Extension Of First-order Logic Research Articles

Logics with Concrete Domains: First-Order Properties, Abstract Expressive Power, and (Un)Decidability

Concrete domains have been introduced in description logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. The primary research goal in this context was to find restrictions on the concrete domain such that its integration into certain DLs preserves decidability or tractability. In this article, we complement these investigations by studying the abstract expressive power of both first-order and description logics extended with concrete domains, i.e., we analyze which classes of first-order interpretations can be expressed using these logics, compared to what their counterparts without concrete domains can express. We demonstrate that, under natural conditions on the concrete domain D (which also play a role for decidability), extensions of first-order logic (FOL) or the well-known DL ALC with D share important formal characteristics with FOL, such as the compactness and the Löwenheim-Skolem properties. Nevertheless, these conditions do not ensure that the abstract expressive power of the extensions we consider is contained in that of FOL, though in some cases it is. To be more precise, we show, on the one hand, that unary concrete domains leave the abstract expressive power within FOL if we are allowed to introduce auxiliary predicates. As a by-product, we obtain (semi-)decidability results for some fragments of FOL extended with the concrete domains considered in this article. On the other hand, we show that the ability to express equality between elements of D , another condition employed in the context of showing decidability of ALC ( D ), is sufficient to push the abstract expressive power of most first-order fragments with concrete domains beyond that of FOL. While for such concrete domains D decidability is retained for ALC ( D ), we show that the availability of equality in D causes undecidability of the two-variable fragment of FOL( D ), although the two-variable fragment of FOL is decidable.

Cyclic proofs for the first-order µ-calculus

Abstract We introduce a path-based cyclic proof system for first-order $\mu $-calculus, the extension of first-order logic by second-order quantifiers for least and greatest fixed points of definable monotone functions. We prove soundness of the system and demonstrate it to be as expressive as the known trace-based cyclic systems of Dam and Sprenger. Furthermore, we establish cut-free completeness of our system for the fragment corresponding to the modal $\mu $-calculus.

Positive logics

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

COUNTING TO INFINITY: GRADED MODAL LOGIC WITH AN INFINITY DIAMOND

AbstractWe extend the languages of both basic and graded modal logic with the infinity diamond, a modality that expresses the existence of infinitely many successors having a certain property. In both cases we define a natural notion of bisimilarity for the resulting formalisms, that we dub $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ , respectively. We then characterise these logics as the bisimulation-invariant fragments of the naturally corresponding predicate logic, viz., the extension of first-order logic with the infinity quantifier. Furthermore, for both $\mathtt {ML}^{\infty }$ and $\mathtt {GML}^{\infty }$ we provide a sound and complete axiomatisation for the set of formulas that are valid in every Kripke frame, we prove a small model property with respect to a widened class of weighted models, and we establish decidability of the satisfiability problem.

Omitting types algebraically and more about amalgamation for modal cylindric algebras

AbstractLet α be an arbitrary infinite ordinal, and . In [26] we studied—using algebraic logic—interpolation and amalgamation for an extension of first order logic, call it , with α many variables, using a modal operator of a unimodal logic that contributes to the semantics. Our algebraic apparatus was the class of modal cylindric algebras. Modal cylindric algebras, briefly , are cylindric algebras of dimension α, expanded with unary modalities inheriting their semantics from a unimodal logic such as , or . When modal cylindric algebras based on are just cylindric algebras, that is to say, . This paper is a sequel to [26], where we study algebraically other properties of . We study completeness and omitting types (s) for s by proving several representability results for so‐called dimension complemented and locally finite . Furthermore, we study the notion of atom‐canonicity for , the variety of n‐dimensional modal cylindric algebras. Atom canonicity, a well known persistence property in modal logic, is studied in connection to for , which is restricted to the first n variables. We further continue our study of interpolation in [26] for algebraizable extensions of by studying using both algebraic logic and category theory. Our main results on are Theorems 3.7, 4.4 & 4.6, while our main results on amalgamation are Theorems 5.7, 5.10, 5.13 & 5.16.

Omitting Types in Fragments and Extensions of First Order Logic

Fix \(2 < n < \omega\). Let \(L_n\) denote first order logic restricted to the first n variables. Using the machinery of algebraic logic, positive and negative results on omitting types are obtained for \(L_n\) and for infinitary variants and extensions of \(L_{\omega, \omega}\).

On equivalence of PFP-operator and PFP-quantifier

In this paper we consider two different logic languages. Both of them are extensions of first order logic. The first semantics is obtained by adding a partial fixed point operator. The second semantics is based on considering a partial fixed point as a non-standard quantifier. For this two semantics we demonstrate that they have an equal expressive power. For this purpose we show how to express an arbitrary formula of one logic with a formula of another.

Inner models from extended logics: Part 1

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.

Non-well-founded Proof Theory of Transitive Closure Logic

Supporting inductive reasoning is an essential component is any framework of use in computer science. To do so, the logical framework must extend that of first-order logic. Transitive closure logic is a known extension of first-order logic that is particularly straightforward to automate. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of a single transitive closure operator has the advantage of uniformly capturing all finitary inductive definitions. To further improve the reasoning techniques for transitive closure logic, we here present an infinitary proof system for it, which is an infinite descent –style counterpart to the existing (explicit induction) proof system for the logic. We show that the infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e., cyclic ) proofs provides the basis for an effective system for automating inductive reasoning.

