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...

A notation for probabilities is proposed that differs from the traditional, conventional notation by making explicit the domains and bound variables involved. The notation borrows from the Z notation, and lends itself well to calculational manipulations, with a smooth transition back and forth to set and predicate notation.

Away is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also ...

In this paper we propose higher-order rewrite systems without bound variables. In order to prove their connuence under the assumption of orthogonality, we study a simple proof method which employs a characterization of the diamond property of a parallel reduction. By an application of the proof method, we obtain a new connuence result for orthogonal higher-order conditional rewrite systems.

Every theory we will consider will be in a language that has a name ṅ for each n ∈ ω. For example, the name for 2 ∈ ω in the language LΩ of arithmetic is the term “1 + 1”. The results stated will only apply to countable languages. This requirement is implicit in the hypothesis of the existence of an enumeration of all formulas with one free variable, for example. We also assume that formulas an...