Deterministic Automata and the Monadic Theory of Ordinals < ?2 (original) (raw)

Near coherence of filters. I. Cofinal equivalence of models of arithmetic

Notre Dame Journal of Formal Logic, 1986

We define cofinal equivalence to be the smallest equivalence relation on models of arithmetic such that every model is equivalent to all of its cofinal submodels. It is easy to classify tall models (those with no last sky) up to cofinal equivalence, but an attempt to do the same for short models leads to questions independent of the axioms of set theory. We introduce the set-theoretic principle (NCF) of near coherence of filters, whose effect is to make all short nonstandard models of arithmetic cofinally equivalent. We give several equivalent formulations and several consequences of NCF.

Selection over classes of ordinals expanded by monadic predicates

Annals of Pure and Applied Logic, 2010

A monadic formula ψ(Y ) is a selector for a monadic formula ϕ(Y ) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies ϕ in M. If C is a class of structures and ϕ is a selector for ψ in every M ∈ C, we say that ϕ is a selector for ϕ over C.

Filter models for conjunctive-disjunctive λ-calculi

Theoretical Computer Science, 1996

The distinction between the conjunctive nature of non-determinism as opposed to the disjunctive c haracter of parallelism constitutes the motivation and the starting point of the present w ork.-calculus is extended with both a non-deterministic choice and a parallel operator a notion of reduction is introduced, extending-reduction of the classical calculus. We study type assignment systems for this calculus, together with a denotational semantics which is initially de ned constructing a set semimodel via simple types. We enrich t h e t ype system with intersection and union types, dually re ecting the disjunctive and conjunctive behaviour of the operators, and we build a lter model. The theory of this model is compared both with a Morris-style operational semantics and with a semantics based on a notion of capabilities. 1 Introduction A v ariety of non-deterministic and parallel operators have been added to the-calculus by several authors with di erent aims. One has been the study of non-determinism in the functional setting (see e.g. 7, 14, 2] and more recently 1, 36]), i.e. the study of (computable) multivalued functions. This view is strictly connected with the theory of powerdomains introduced in 38, 4 3 ]. These e orts receive n e w i n terest in connection with recent research activities aiming at a theory of higher-order communicating processes. So it is natural to ask for a theory in which communication embodies functional application. This has been studied by Thomsen in 44] a n d b y Boudol in 15] explicitly, while it is an implicit theme in current research on Milner's-calculus 32]. Non-determinism and parallelism (usually represented by a n i n terleaving operator) are fundamental concepts in process algebra theory. C o m bining them and-calculus can enlighten the theory of higher-order process algebras. Indeed an open problem with the former theory is the lack of a good denotational semantics. It is encouraging that a main step toward a de nition of ? This work has been partially supported by grants from ESPRIT-BRA 7232 GENTZEN and from CNR-GNASAGA.

Filter pairs and natural extensions of logics

arXiv: Logic, 2020

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality kappa\kappakappa, {where k\kk is a regular cardinal}. The corresponding new notion is called kappa\kappakappa-filter pair. We show that any kappa\kappakappa-filter pair gives rise to a logic of cardinality kappa\kappakappa and that every logic of cardinality kappa\kappakappa comes from a kappa\kappakappa-filter pair. We use filter pairs to construct natural extensions for a given logic and work out the relationships between this construction and several others proposed in the literature. Conversely, we describe the class of filter pairs giving rise to a fixed logic in terms of the natural extensions of that logic.

Axiomatization of the Monadic Theory of Ordinals < Ω2

Mathematical Logic Quarterly, 1983

Let K be a class of structures for 9 and define ?INK = 3'' n K for each set of sentences 91. Then p K z is still a closure operator and we may define K-consequence, Ktheory, etc. as above using p K z in place pz. If Kl 5 K2 then any K,-consequence is a K1-consequence, any K,-theory is a K,-theory, any K,-axiom system is a K,-axiom system, etc. In general these implications are not reversible. If K is the collection of all standard structures we replace K-consequence by standard-consequence, K-theory by standard-theory, etc. 2. Standard axiomatization In this section we exhibit standard axiom systems for MT[ <w,] (= MT[a, <]) and the MT[a, < ] , a < w 2. We begin with the following collection of formulas and sentences as taken from [el, the names of which indicate their interpretation in standard structures.