We present a novel analysis for sorted logic, which determines if a given sort is monotone. The domain of a monotone sort can always be extended with an extra element. We use this analysis to significantly improve well-known translations between unsorted and many-sorted logic, making use of the fact that it is cheaper to translate monotone sorts than non-monotone sorts. Many interesting problem...

In this paper I extend the standard first-order resolution method with special reasoning mechanisms for sorts. Sorts are unary predicates. Literals built from unary predicates are called sort literals. Negative sort literals can be compiled into restrictions of the relevant variables to sorts or can be deleted if they fulfill special conditions. Positive sort literals define the sort-theory. So...

We show that uni cation in certain extensions of shallow equational theories is decidable Our extensions generalize the known classes of shallow or standard equational theories In order to prove de cidability of uni cation in the extensions a class of Horn clause sets called sorted shallow equational theories is introduced This class is a natural extension of tree automata with equality constra...

We give a machine-checked proof of the equivalence of the usual, two-sorted presentation of System F and its single-sorted pure type system variant λ2. This is established by reducing the typability problem of F to λ2 and vice versa. The systems are formulated using higher-order abstract syntax and the proof is executed in the Abella proof system. We compare and contrast this proof to our earli...

Sorted term-position algebras are an extension of term algebras. In addition to sorted terms with constructor and selector functions, they provide term positions as algebra elements and functions that relate term positions. This paper describes possible applications of termposition algebras and investigates their formal specification in existing specification frameworks. In particular, it prese...