Zero-one law and definability of linear order

2009 ◽  
Vol 74 (1) ◽  
pp. 105-123
Author(s):  
Hannu Niemistö
Keyword(s):  
Linear Order ◽  
Quantifier Elimination ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Almost Everywhere ◽  
Asymptotic Probability ◽  
Finite Variable Logic ◽  
Second Order Logic

§1. Introduction. A logic ℒ has a limit law, if the asymptotic probability of every query definable in ℒ converges. It has a 0–1-law if the probability converges to 0 or 1. The 0–1-law for first-order logic on relational vocabularies was independently found by Glebski et al. [6] and Fagin [5]. Later it has been shown for many other logics, for instance for fragments of second order logic [12], for finite variable logic [13] and for FO extended with the rigidity quantifier [3]. Lynch [14] has shown a limit law for first-order logic on vocabularies with unary functions.We say that two formulas or two logics are almost everywhere equivalent, if they are equivalent on a class of structures whose asymptotic probability measure is one [7]. A 0–1-law is usually proved by showing that every quantifier of the logic has almost everywhere quantifier elimination, i.e., every formula with just one quantifier in front of it is almost everywhere equivalent to a quantifier-free formula. Besides proving 0–1-law, this implies that the logic is (weakly) almost everywhere equivalent to first-order logic.The aim of this paper is to study, whether a logic with a 0–1-law can have greater expressive power than FO in the almost everywhere sense and to what extent. In particular, we are interested on the definability of linear order. Because a 0–1-law determines the almost everywhere expressive power of the sentences of the logic completely, but does not say anything about formulas explicitly, we have to assume some regularity on logics. We will therefore mostly consider extensions of first-order logic with generalized quantifiers.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

Towards a characterization of order-invariant queries over tame graphs

2009 ◽  
Vol 74 (1) ◽  
pp. 168-186 ◽  
Author(s):  
Michael Benedikt ◽  
Luc Segoufin
Keyword(s):  
Linear Order ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Graphs ◽  
Bounded Treewidth ◽  
First Order ◽  
A New Technique ◽  
Order Invariant

AbstractThis work deals with the expressive power of logics on finite graphs with access to an additional “arbitrary” linear order. The queries that can be expressed this way are the order-invariant queries for the logic. For the standard logics used in computer science, such as first-order logic, it is known that access to an arbitrary linear order increases the expressiveness of the logic. However, when we look at the separating examples, we find that they have satisfying models whose Gaifman Graph is complex – unbounded in valence and in treewidth. We thus explore the expressiveness of order-invariant queries over well-behaved graphs. We prove that first-order order-invariant queries over strings and trees have no additional expressiveness over first-order logic in the original signature. We also prove new upper bounds on order-invariant queries over bounded treewidth and bounded valence graphs. Our results make use of a new technique of independent interest: the application of algebraic characterizations of definability to show collapse results.

The relative expressive power of some logics extending first-order logic

1979 ◽  
Vol 44 (2) ◽  
pp. 129-146 ◽  
Author(s):  
John Cowles
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Finite Sets ◽  
Limited Ability ◽  
First Order ◽  
Second Order Logic

In recent years there has been a proliferation of logics which extend first-order logic, e.g., logics with infinite sentences, logics with cardinal quantifiers such as “there exist infinitely many…” and “there exist uncountably many…”, and a weak second-order logic with variables and quantifiers for finite sets of individuals. It is well known that first-order logic has a limited ability to express many of the concepts studied by mathematicians, e.g., the concept of a wellordering. However, first-order logic, being among the simplest logics with applications to mathematics, does have an extensively developed and well understood model theory. On the other hand, full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. Indeed, the search for a logic with a semantics complex enough to say something, yet at the same time simple enough to say something about, accounts for the proliferation of logics mentioned above. In this paper, a number of proposed strengthenings of first-order logic are examined with respect to their relative expressive power, i.e., given two logics, what concepts can be expressed in one but not the other?For the most part, the notation is standard. Most of the notation is either explained in the text or can be found in the book [2] of Chang and Keisler. Some notational conventions used throughout the text are listed below: the empty set is denoted by ∅.

Beyond first-order logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Fixed Point ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Compactness Theorem ◽  
First Order Logic ◽  
First Order ◽  
Skolem Theorem ◽  
Additional Rule ◽  
Second Order Logic

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

scholarly journals Strong Logics of First and Second Order

2010 ◽  
Vol 16 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Peter Koellner
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Close Correspondence ◽  
First Order ◽  
Second Order Logic

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

scholarly journals Taking Plurals at Face Value

2021 ◽  
pp. 8-30
Author(s):  
Salvatore Florio ◽  
Øystein Linnebo
Keyword(s):  
Philosophy Of Mathematics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Logical System ◽  
Plural Logic ◽  
First Order ◽  
Philosophical Significance ◽  
Pure Logic

Plural logic is a logical system in which plural terms and predicates figure as primitive expressions alongside the singular resources of ordinary first-order logic. The philosophical significance of this system depends on two of its alleged features: being pure logic and providing more expressive power than first-order logic. This chapter first introduces the language and axioms of plural logic and then analyzes this logic’s main philosophical applications in metaphysics, philosophy of mathematics, and semantics.

scholarly journals Using hybrid logic for querying dependency treebanks

2012 ◽  
Vol 7 ◽  
Author(s):  
Anders Søgaard ◽  
Søren Lind Kristiansen
Keyword(s):  
General Purpose ◽  
Second Order ◽  
Logic Model ◽  
Order Logic ◽  
Hybrid Logic ◽  
First Order Logic ◽  
Model Checker ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Existing logic-based querying tools for dependency treebanks use first order logic or monadic second order logic. We introduce a very fast model checker based on hybrid logic with operators ↓, @ and A and show that it is much faster than an existing querying tool for dependency treebanks based on first order logic, and much faster than an existing general purpose hybrid logic model checker. The querying tool is made publicly available.

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