The modal logic of continuous functions on cantor space

Let be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality and a temporal modality , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language by interpreting in dynamic topological systems, i.e. ordered pairs , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.     

The modal logic of forcing

A set theoretical assertion is forceable or possible, written , if holds in some forcing extension, and necessary, written , if holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory .

     

Generating functions, Fibonacci numbers and rational knots

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces.     

The modal logic of the countable random frame

 We study the modal logic M L ^r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L ^r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. Received: 1 May 2000 / Revised version: 29 July 2001 / Published online: 2 September 2002 Mathematics Subject Classification (2000): 03B45, 03B70, 03C99 Key words or phrases: Modal logic – Random frames – Almost sure frame validity – Countable random frame – Axiomatization – Completeness     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.     

The homotopy type of rational functions

During the preparation of this work each of the authors was supported by an NSF grant, the second author by an NSF-PYI award, and the first and fourth authors by the S.F.B. 170 in G?ttingen     

On the product sets of rational numbers

A new lower bound on the size of product sets of rational numbers is obtained. An upper estimate for the multiplicative energy of two sets of rational numbers is also found.     

Path calculus in the modal logic S4

We describe the tableau and inverse calculi for the propositional modal logic S4. The formulas are treated as sets of paths. We obtain the upper bound for the number of applications of rules in the deduction tree.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 117–126, January–March, 2005.Translated by R. Lapinskas     

Classification of extensions of the modal logic S4

We introduce a natural classification of normal extensions of the modal logic S4 in accordance to the volumes of clusters in the Kripke frames and prove the decidability of the classification. We distinguish the main logics in this classification and establish their important properties: finite axiomatizability, finite approximability, and recognizability.