Syntactic Characterization Research Articles

Weakly Schreier extensions for general algebras

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term theta ). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the theta appearing in their syntactic characterisation). Restricting again to the case of monoids, a different choice of theta leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.

Semi-Oblivious Chase Termination for Linear Existential Rules: An Experimental Study

The chase procedure is a fundamental algorithmic tool in databases that allows us to reason with constraints, such as existential rules, with a plethora of applications. It takes as input a database and a set of constraints, and iteratively completes the database as dictated by the constraints. A key challenge, though, is the fact that it may not terminate, which leads to the problem of checking whether it terminates given a database and a set of constraints. In this work, we focus on the semi-oblivious version of the chase, which is well-suited for practical implementations, and linear existential rules, a central class of constraints with several applications. In this setting, there is a mature body of theoretical work that provides syntactic characterizations of when the chase terminates, algorithms for checking chase termination, and precise complexity results. Our main objective is to experimentally evaluate the existing chase termination algorithms with the aim of understanding which input parameters affect their performance, clarifying whether they can be used in practice, and revealing their performance limitations.

Algebra of Data Reconciliation

With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

From partially ordered monoids to partially ordered groups via free nuclear preimages

Two fundamental constructions operating on residuated lattices and partially ordered monoids (pomonoids) are nuclear images and conuclear images. Nuclear images allow us to construct many of the ordered algebras which arise in non-classical logic (such as pomonoids, semilattice-ordered monoids, and residuated lattices) from cancellative ordered algebras. Conuclear images then allow us to construct some of these cancellative algebras from partially ordered or lattice-ordered groups. Among other things, we show that finite (commutative) integral residuated lattices are precisely the finite nuclear images of (commutative) cancellative integral residuated lattices and that (commutative) integrally closed pomonoids are precisely the nuclear images of subpomonoids of partially ordered (Abelian) groups. The key construction is the free nuclear preimage of a pomonoid. As a by-product of our study of free nuclear preimages, we obtain a syntactic characterization of quasivarieties of pomonoids and semilattice-ordered monoids closed under nuclear images.

A tetrachotomy of ontology-mediated queries with a covering axiom

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by description logic ontologies and constructing their optimal rewritings to standard database queries. Originated in ontology-based data access and datalog optimisation, this problem is known to be computationally very complex in general, with no explicit syntactic characterisations available. In this article, aiming to understand the fundamental roots of this difficulty, we strip the problem to the bare bones and focus on Boolean conjunctive queries mediated by a simple covering axiom stating that one class is covered by the union of two other classes. We show that, on the one hand, these rudimentary ontology-mediated queries, called disjunctive sirups (or d-sirups), capture many features and difficulties of the general case. For example, answering d-sirups is Π2p-complete for combined complexity and can be in ▪ or L-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups being 2ExpTime-hard); some d-sirups only have exponential-size resolution proofs, some only double-exponential-size positive existential FO-rewritings and single-exponential-size nonrecursive datalog rewritings. On the other hand, we prove a few partial sufficient and necessary conditions of FO- and (symmetric/linear-) datalog rewritability of d-sirups. Our main technical result is a complete and transparent syntactic ▪/NL/P/coNP tetrachotomy of d-sirups with disjoint covering classes and a path-shaped Boolean conjunctive query. To obtain this tetrachotomy, we develop new techniques for establishing P- and coNP-hardness of answering non-Horn ontology-mediated queries as well as showing that they can be answered in NL.

Subjective Information and Survival in a Simulated Biological System.

Information transmission and storage have gained traction as unifying concepts to characterize biological systems and their chances of survival and evolution at multiple scales. Despite the potential for an information-based mathematical framework to offer new insights into life processes and ways to interact with and control them, the main legacy is that of Shannon’s, where a purely syntactic characterization of information scores systems on the basis of their maximum information efficiency. The latter metrics seem not entirely suitable for biological systems, where transmission and storage of different pieces of information (carrying different semantics) can result in different chances of survival. Based on an abstract mathematical model able to capture the parameters and behaviors of a population of single-celled organisms whose survival is correlated to information retrieval from the environment, this paper explores the aforementioned disconnect between classical information theory and biology. In this paper, we present a model, specified as a computational state machine, which is then utilized in a simulation framework constructed specifically to reveal emergence of a “subjective information”, i.e., trade-off between a living system’s capability to maximize the acquisition of information from the environment, and the maximization of its growth and survival over time. Simulations clearly show that a strategy that maximizes information efficiency results in a lower growth rate with respect to the strategy that gains less information but contains a higher meaning for survival.

Weakly and locally positive Robinson theories

AbstractWe introduce the notions of weakly and locally positive Robinson theories. We give a characterization of weakly positive Robinson theories by the amalgamation property, and a syntactic characterization of locally positive Robinson theories.

