Linear interpolation for the higher-order matching problem

Abstract

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].

Higher-Order Matching, Games and Automata

“…, A n , 0), 0). This appears to raise order by 2 as with the reduction of matching to pairs of interpolation equations in Schubert [12]. However, we only need to consider potential solution terms (in normal form with the right type) λz.zt 1 .…”

“…It first appeals to Padovani's and Schubert's reduction of matching to the conceptually simpler (dual) interpolation problem [12,10]. It is then inspired by model-checking games (such as in [15]) where a model, a transition graph, is traversed relative to a property and players make choices at appropriate positions.…”

Deterministic second-order patterns

“…However, second-order matching in general is nondeterministic [9] (there is more than a single match). It is orthodox to restrict the form of higher-order patterns to generate the desirable matches satisfying certain properties such as decidability [12] and finiteness [6].…”

Dependency Tree Automata

“…, A n , 0), 0). This appears to raise order by 2 as with the reduction of matching to pairs of interpolation equations in [10]. However, we only need to consider potential solution terms (in normal form with the right type) λz.zt 1 .…”