The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions (by a monadic formula with parameters)? This continues [LiSh539] where the question was asked only with respect to choice functions. Here we define a subclass of the class of tame trees...

Ternary exclusive or is the (two valued) truth function that is true just in case exactly one of its three arguments is true. This is an interesting truth function, not definable in terms of the binary exclusive or alone, although the binary case is definable in terms of the ternary case. This article investigates the types of truth functions that can be defined by ternary exclusive or, and rel...

I show that in an o-minimal structure on the real field, satisfying certain conditions, every closed definable set is the zero set of a smooth definable function. The conditions are shown to hold in the Pfaffian closure of a polynomially bounded o-minimal structure having smooth cell decomposition.

We consider a typed lambda-calculus with no function types, only alternating sum and product types, so that closed terms represent strategies. We add nondeterminism and consider strategies up to lower (i.e. divergence-insensitive) bisimilarity. We investigate the question: when is a function on strategies definable by an open term (with sufficiently large nondeterminism)? The answer is: when it...

Consider a polynomially bounded, o-minimal structure on the field R of reals. A smooth (i.e. of class C∞) definable function φ : U −→ R on an open set U in R determines two closed subsets W := {u ∈ U : φ(u) ≤ 0} and Z := {u ∈ U : φ(u) = 0}. We shall investigate the links of the sets W and Z at the points u ∈ U , which are well-defined up to a definable homeomorphism. It is proven that the Euler...

Given a group (G, ·), G ⊆M, definable in a first order structureM = (M, . . .) equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V ⊆ G and define a new topology τ on G with which (G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M. Likewise we topologize tra...

Let G be a compact affine definable C group and let r be∞ or ω. We prove that the representative definable C functions on G is dense in the space of continuous functions on G. Moreover we compare the category of compact affine definable C groups D with that of compact real algebraic groups A.

Proof. Suppose the first condition holds. Let B ⊇ C and let a be any realization of p|B. For γ ∈ Γ(Ba), we can write γ as f(a) for some B-definable function. Then γ |= f∗p|B. Also, p is B-invariant and f is B-definable, so the type f∗p is B-invariant. Since it is constant, it must contain the formula x = γ0 for some γ0, and γ0 must be B-definable. Therefore the formula x = γ0 is in f∗p|B, and s...