We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

This paper and its sequel [17] deal with a range of questions about the subgroup structure of infinite symmetric groups. Our concern is with such questions as the following. How can an infinite symmetric group be expressed as the union of a chain of proper subgroups? What are the subgroups that supplement the normal subgroups

In this paper, we develop a system of typed lambda-calculus for the Zermelo-Fraenkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambda-calculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connective #. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property : every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard[4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming

We consider relational databases organized over an ordered domain with some additional relations---a typical example is the ordered domain of rational numbers together with the operation of addition. In the focus of our study are the first-order (FO) queries that are invariant under order-preserving "permutations"---such queries are called ordergeneric. It has recently been discovered that for some domains ordergeneric FO queries fail to express more than pure order queries. For example, every order-generic FO query over rational numbers with + can be rewritten without +. For some other domains, however, this is not the case. We provide very general conditions on the FO theory of the domain that ensure the collapse of order-generic extended FO queries to pure order queries over this domain: the Pseudo-finite Homogeneity Property and a stronger Isolation Property. We further distinguish one broad class of domains satisfying the Isolation Property, the so-called quasi-o -...

We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.’s). This class seems to be the natural context for studying structures like measurable function spaces, equidecomposability types of spaces, partially ordered abelian groups and cardinal algebras. Then, we can prove that roughly speaking, spaces of measures with values in complete P.O.M.’s are complete P.O.M.’s. Furthermore, this notion of completeness yields us an ‘arithmetical ’ characterization of injective P.O.M.’s.

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Abstract. Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2)Kcomputes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K�κ is universal for mice of height ≤ κ whenever κ ≥ℵ2,5)ifthereisaκ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and � <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdös cardinal suffices to develop the basic theory of K. 1.

Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 2ω asymptotic cones up to homeomorphism. 1.

Abstract. In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2 ω = ω2 and that L(P(ω1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that �(κ) fails for all regular κ> ω1. 1.