λ-Definability on free algebras

Computability and λ-Definability

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The Undecidability of λ-Definability

In this article, we shall show that the Plotkin-Statman conjecture [?, ?] is false. The conjecture was that, in a model of the simply typed λ-calculus with only finitely many elements at each type, definability (by a closed term of the calculus) is decidable. This conjecture had been shown to imply many things, for example, Statman [?] (see also Wolfram’s book [?]) has shown it implies the deci...

On free annotated algebras

First-Order Logic Definability of Free Languages

Operator Precedence Grammars (OPGs) define a deterministic class of context-free languages, which extend input-driven languages and still enjoy many properties: they are closed w.r.t. Boolean operations, concatenation and Kleene star; the emptiness problem is decidable; they are recognized by a suitable model of pushdown automaton; they can be characterized in terms of a monadic second-order lo...

Epimorphisms in Cylindric Algebras and Definability in Finite Variable Logic

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of n-dimensional cylindric algebras and the class of representable algebras in CAn for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Né...