Andrej Dudenhefner

Combinatory Logic Synthesizer

We present Combinatory Logic Synthesizer CLS, a type-based tool to automatically compose larger systems from repositories of components. We overview its underlying theory, combinatory logic with intersection types, and exemplify its application to synthesis. We describe features and architecture of the tool and our plans for its ongoing and future development. Finally, we present some use cases in ongoing work, especially in the context of synthesis for Object Oriented Software.

Intersection type calculi of bounded dimension

A notion of dimension in intersection typed λ-calculi is presented. The dimension of a typed λ-term is given by the minimal norm of an elaboration (a proof theoretic decoration) necessary for typing the term at its type, and, intuitively, measures intersection introduction as a resource. Bounded-dimensional intersection type calculi are shown to enjoy subject reduction, since terms can be elaborated in non-increasing norm under β-reduction. We prove that a multiset interpretation (corresponding to a non-idempotent and non-linear interpretation of intersection) of dimensionality corresponds to the number of simultaneous constraints required during search for inhabitants. As a consequence, the inhabitation problem is decidable in bounded multiset dimension, and it is proven to be EXPSPACE-complete. This result is a substantial generalization of inhabitation for the rank 2-fragment, yielding a calculus with decidable inhabitation which is independent of rank. Our results give rise to a new criterion (dimensional bound) for subclasses of intersection type calculi with a decidable inhabitation problem, which is orthogonal to previously known criteria, and which should have immediate applications in synthesis. Additionally, we give examples of dimensional analysis of fragments of the intersection type system, including conservativity over simple types, rank 2-types, and normal form typings, and we provide some observations towards dimensional analysis of other systems. It is suggested (for future work) that our notion of dimension may have semantic interpretations in terms of of reduction complexity.

Combinatory Process Synthesis

We report on a type-theoretic method for functional synthesis of processes from repositories of components. Our method relies on the existing framework for composition synthesis based on combinatory logic, (CL)S. Simple types for BPMN 2.0 components and a taxonomy of domain specific concepts are used to assign types to BPMN 2.0 fragments and functional fragment constructors. Both serve as input for the automatic creation of meaningful processes. Staging synthesis into two levels provides a separation of concerns between the easy task of extracting fragments from existing processes and the more sophisticated task of deducing functional fragment transformations.

Typing Classes and Mixins with Intersection Types

We study an assignment system of intersection types for a lambda-calculus with records and a recordmerge operator, where types are preserved both under subject reduction and expansion. The calculus is expressive enough to naturally represent mixins as functions over recursively defined classes, whose fixed points, the objects, are recursive records. In spite of the double recursion that is involved in their definition, classes and mixins can be meaningfully typed without resorting to neither recursive nor F-bounded polymorphic types. We then adapt mixin construct and composition to Java and C#, relying solely on existing features in such a way that the resulting code remains typable in the respective type systems. We exhibit some example code, and study its typings in the intersection type system via interpretation into the lambdacalculus with records we have proposed.

Mixin Composition Synthesis Based on Intersection Types

We present a method for synthesizing compositions of mixins using type inhabitation in intersection types. First, recursively defined classes and mixins, which are functions over classes, are expressed as terms in a lambda calculus with records. Intersection types with records and record-merge are used to assign meaningful types to these terms without resorting to recursive types. Second, typed terms are translated to a repository of typed combinators. We show a relation between record types with record-merge and intersection types with constructors. This relation is used to prove soundness and partial completeness of the translation with respect to mixin composition synthesis. Furthermore, we demonstrate how a translated repository and goal type can be used as input to an existing framework for composition synthesis in bounded combinatory logic via type inhabitation. The computed result corresponds to a mixin composition typed by the goal type.

Typability in bounded dimension

Recently (authors, POPL 2017), a notion of dimensionality for intersection types was introduced, and it was shown that the bounded-dimensional inhabitation problem is decidable under a non-idempotent interpretation of intersection and undecidable in the standard set-theoretic model. In this paper we study the typability problem for bounded-dimensional intersection types and prove that the problem is decidable in both models. We establish a number of bounding principles depending on dimension. In particular, it is shown that dimensional bound on derivations gives rise to a bounded width property on types, which is related to a generalized subformula property for typings of arbitrary terms. Using the bounded width property we can construct a nondeterministic transformation of the typability problem to unification, and we prove that typability in the set-theoretic model is PSPACE-complete, whereas it is in NP in the multiset model.

The Intersection Type Unification Problem.

The 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 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 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.