On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

Analog reducibility

Abstract In this paper we introduce and characterize two ‘analog reducibility’ notions for $[0,1]$-valued oracles on $\omega $ obtained by applying the syntactic characterizations of Turing and enumeration reducibility in terms of (positive) relatively $\varSigma _1$ and $\varPi _1$ formulas to formulas in continuous logic (Ben Yaacov, Berenstein, Henson and Usvyatsov, 2008, Model Theory for Metric Structures, vol. 2 of London Mathematical Society Lecture Note Series, pp. 315–427. Cambridge University Press.). The resulting analog and analog enumeration degree structures, $\mathscr {D}_a$ and $\mathscr {D}_{ae}$, naturally extend $\mathscr {D}_T$ and $\mathscr {D}_e$ in a compatible way. To show that these extensions are proper we prove that a sufficiently generic total $[0,1]$-valued oracle does not ‘analog enumerate’ any non-c.e. discrete set and that a sufficiently generic positive $[0,1]$-valued oracle neither ‘analog enumerates’ a non-c.e. discrete set nor ‘analog computes’ a non-trivial total $[0,1]$-valued oracle. We also provide a characterization of the continuous degrees among $\mathscr {D}_{ae}$ as precisely $\mathscr {D}_e \cap \mathscr {D}_a$. Finally we characterize a generalization of r.i.c.e. relations to metric structures via $\varSigma _1$ formulas in the ‘hereditarily compact superstructure’, which was the original motivation for the concepts in this paper.

An operational guide to monitorability with applications to regular properties

Monitorability underpins the technique of runtime verification because it delineates what properties can be verified at runtime. Although many monitorability definitions exist, few are defined explicitly in terms of the operational guarantees provided by monitors, i.e. the computational entities carrying out the verification. We view monitorability as a spectrum, where the fewer guarantees that are required of monitors, the more properties become monitorable. Accordingly, we present a monitorability hierarchy based on this trade-off. For regular specifications, we give syntactic characterisations in Hennessy–Milner logic with recursion for its levels. Finally, we map existing monitorability definitions into our hierarchy. Hence, our work gives a unified framework that makes the operational assumptions and guarantees of each definition explicit. This provides a rigorous foundation that can inform design choices and correctness claims for runtime verification tools.

Belief change and 3-valued logics: Characterization of 19,683 belief change operators

In this work we introduce a 3-valued logic with modalities, with the aim of having a clear and precise representation of epistemic states, thus the formulas of this logic will be our epistemic states. Indeed, these formulas are identified with ranking functions of 3 values, a generalization of total preorders of three levels. In this framework we analyze some types of changes of these epistemic structures and give syntactical characterizations of them in the introduced logic. In particular, we introduce and study carefully a new operator called Cautious Improvement operator. We also characterize all operators that are definable in this framework.

CONTINUOUS SENTENCES PRESERVED UNDER REDUCED PRODUCTS

AbstractAnswering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

Semi-Oblivious Chase Termination: The Sticky Case

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is undecidable, it is natural to ask whether known well-behaved classes of TGDs, introduced in different contexts such as ontological reasoning, ensure decidability. We consider a prominent paradigm that led to a robust TGD-based formalism, called stickiness. We show that for sticky sets of TGDs, all-instances chase termination is decidable if we focus on the (semi-)oblivious chase, and we pinpoint its exact complexity: PSpace-complete in general, and NLogSpace-complete for predicates of bounded arity. These complexity results are obtained via a graph-based syntactic characterization of chase termination that is of independent interest.

Belief Changes and Cognitive Development: Doxastic Logic {mathsf {LCB}}

We present the logic {mathsf {LCB}} which is expressed in a propositional language constantly enriched by new atomic expressions. Our formal framework is the propositional doxastic logic {mathsf {KD45}} with the belief operator {mathcal {B}}, extended by the {mathcal {C}} operator, to be read it changes that ... . We describe the changing beliefs of an agent who uses progressively expanding language. The approach presented here allows us to weaken pragmatic objections to the so-called principle of negative retrospection accepted in {mathsf {KD45}} and the problem of logical omniscience. In what follows, we present the expanding propositional language used in our formalism, interpreted using an epistemic version of Kripke semantics. Next, we give a syntactic characterization of logic {mathsf {LCB}} and prove the soundness and completeness of {mathsf {LCB}} in respect to our semantics. Finally, we compare the idea of expanding language with the notion of agent awareness and we relate our formalism to two epistemic temporal logics.

Filtered Belief Revision and Generalized Choice Structures

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the choice structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of (a generalized notion of) choice structure in terms of belief revision by adding: (1) the possibility that an item of information'' might be discarded as not credible (thus dropping the AGM success axiom) and (2) the possibility that an item of information, while not accepted as fully credible, may still be taken seriously'' (we call such items of information allowable''). We establish a correspondence between generalized choice structures (GCS) and AGM belief revision; furthermore, we provide a syntactic characterization of the proposed notion of belief revision, which we call filtered belief revision.

