatory logic

AbstractTranslations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schönfinkel [9]. The algorithm introduces at most one combinator for each abstraction in the initial Lambda term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus.

The "Relevance" of Intersection and Union Types

The aim of this paper is to investigate a Curry-Howard interpre- tation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable dis- junctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that ∨-∧- typed combinatory logic terms are proved to completely codify the associated logical proofs.

Lambda terms definable as combinators

It is well known that for each λ-term there is a corresponding combinatory term formed using the combinators K and S instead of the λ-operator. Similarly for every combinatory term there is a λ-term. For weaker sets of combinators such as B, C and K or B, B′, I and W we show how such a correspondence or “translation” can be formulated and we determine in the case of several such sets of combinators the sets of λ-terms that can be translated using them. As combinators can represent Hilbert-style proofs of theorems of implicational logic and λ-terms natural deduction style proofs, this work allows us to formulate natural deduction systems equivalent to the Hilbert-style systems for various implicational logics and can form a basis for proof generating programs for these logics.

Normal Forms in Combinatory Logic

Let $R$ be a convergent term rewriting system, and let $CR$-equality on (simply typed) combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long $\beta$-normal forms in the lambda calculus. As a consequence, the translates of long $\beta R$-normal forms are easily seen to serve as canonical representatives for $CR$-equivalence classes of combinatory logic terms for non-empty, as well as for empty, $R$.

Higher-order unification via combinators

We present an algorithm for unification in the simply typed lambda calculus which enumerates complete sets of unifiers using a finitely branching search space. In fact, the types of terms may contain type variables, so that a solution may involve type-substitution as well as term substitution. The problem is first translated into the problem of unification with respect to exstensional equality in combinatory logic, and the algorithm is defined in terms of transformations on systems of combinatory terms. These transformations are based on a new method (itself based on systems) for deciding extensional equality between typed combinatory logic terms.

The converse principal type-scheme theorem in lambda calculus

A principal type-scheme of a λ-term is the most general type-scheme for the term. The converse principal type-scheme theorem (J.R. Hindley, The principal typescheme of an object in combinatory logic, Trans. Amer. Math. Soc.146 (1969) 29–60) states that every type-scheme of a combinatory term is a principal type-scheme of some combinatory term.

Some improvements to Turner's algorithm for bracket abstraction

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.There are several ways in which these problems can be alleviated:(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.(2) The use of an extra clause (c), appropriate to βη reduction.(3) The introduction of a finite number of extra combinators.The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

An abstraction algorithm for combinatory logic

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets {S, K}, {B, C, K, W}, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).

An abstraction algorithm for combinatory logic

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets {S, K}, {B, C, K, W}, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).