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 is a class typed by the goal type and generated by a mixin composition applied to an existing class

Third-order matching in λ→\lambda\rightarrow-Curry is undecidable

Given closed untyped $\lambda$-terms $\lambda x1... xk.s$ and $t$, which can be assigned some types $S1->...->Sk->T$ and $T$ respectively in the Curry-style systems of type assignment (essentially due to R.~Hindley) $\lambda->$-Curry [Barendregt 92], $\lambda^{->}_t$ [Mitchell 96], $TA_\lambda$ [Hindley97], it is undecidable whether there exist closed terms $s1,...,sk$ of types $S1,...,Sk$ such that $s[s1/x1,...,sk/xk]=_{\beta\eta}t$, even if the orders of $si$'s do not exceed 3. This undecidability result should be contrasted to the decidability of the third-order matching in the Church-style simply typed lambda calculus with a single constant base type [Dowek 92]. The proof is by reduction from the recursively inseparable sets of invalid and finitely satisfiable sentences of the first-order theory of binary relation [Trakhtenbrot 53, Vaught 60]

The D-Completeness of T→

A Hilbert-style version of an implicational logic can be represented by a set of axiom schemes and modus ponens or by the corresponding axioms, modus ponens and substitution. Certain logics, for example the intuitionistic implicational logic, can also be represented by axioms and the rule of condensed detachment, which combines modus ponens with a minimal form of substitution. Such logics, for example intuitionistic implicational logic, are said to be D-complete. For certain weaker logics, the version based on condensed detachment and axioms (the condensed version of the logic) is weaker than the original. In this paper we prove that the relevant logic T→, and any logic of which this is a sublogic, is D-complete

Retractions in Intersection Types

This paper deals with retraction - intended as isomorphic embedding - in intersection types building left and right inverses as terms of a lambda calculus with a bottom constant. The main result is a necessary and sufficient condition two strict intersection types must satisfy in order to assure the existence of two terms showing the first type to be a retract of the second one. Moreover, the characterisation of retraction in the standard intersection types is discussed.Comment: In Proceedings ITRS 2016, arXiv:1702.0187