Standard Breton, Traditional Dialects, and How They Differ Syntactically

This article provides a first attempt of a syntactic characterization of the different Breton varieties spoken in the twenty-first century. Standard Breton is addressed as one of the modern dialects spoken in Brittany, and its syntax is compared with that of traditional varieties. I first establish a baseline and inventory the syntactic parameters that differentiate the traditional dialects from each other: Kerne, Leon, Goelo, Treger (KLT in the West) and Gwenedeg (South East). I show that a robust body of syntactic variation characterizes traditional dialects. I next compare these with the Standard variety that emerged during the twentieth century, and show that if Standard Breton has original features of its own, it varies less with respect to traditional varieties than do traditional varieties among themselves.

Strong Equivalence for Epistemic Logic Programs Made Easy

Epistemic Logic Programs (ELPs), that is, Answer Set Programming (ASP) extended with epistemic operators, have received renewed interest in recent years, which led to a flurry of new research, as well as efficient solvers. An important question is under which conditions a sub-program can be replaced by another one without changing the meaning, in any context. This problem is known as strong equivalence, and is well-studied for ASP. For ELPs, this question has been approached by embedding them into epistemic extensions of equilibrium logics. In this paper, we consider a simpler, more direct characterization that is directly applicable to the language used in state-of-the-art ELP solvers. This also allows us to give tight complexity bounds, showing that strong equivalence for ELPs remains coNP-complete, as for ASP. We further use our results to provide syntactic characterizations for tautological rules and rule subsumption for ELPs.

An Operational Guide to Monitorability

Monitorability delineates what properties can be verified at runtime. Although many monitorability definitions exist, few are defined explicitly in terms of the guarantees provided by monitors, i.e., the computational entities carrying out the verification. We view monitorability as a spectrum: the fewer monitor guarantees that are required, the more properties become monitorable. We present a monitorability hierarchy and provide operational and syntactic characterisations for its levels. Existing monitorability definitions are mapped into our hierarchy, providing a unified framework that makes the operational assumptions and guarantees of each definition explicit. This provides a rigorous foundation that can inform design choices and correctness claims for runtime verification tools.

A modal extension of Jaśkowski’s discussive logic $\textbf{D}_\textbf{2}$

Abstract In Jaśkowski’s model of discussion, discussive connectives represent certain interactions that can hold between debaters. However, it is not possible within the model for participants to use explicit modal operators. In the paper we present a modal extension of the discussive logic $\textbf{D}_{\textbf{2}}$ that formally corresponds to an extended version of Jaśkowski’s model of discussion that permits such a use. This logic is denoted by $\textbf{m}\textbf{D}_{\textbf{2}}$. We present philosophical motivations for the formulation of this logic. We also give syntactic characterizations of the logic and propose a comparison with certain other modal systems. In particular, we prove that $\textbf{m}\textbf{D}_{\textbf{2}}$ is neither normal nor regular. On the basis of the axiomatization of $\textbf{D}_{\textbf{2}}$, we give an axiomatization of $\textbf{m}\textbf{D}_{\textbf{2}}$. We also give another axiomatization which is not based on the axiomatization of $\textbf{D}_{\textbf{2}}$. Furthermore, we give a natural Kripke-style semantics for $\textbf{m}\textbf{D}_{\textbf{2}}$ and prove the respective adequacy theorems.

Subrecursive neural networks

It has been known for discrete-time recurrent neural networks (NNs) that binary-state models using the Heaviside activation function (with Boolean outputs 0 or 1) are equivalent to finite automata (level 3 in the Chomsky hierarchy), while analog-state NNs with rational weights, employing the saturated-linear function (with real-number outputs in the interval [0,1]), are Turing complete (Chomsky level 0) even for three analog units. However, it is as yet unknown whether there exist subrecursive (i.e. sub-Turing) NN models which occur on Chomsky levels 1 or 2. In this paper, we provide such a model which is a binary-state NN extended with one extra analog unit (1ANN). We achieve a syntactic characterization of languages that are accepted online by 1ANNs in terms of so-called cut languages which are combined in a certain way by usual operations. We employ this characterization for proving that languages accepted by 1ANNs with rational weights are context-sensitive (Chomsky level 1) and we present explicit examples of such languages that are not context-free (i.e. are above Chomsky level 2). In addition, we formulate a sufficient condition when a 1ANN recognizes a regular language (Chomsky level 3) in terms of quasi-periodicity of parameters derived from its real weights, which is satisfied e.g. for rational weights provided that the inverse of the real self-loop weight of the analog unit is a Pisot number.