Abstract. We extend a Java-like language with immutability specifications and a static type system for verifying immutability. A class modifier immutable specifies that all class instances are immutable objects. Ownership types specify the depth of object states and enforce encapsulation of representation objects. The type system guarantees that the state of immutable objects does not visibly mutate during a program run. Provided immutability-annotated classes and methods are final, this is true even if immutable classes are composed with untrusted classes that follow Java's type system, but not our immutability type system.

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Abstract. We stratify intuitionistic first-order logic over (∀, →) into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the ∆2 level is undecidable and that Σ1 is Expspace-complete. We also prove that the arity-bounded fragment of Σ1 is complete for co-Nexptime.

Abstract. We present here a particular case of the higher order matching problem --the linear interpolation problem. The problem consists in solving a collection of higher order matching equations of the shape xM1... Mk = N, where x is the only unknown quantity. We prove recursive equivalence of the higher order matching problem and the linear interpolation problem. We also investigate decidability of a special case of the fifth order linear interpolation problem. The restriction we consider consists in that arguments of variables from the main abstraction in terms M1,. 9 9 Mk cannot contain variables from the main abstraction. In this paper, we present the linear interpolation problem. This problem is interesting since to construct a solution for such a problem we deal with a single object, not a set of objects as in the case of the matching problem in general formulation. Moreover, V. Padovani investigates a similar problem in his paper [Pad96]. The Padovani's problem consists in solving the pair of sets {~, gr} of interpolation equations. A solution of such a problem is a concretisation of unknown quantities which satisfies each equation in the set ~ and does not satisfy any equation in the set ~. Decidability of the problem implies decidability of the matching problem as proven in [Pad96].1 This work has been partly supported by ESPRIT BRA 7232 GENTZEN, and KBN 8 TllC 034 10 grants. 442In the second part of the paper, we look into decidability of a special case of the fifth order linear interpolation problem. The restriction we consider is that arguments of variables from the main abstraction in terms M1, 9 9 Mk cannot contain occurrences of variables from the main abstraction.This issue is interesting, since it gives constructors of proof-checkers and proof-assistants possibility of solving some fifth order matching equations. This paper is organised as follows --in Section 2 we present some basic definitions and define some useful notation, in Section 3 we prove recursive equivalence of the higher-order matching problem and the interpolation problem, and in Section 4 we prove our decidability result.The present paper contains only a sketch of the proof. More details can be found in the technical report [Sch96].