Existential Quantification Research Articles

Polynomial rewritings from expressive Description Logics with closed predicates to variants of Datalog

In many scenarios, complete and incomplete information coexist. For this reason, the knowledge representation and database communities have long shown interest in simultaneously supporting the closed- and the open-world views when reasoning about logic theories. Here we consider the setting of querying possibly incomplete data using logic theories, formalized as the evaluation of an ontology-mediated query (OMQ) that pairs a query with a theory, sometimes called an ontology, expressing background knowledge. This can be further enriched by specifying a set of closed predicates from the theory that are to be interpreted under the closed-world assumption, while the rest are interpreted with the open-world view. In this way we can retrieve more precise answers to queries by leveraging the partial completeness of the data.The central goal of this paper is to understand the relative expressiveness of ontology-mediated query languages in which the ontology part is written in the expressive Description Logic (DL) ALCHOI and includes a set of closed predicates. We consider a restricted class of conjunctive queries. Our main result is to show that every query in this non-monotonic query language can be translated in polynomial time into Datalog with negation as failure under the stable model semantics. To overcome the challenge that Datalog has no direct means to express the existential quantification present in ALCHOI, we define a two-player game that characterizes the satisfaction of the ontology, and design a Datalog query that can decide the existence of a winning strategy for the game. If there are no closed predicates—in the case of querying an ALCHOI knowledge base—our translation yields a positive disjunctive Datalog program of polynomial size. To the best of our knowledge, unlike previous translations for related fragments with expressive (non-Horn) DLs, these are the first polynomial time translations.

SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE

AbstractThe 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

EA solutions and EA solvability to general interval linear systems

ABSTRACT S. Shary introduced the notion of quantifier solutions to interval systems, in which the uncertainty on the coefficients of a system is modelled by a selecting predicate, allowing an alternation in an arbitrary order of the logical quantifiers (∀ and ∃). AE solutions, a particular type of quantifier solutions, in which all the occurrences of the universal quantifier ∀ precede the occurrences of the existential quantifier ∃, have been studied by many authors. In this work we deal with EA solutions, another type of quantifier solutions to general interval linear systems, such that, in the selecting predicate, all the occurrences of the existential quantifier ∃ precede the occurrences of the universal quantifier ∀. The EA solutions can be considered as a dual form of AE solutions. We provide a full characterization of EA solutions. Moreover, we develop a necessary condition for EA solvability of the general interval linear systems, as well as a necessary and sufficient condition for a certain subcase.

Certifying graph-manipulating C programs via localizations within data structures

We develop powerful and general techniques to mechanically verify realistic programs that manipulate heap-represented graphs. These graphs can exhibit well-known organization principles, such as being a directed acyclic graph or a disjoint-forest; alternatively, these graphs can be totally unstructured. The common thread for such structures is that they exhibit deep intrinsic sharing and can be expressed using the language of graph theory. We construct a modular and general setup for reasoning about abstract mathematical graphs and use separation logic to define how such abstract graphs are represented concretely in the heap. We develop a Localize rule that enables modular reasoning about such programs, and show how this rule can support existential quantifiers in postconditions and smoothly handle modified program variables. We demonstrate the generality and power of our techniques by integrating them into the Verified Software Toolchain and certifying the correctness of seven graph-manipulating programs written in CompCert C, including a 400-line generational garbage collector for the CertiCoq project. While doing so, we identify two places where the semantics of C is too weak to define generational garbage collectors of the sort used in the OCaml runtime. Our proofs are entirely machine-checked in Coq.

An investigation of numeral quantifiers in English

There exist a broad range of theories which model quantifier scope (Ruys & Winter 2011). The empirical coverage and predictions of three theories: Quantifier Raising (May 1985) with Scope Economy (Fox 2000), Feature Checking (Beghelli & Stowell 1997), and Choice Functions (Winter 1997) are tested using a sentence-picture verification task. Speakers rated the quality of sentences with bare and modified numeral quantifiers in surface and inverse scope conditions. In Experiment 1, it was discovered that bare numerals marginally resisted inverse scope, contra the predictions of (Beghelli & Stowell 1997). Experiment 2 revealed that the scope of numerals is clause-bound, in argument against the theory of Choice Functions which predict extra-clausal scope for existential quantifiers. The results suggest that a modified QR theory best accounts for the observed data.

