1,205 research outputs found
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
Counting Incompossibles
We often speak as if there are merely possible people—for example, when we make such claims as that most possible people are never going to be born. Yet most metaphysicians deny that anything is both possibly a person and never born. Since our unreflective talk of merely possible people serves to draw non-trivial distinctions, these metaphysicians owe us some paraphrase by which we can draw those distinctions without committing ourselves to there being merely possible people. We show that such paraphrases are unavailable if we limit ourselves to the expressive resources of even highly infinitary first-order modal languages. We then argue that such paraphrases are available in higher-order modal languages only given certain strong assumptions concerning the metaphysics of properties. We then consider alternative paraphrase strategies, and argue that none of them are tenable. If talk of merely possible people cannot be paraphrased, then it must be taken at face value, in which case it is necessary what individuals there are. Therefore, if it is contingent what individuals there are, then the demands of paraphrase place tight constraints on the metaphysics of properties: either (i) it is necessary what properties there are, or (ii) necessarily equivalent properties are identical, and having properties does not entail even possibly being anything at all
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
One Quantifier Alternation in First-Order Logic with Modular Predicates
Adding modular predicates yields a generalization of first-order logic FO
over words. The expressive power of FO[<,MOD] with order comparison and
predicates for has been investigated by Barrington,
Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated
by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that
definability in the two-variable fragment FO2[<,MOD] is decidable. In this
paper we continue this line of work.
We give an effective algebraic characterization of the word languages in
Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex
normal form with two blocks of quantifiers starting with an existential block.
In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD]
which is closed under negation, has the same expressive power as two-variable
logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and
Wilke to modular predicates. As a byproduct, we obtain another decidable
characterization of FO2[<,MOD]
- …