In defence of Higher-Level Plural Logic: drawing conclusions from natural language

Plural Logic is an extension of First-Order Logic which has, as well as singular terms and quantifiers, their plural counterparts. Analogously, Higher-Level Plural Logic is an extension of Plural Logic which has, as well as plural terms and quantifiers, higher-level plural ones. Roughly speaking, higher-level plurals stand to plurals like plurals stand to singulars; they are pluralised plurals. Allegedly, Higher-Level Plural Logic enjoys the expressive power of a simple type theory while committing us to nothing more than the austere ontology of First-Order Logic. Were this true, Higher-Level Plural Logic would be a useful tool, with various applications in philosophy and linguistics. However, while the notions of plural reference and quantification enjoy widespread acceptance today, their higher-level counterparts have been received with a lot of scepticism. In this paper, I argue for the legitimacy of Higher-Level Plural Logic by providing evidence to the effect that natural languages contain higher-level plural expressions and showing that it is likely that they do so in an indispensable manner. Since the arguments I put forward are of the same sort advocates of Plural Logic have employed to defend their position, I conclude that the commonly held view that Plural Logic is legitimate, but not so its higher-level plural extensions is untenable.

Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts

The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic (IFL) is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact of IFL, that first-order logic (FOL) can adequately express uniformity concepts in real analysis, whereas IFL (understood as a non-trivial extension of FOL) cannot. This not only radically contradicts Hintikka’s particular claim in that article, but also undermines his whole enterprise of founding mathematics on his logic system.

Nominal Unification from a Higher-Order Perspective

Nominal logic is an extension of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to lambda-terms, in nominal terms, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of the lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be quadratically reduced to a particular fragment of higher-order unification problems: higher-order pattern unification. We also prove that the translation preserves most generality of unifiers.

Recursive proofs for inductive tree data-structures

We develop logical mechanisms and procedures to facilitate the verification of full functional properties of inductive tree data-structures using recursion that are sound, incomplete, but terminating. Our contribution rests in a new extension of first-order logic with recursive definitions called Dryad, a syntactical restriction on pre- and post-conditions of recursive imperative programs using Dryad, and a systematic methodology for accurately unfolding the footprint on the heap uncovered by the program that leads to finding simple recursive proofs using formula abstraction and calls to SMT solvers. We evaluate our methodology empirically and show that several complex tree data-structure algorithms can be checked against full functional specifications automatically, given pre- and post-conditions. This results in the first automatic terminating methodology for proving a wide variety of annotated algorithms on tree data-structures correct, including max-heaps, treaps, red-black trees, AVL trees, binomial heaps, and B-trees.

Algorithmic correspondence and completeness in modal logic. V. Recursive extensions of SQEMA

The previously introduced algorithm SQEMA computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend SQEMA with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points FO μ. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid μ-calculus. In particular, we prove that the recursive extension of SQEMA succeeds on the class of ‘recursive formulae’. We also show that a certain version of this algorithm guarantees the canonicity of the formulae on which it succeeds.

Grounding FO and FO(ID) with Bounds

Grounding is the task of reducing a first-order theory and finite domain to an equivalent propositional theory. It is used as preprocessing phase in many logic-based reasoning systems. Such systems provide a rich first-order input language to a user and can rely on efficient propositional solvers to perform the actual reasoning. Besides a first-order theory and finite domain, the input for grounders contains in many applications also additional data. By exploiting this data, the size of the grounder's output can often be reduced significantly. A common practice to improve the efficiency of a grounder in this context is by manually adding semantically redundant information to the input theory, indicating where and when the grounder should exploit the data. In this paper we present a method to compute and add such redundant information automatically. Our method therefore simplifies the task of writing input theories that can be grounded efficiently by current systems. We first present our method for classical first-order logic (FO) theories. Then we extend it to FO(ID), the extension of FO with inductive definitions, which allows for more concise and comprehensive input theories. We discuss implementation issues and experimentally validate the practical applicability of our method.

FO(ID) as an extension of DL with rules

There are many interesting Knowledge Representation questions surrounding rule languages for the Semantic Web. The most basic one is of course: which kind of rules should be used and how do they integrate with existing Description Logics? Similar questions have already been addressed in the field of Logic Programming, where one particular answer has been provided by the language of FO(ID). FO(ID) is an extension of first-order logic with a rule-based representation for inductive definitions. By offering a general integration of first-order logic and Logic Programs, it also induces a particular way of extending Description Logics with rules. The goal of this paper is to investigate this integration and discover whether there are interesting extensions of DL with rules that can be arrived at by imposing appropriate restrictions on the highly expressive FO(ID).

Properties of Almost All Graphs and Generalized Quantifiers

We study 0-1 laws for extensions of first-order logic by Lindstrom quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q] - the extension of first-order logic by means of the quantifier Q - to have a 0-1 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 0-1 law. We also show that extensions of first-order logic with quantifiers for Hamiltonicity, regularity and self-complementarity of graphs do not have a 0-1 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 0-1 law. It is a consequence of our results that there is no such regular logic (in the sense of abstract model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantifiers, and show that the extensions of first-order logic obtained by adding such sequences generated by quantifiers that are closed under substructures have 0-1 laws. The positive results also extend to the infinitary logic with finitely many variables.

Zero-one law and definability of linear order

§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.

A Simple Nominal Type Theory

Nominal logic is an extension of first-order logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic that extend standard type theories for propositional, first- or higher-order logic. This has proven difficult, largely because of complex interactions between nominal logic's name-abstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory that enjoys properties such as type soundness and strong normalization, and that can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is a first step towards understanding the constructive content of nominal logic and incorporating it into existing constructive logics and type theories.