The experiential aspect of Mandarin Chinese (-guo): Semantics and pragmatics

The study of the Chinese experiential aspect expressed by -guo is an important research topic in both Chinese theoretical linguistics and the semantics-pragmatics interface more broadly. Previous studies propose that the semantics of -guo has the property of what is called repeatability, and/or discontinuity. In this article, I show that previous theories do not adequately explain the distributional pattern of -guo. The semantics of -guo therefore remains unclear. Based on a comprehensive observation of the experiential aspect marker -guo, I propose that (1) -guo semantically is an existential quantifier indicating a non-empty set of a type of eventuality in a certain time frame which is presupposed to be before a reference time, and with this new proposal, the so-called perfective -guo described by previous studies can be explained in the same framework, and thus the two -guos can be unified; (2) the discontinuity property can be accounted for from a pragmatic perspective under the notion of Question Under Discussion (Roberts, 1996).

On the satisfiability problem for fragments of two-variable logic with one transitive relation

Abstract We study the satisfiability problem for two-variable first-order logic over structures with one transitive relation. We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model property nor the tree model property, to show decidability we introduce a novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation; hence, contrary to our previous claim, [FO$^2$ with one transitive relation is decidable, STACS 2013: 317-328], the status of the latter problem remains open.

Quantification of process flexibility via space projection

AbstractFlexibility index is an effective indicator for characterizing the operational flexibility of a process design model. In this work, the concept of δ‐space projection is proposed and defined from the view of solution space, and the flexibility index can be quantified without solving any optimization problems. First, the flexibility index problem is reformulated as an existential quantifier model. Then, the projection operator in the cylindrical algebraic decomposition (CAD) method is introduced to project the solution space onto the one‐dimensional feasible space of the flexibility index. Subsequently, all the candidate values of the flexibility index, which may refer to the locations of the critical points, that is, vertexes and tangency points, are extracted. Last, two δ‐checking rules based on the traditional and improved CAD methods are proposed for finding the flexibility index. The case studies show that the proposed method can exactly find the flexibility index, regardless of convex or nonconvex systems.

Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball

We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

Implicit quantification made explicit: How to interpret blank nodes and universal variables in Notation3 Logic

Since the invention of Notation3 Logic, several years have passed in which the theory has been refined and applied in different reasoning engines like Cwm, EYE, and FuXi. But despite these developments, a clear formal definition of Notation3’s semantics is still missing. This does not only form an obstacle for the formal investigation of that logic and its relations to other formalisms, it has also practical consequences: in many cases the interpretations of the same formula differ between reasoning engines. In this paper we tackle one of the main sources of that problem, namely the uncertainty about implicit quantification. This refers to Notation3’s ability to use bound variables for which the universal or existential quantifiers are not explicitly stated, but implicitly assumed. We provide a tool for clarification through the definition of a core logic for Notation3 that only supports explicit quantification. We specify an attribute grammar which maps Notation3 formulas to that logic according to the different interpretations and thereby define the semantics of Notation3. This grammar is then implemented and used to test the impact of the differences between interpretations on practical cases. Our dataset includes Notation3 implementations from former research projects and test cases developed for the reasoner EYE. We find that 31% of these files are understood differently by different reasoners. We further analyse these cases and categorise them in different classes of which we consider one most harmful: if a file is manually written by a user and no specific built-in predicates are used (13% of our critical files), it is unlikely that this user is aware of possible differences. We therefore argue the need to come to an agreement on implicit quantification, and discuss the different possibilities.

Cross-Linguistic Variation in the Meaning of Quantifiers: Implications for Pragmatic Enrichment.

One of the most studied scales in the literature on scalar implicatures is the quantifier scale. While the truth of some is entailed by the truth of all, some is felicitous only when all is false. This opens the possibility that some would be felicitous if, e.g., almost all of the objects in the restriction of the quantifier have the property ascribed by the nuclear scope. This prediction from the standard theory of quantifier interpretation clashes with native speakers’ intuitions. In Experiment 1 we report a questionnaire study on the perception of quantifier meanings in English, French, Slovenian, and German which points to a cross-linguistic variation with respect to the perception of numerical bounds of the existential quantifier. In Experiment 2, using a picture choice task, we further examine whether the numerical bound differences correlate with differences in pragmatic interpretations of the quantifier some in English and quelques in French and interpret the results as supporting our hypothesis that some and its cross-linguistic counterparts are subjected to different processes of pragmatic enrichment.

