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.Comment: In Proceedings ITRS 2012, arXiv:1307.784

The Algebraic Intersection Type Unification Problem

The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type unification problem is decidable. We give the first nontrivial lower bound for the problem by showing (our main result) that it is exponential time hard. Furthermore, we show that this holds even under rank 1 solutions (substitutions whose codomains are restricted to contain rank 1 types). In addition, we provide a fixed-parameter intractability result for intersection type matching (one-sided unification), which is known to be NP-complete. We place the algebraic intersection type unification problem in the context of unification theory. The equational theory of intersection types can be presented as an algebraic theory with an ACI (associative, commutative, and idempotent) operator (intersection type) combined with distributivity properties with respect to a second operator (function type). Although the problem is algebraically natural and interesting, it appears to occupy a hitherto unstudied place in the theory of unification, and our investigation of the problem suggests that new methods are required to understand the problem. Thus, for the lower bound proof, we were not able to reduce from known results in ACI-unification theory and use game-theoretic methods for two-player tiling games

Finite State Machine Synthesis for Evolutionary Hardware

This article considers application of genetic algorithms for finite machine synthesis. The resulting genetic finite state machines synthesis algorithm allows for creation of machines with less number of states and within shorter time. This makes it possible to use hardware-oriented genetic finite machines synthesis algorithm in autonomous systems on reconfigurable platforms

Normal-order reduction grammars

We 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 $S K$-combinators reducing in $n$ normal-order reduction steps. Finally, we investigate the size of generated grammars, giving a primitive recursive upper bound

Asymptotically almost all \lambda-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator