Monadic Sentences Research Articles

A Disproof the Le Bars Conjecture about the Zero–One Law for Existential Monadic Second-Order Sentences

The Le Bars conjecture (2001) states that the binomial random graph G(n, $$\frac{1}{2}$$ ) obeys the zero–one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge.

Spectra of short monadic sentences about sparse random graphs

A random graph G(n, p) is said to obey the (monadic) zero–one k-law if, for any monadic formula of quantifier depth k, the probability that it is true for the random graph tends to either zero or one. In this paper, following J. Spencer and S. Shelah, we consider the case p = n −α. It is proved that the least k for which there are infinitely many α such that a random graph does not obey the zero–one k-law is equal to 4.

Separating intermediate predicate logics of well-founded and dually well-founded structures by monadic sentences

We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than ωω. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity.

Selection over classes of ordinals expanded by monadic predicates

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 . For a monadic formula φ ( X , Y ) and ordinals α ≤ ω 1 and δ < ω ω , we decide whether there exists a monadic formula ψ ( X , Y ) such that for every P ⊆ α of order-type smaller than δ , ψ ( P , Y ) selects φ ( P , Y ) in ( α , < ) . If so, we construct such a ψ . We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S ⊆ ω ω such that in the structure ( ω ω , < , S ) every formula has a selector. Given a monadic sentence π and a monadic formula φ ( Y ) , we decide whether φ has a selector over the class of countable ordinals satisfying π , and if so, construct one for it.

On decidability properties of local sentences

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences [J.-P. Ressayre, Formal languages defined by the underlying structure of their words, J. Symbolic Logic 53 (4) (1988) 1009–1026]. Another stretching theorem of Finkel and Ressayre implies that one can decide, for a given local sentence ϕ and an ordinal α < ω ω , whether ϕ has a model of order type α . This result is very similar to Büchi's one who proved that the monadic second order theory of the structure ( α , < ) , for a countable ordinal α , is decidable. It is in fact an extension of that result, as shown in [O. Finkel, finite languages, Theoret. Comput. Sci. 255 (1–2) (2001) 223–261] by considering the expressive power of monadic sentences and of local sentences over languages of words of length α . The aim of this paper is twofold. We wish first to attract the reader's attention on these powerful decidability results proved using methods of model theory and which should find some applications in computer science and we prove also here several additional results on local sentences. The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal ω α (the α th infinite cardinal), whether a local sentence ϕ has a model of order type ω α . Secondly we show that this result cannot be extended to the general case. Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type ω 2 is not determined by the axiomatic system ZFC + GCH , where GCH is the generalized continuum hypothesis. Next we prove that for all integers n , p ⩾ 1 , if n < p then the local theory of ω n , i.e. the set of local sentences having a model of order type ω n , is recursive in the local theory of ω p and also in the local theory of α where α is any ordinal of cofinality ω n .

On Decidability Properties of Local Sentences

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences [J.-P. Ressayre, Formal Languages defined by the Underlying Structure of their Words, Journal of Symbolic Logic 53 (4) (1988) 1009-1026]. Another stretching theorem of Finkel and Ressayre implies that one can decide, for a given local sentence φ and an ordinal α<ωω, whether φ has a model of order type α. This result is very similar to Büchi's one who proved that the monadic second order theory of the structure (α,<), for a countable ordinal α, is decidable. It is in fact an extension of that result, as shown in [O. Finkel, Locally Finite Languages, Theoretical Computer Science 255 (1-2) (2001) 223-261] by considering the expressive power of monadic sentences and of local sentences over languages of words of length α. The aim of this paper is twofold. We wish first to attract the reader's attention on these powerful decidability results proved using methods of model theory and which should find some applications in computer science and we prove also here two additional results on local sentences.The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinalωα (the α-th cardinal), whether a local sentence φ has a model of order type ωα.Secondly we show that this result can not be extended to the general case. Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type ω2 is not determined by the axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesis.

On random models of finite power and monadic logic

For any property θ of a model (or graph), let μ n( θ) be the fraction of models of power n which satisfy θ, and let μ( θ) = lim n →∞ μ n( θ) if this limit exists. For first-order properties θ, it is known that μ(θ) must be 0 or 1. We answer a question of K. Compton by proving in a strong way that this 0–1 law can fail if we allow monadic quantification (that is, quantification over sets) in defining the sentence θ. In fact, by producing a monadic sentence which codes arithmetic on n with probability μ = 1, we show that every recursive real is μ(θ) for some monadic θ.

On the monadic theory ofω 1 without A.C.

Buchi inLecture Notes in Mathematics, Decidable Theories II (1973) by using A.C. characterized the theoriesMT[β, <] forβ<ω1 and showed thatMT[ω1, <] is decidable. We extend Buchi’s results to a larger class of models of ZF (without A.C.) by proving the following under ZF only: (1) There is a choice function which chooses a “good” run of an automaton on countable input (Lemma 5.1). It follows that Buchi’s results cocerning countable ordinals are provable within ZF. (2) Let U.D. be the assertion that there exists a uniform denumeration ofω1 (i.e. a functionf: ω1 → ω1ω such that for everyα<ω1,f(α) is a function fromω ontoα). We show that U.D. can be stated as a monadic sentence, and thereforeω1 is characterizable by a sentence. (3) LetF be the filter of the cofinal closed subsets ofω1. We show that if U.D. holds thenMT[ω1, <] is recursive in the first order theory of the boolean algebraP (ω1)/F. (We can effectively translate each monadic sentence Σ to a boolean sentenceσ such that [ω1, <] ⊨ Σ iffP(ω1)/F⊨σ). (4) As every complete boolean algebra theory is recursive we have that in every model of ZF+U.D.,MT[ω1, <] is recursive. All our proofs are within ZF. Buchi’s work is often referred to. Following Buchi, the main tool is finite automata. We don’t deal withMT[ω1, <] forω1 which doesn’t satisfy U.D.

Concerning measures in first order calculi

~0. Introduction. The idea of treating probability as a real valued function defined on sentences is an old one (see ['6] and [7], where other references can be found). Carnap's at tempt to set up a theory of probabili ty which will have a logical status analogous to that o f two valued logic, is closely connected with it, ef. [1]. So far the sentences were used mainly from a algebraic point of view, that is, the operations that were involved were those of the sentential calculus. (Th e work of Carnap and his collaborators does, however, touch on probabilities which are defined for special cases of first order monadic sentences.) A measure on a sentential calculus which assigns real values to sentences is essentially the same as a measure on the Lindenbaum-Tarski algebra of that calculus, thus its investigation falls under the study of measures on Boolean algebras. These were studied quite a lot; see [3, 5] were other references are given. In this work the notions of a measure on a first order calculus, and of a measuremodel, are introduced and investigated. This is done not from a point of view concerning the foundations of probability but with an eye to mathematical logic and measure theory; the concepts with which we shall deal form a natural generalization of the concepts of a theory and a model in the usual sense. In §1 the not ion of a measure on a first order calculus is introduced. In §2 the notion of a measure-model is defined and a theorem analogous to the completeness theorem is proved. In §3 the case of a calculus with an equality is treated. §4 is concerned with measure-models in which the measure is invariant under permutations of the individuals, and §5 contains a specific example of such a model. Whereas the propositions of §§1, 2 are analogous to similar ones concerning theories and models (in the usual sense), §§4, 5 deal with situations which are typical to measures and measure-models and have no analogous counterpart.