Modals under epistemic tension

According to Kratzer’s influential account of epistemic must and might, these operators involve quantification over domains of possibilities determined by a modal base and an ordering source. Recently, this account has been challenged by invoking contexts of ‘epistemic tension’: i.e., cases in which an assertion that must $$\phi $$ is conjoined with the possibility that $$\lnot \phi $$ , and cases in which speakers try to downplay a previous assertion that must $$\phi $$ , after finding out that $$\lnot \phi $$ . Epistemic tensions have been invoked from two directions. Von Fintel and Gillies (Nat Lang Semant 18(4):351–383, 2010) propose a return to a simpler modal logic-inspired account: must and might still involve universal and existential quantification, but the domains of possibilities are determined solely by realistic modal bases. In contrast, Lassiter (Nat Lang Semant 24(2):117–163, 2016), following Swanson (Interactions with context. Ph.D. thesis, MIT, 2006; and in A. Eagan and B. Weatherstone, eds., Epistemic Modality, Oxford UP, 2011), proposes a more revisionary account which treats must and might as probabilistic operators. In this paper, we present a series of experiments to obtain reliable data on the degree of acceptability of various contexts of epistemic tension. Our experiments include novel variations that, we argue, are required to make progress in this debate. We show that restricted quantificational accounts a la Kratzer fit the overall pattern of results better than either of their recent competitors. In addition, our results help us identify the key components of restricted quantificational accounts, and on that basis propose some refinements and general constraints that should be satisfied by any account of the modal auxiliaries.

Fast Query Answering over Existential Rules

Enhancing Datalog with existential quantification gives rise to Datalog ∃ , a powerful knowledge representation language widely used in ontology-based query answering. In this setting, a conjunctive query is evaluated over a Datalog ∃ program consisting of extensional data paired with so-called “existential” rules. Owing to their high expressiveness, such rules make the evaluation of queries undecidable, even when the latter are atomic. Decidable generalizations of Datalog by existential rules have been proposed in the literature (such as weakly acyclic and weakly guarded); but they pay the price of higher computational complexity, hindering the implementation of effective systems. Conversely, the results in this article demonstrate that it is definitely possible to enable fast yet powerful query answering over existential rules that strictly generalize Datalog by ensuring decidability without any complexity overhead. On the theoretical side, we define the class of parsimonious programs that guarantees decidability of atomic queries. We then strengthen this class to strongly parsimonious programs ensuring decidability also for conjunctive queries. Since parsimony is an undecidable property, we single out Shy, an easily recognizable class of strongly parsimonious programs that generalizes Datalog while preserving its complexity even under conjunctive queries. Shy also generalizes the class of linear existential programs, while it is uncomparable to the other main classes ensuring decidability. On the practical side, we exploit our results to implement DLV ∃ , an effective system for query answering over parsimonious existential rules. To assess its efficiency, we carry out an experimental analysis, evaluating DLV ∃ performances for ontology-based query answering on both real-world and synthetic ontologies.

Monadic NM-algebras

Abstract In this paper, we investigate universal and existential quantifiers on NM-algebras. The resulting class of algebras will be called monadic NM-algebras. First, we show that the variety of monadic NM-algebras is algebraic semantics of the monadic NM-predicate logic. Moreover, we discuss the relationship among monadic NM-algebras, modal NM-algebras and rough approximation spaces. Second, we introduce and investigate monadic filters in monadic NM-algebras. Using them, we prove the subdirect representation theorem of monadic NM-algebras, and characterize simple and subdirectly irreducible monadic NM-algebras. Finally, we present the monadic NM-logic and prove its (chain) completeness with respect to (strong) monadic NM-algebras. These results constitute a crucial first step for providing an algebraic foundation for the monadic NM-predicate logic.

