Let S be some type system. A typing in S for a typable term M is the collection of all of the information other than M which appears in the final judgement of a proof derivation showing that M is typable. For example, suppose there is a derivation in S ending with the judgement A M : # meaning that M has result type # when assuming the types of free variables are given by A. Then (A, #) is a typing for M .

... this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each second-order variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i)#(iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size <n, (ii) the arity of all second-order variables is <n, (iii) the number of occurrences of second-order variables is #5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables.

Abstract. There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions. 1

this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of -terms in . Quite apart from the new light it sheds on fi-reduction, such an analysis turns out to have several other benefits

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Abstract. Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F, the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable. We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. 1

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