ion Let τ = λx.σ. By the induction hypothesis, σ defines a function of its free variables. Let those free variables be of types D0, . . . ,Dn where Dn is the type of x. Then σ defines an approximable function g : D0 × . . .×Dn → D′ where D′ is the type of σ. Using Theorem 3.12, the function curry(g) : D0 × . . .×Dn−1 → (Dn → D′) yields an approximable function, but this is just the function def...

The paper considers the set ML1 of rst-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using the mosaic technique, we prove a general satissability criterion for formulas in ML1 which reduces the modal satissability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragm...

Van Lambalgen (1991) proposed a translation from a language containing a generalized quanti er Q into a rst-order language enriched with a family of predicates R i , for every arity i (or an in nitary predicate R) which takes Qx (x; y 1 ; : : : ; y n ) to 8x(R(x; y 1 ; : : : ; y n ) ! (x;y 1 ; : : : ; y n )) (y 1 ; : : : ; y n are precisely the free variables of Qx ). The logic of Q (without or...

Second order uni cation is undecidable in general Miller showed that uni cation of so called higher order patterns is decidable and unitary We show that the uni cation of a linear higher order pat tern s with an arbitrary second order term that shares no variables with s is decidable and nitary A few extensions of this uni cation problem are still decidable unifying two second order terms where...

This paper serves as a self-contained, tutorial introduction to combinatory models of the untyped lambda calculus. We focus particularly on the interpretation of free variables. We argue that free variables should not be interpreted as elements in a model, as is usually done, but as indeterminates. We claim that the resulting interpretation is more natural and leads to a closer correspondence b...

In [9] we introduced a new framework for asymptotic probabilities. in which a r i-add~t~ve measure is defined on the sample space of all sequences A =< dl.d?. . d, > of finite models. where the uniberse of sf,, is { I . 2. ...n ) . In this framework we Investigated the strong 0-1 law for sentences. whlch states that each sentence either holds in d,, eventually almost surely or fails In d,, even...

In this paper we prove decidability of two kinds of branching time temporal logics. First we show that the non-local version of propositional PCTL∗, in which truth values of atoms may depend on the branch of evaluation, is decidable. Then we use this result to establish decidability of various fragments of quantified PCTL∗, where the next-time operator can be applied only to formulas with at mo...

Let L be a first-order language and let M = 〈M, I〉 be a model for L. Then every formula φ(x1, . . . , xn) of L determines a unique n-place predicate: φ = {〈a1, . . . , an〉 ∈M n : M |= φ[a1, . . . , an]} Here, of course, the notation φ(x1, . . . , xn) is used to indicate that the free variables of φ constitute a subset of {x1, . . . , xn} (see [1] p. 24). A n-placed relation R ⊆M is said to be d...

be morphisms of monoids, considered as functors. Let the functor f is left adjoint to the functor g. Is it true then that f (or, what is the same, g) is always an isomorphism? In [1], p.136, this question was posed as an open question. Here I answer this question and the answer is no. To prove this, I will construct a Birkhoff variety of algebras, which is naturally equivalent to the category o...