Practical Subtyping for Curry-Style Languages

We present a new, syntax-directed framework for Curry-style type systems with subtyping. It supports a rich set of features, and allows for a reasonably simple theory and implementation. The system we consider has sum and product types, universal and existential quantifiers, and inductive and coinductive types. The latter two may carry size invariants that can be used to establish the termination of recursive programs. For example, the termination of quicksort can be derived by showing that partitioning a list does not increase its size. The system deals with complex programs involving mixed induction and coinduction, or even mixed polymorphism and (co-)induction. One of the key ideas is to separate the notion of size from recursion. We do not check the termination of programs directly, but rather show that their (circular) typing proofs are well-founded. Termination is then obtained using a standard (semantic) normalisation proof. To demonstrate the practicality of the system, we provide an implementation accepting all the examples discussed in the article.

Implicit Quantification Made Explicit: How to Interpret Blank Nodes and Universal Variables in Notation3 Logic

Since the invention of Notation3 Logic, several years have passed in which the theory has been refined and applied in different reasoning engines like Cwm, EYE, and FuXi. But despite these developments, a clear formal definition of Notation3’s semantics is still missing. This does not only form an obstacle for the formal investigation of that logic and its relations to other formalisms, it has also practical consequences: in many cases the interpretations of the same formula di er between reasoning engines. In this paper we tackle one of the main sources of that problem, namely the uncertainty about implicit quantification. This refers to Notation3’s ability to use bound variables for which the universal or existential quantifiers are not explicitly stated, but implicitly assumed. We provide a tool for clarification through the definition of a core logic for Notation3 that only supports explicit quantification. We specify an attribute grammar which maps Notation3 formulas to that logic according to the different interpretations and thereby define the semantics of Notation3. This grammar is then implemented and used to test the impact of the differences between interpretations on practical cases. Our dataset includes Notation3 implementations from former research projects and test cases developed for the reasoner EYE. We find that 31% of these files are understood differently by different reasoners. We further analyse these cases and categorize them in different classes of which we consider one most harmful: if a file is manually written by a user and no specific built-in predicates are used (13% of our critical files), it is unlikely that this user is aware of possible differences. We therefore argue the need to come to an agreement on implicit quantification, and discuss the different possibilities.

An enquiry into the ontological and logical foundations of sustainability: Toward a conceptual integration of the interface ‘Nature/Humanity’

Non-technical summary Implementing the logical and ontological principles (dualism, mechanicism, reductionism, law of excluded middle, etc.) of modernity has brought forth an unsustainable world. An overcoming of these principles is proposed by mesology (Umweltlehre, fûdoron), centring on the concept of trajection and the existential operator as (als, en tant que).

R-impersonal interpretation in Italian Sign Language (LIS)

Abstract In this paper, we examine agent backgrounding in Italian Sign Language (LIS). Specifically, we are interested in identifying and describing the strategies used by LIS signers to reduce referentiality. On the basis of low-referential contexts (cf. questionnaire in the Introduction chapter), we recorded target sentences containing potential markers of agent backgrounding and asked three LIS native signers to provide felicity judgments on them using a 7-point scale. We discuss agent-backgrounding strategies of different types: (i) manual, (ii) non-manual, and (iii) syntactic. Overall, our study shows that the combination of raised eyebrows and mouth-corners down associated with the existential quantifier someone and the sign person makes the agent-backgrounding reading more prominent. Other strategies that can be used in LIS to reduce referentiality are free relatives, perspective shift, and null subject. We also investigate in more detail the semantic status of someone, person, and the null subject through well-established tests from the literature.

UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS

AbstractWe introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

Determiners and quantifiers in Spanish: types, tests and theories

This article provides an overview about the main facts and analytical options in the domain of determiners and quantifiers in Spanish. It covers the main classification of determiners and their basic syntactic and semantic properties (§1), the differences in behaviour between quantifiers and determiners in the strict sense (§2), the notion of definiteness and the contrasts in the use of the definite and indefinite articles (§3), the notion of specificity (§4) and the main types of quantifiers and how they can be identified (§5). In terms of analytical problems, it discusses whether determiners should be considered heads or not (§6), the areas within the determiner and quantifier domain (§7), the nature of the indefinite article as an element that shares properties with existential quantifiers (§8), the problems posed by proper names (§9) and the possible existence of phonologically null determiners in Spanish (§10). Conclusions are presented in §11.