The Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all positive existential formulas, can express the first-order translations of basic modal logic and of many description logics, along with many sentences that arise in databases. It has been shown that the syntax of GNFO is restrictive enough so that computational problems such as validity and satisfiability are still decidable. This suggests that, in spite of its expressive power, GNFO formulas are amenable to novel optimizations. In this paper we study the model theory of GNFO formulas. Our results include effective preservation theorems for GNFO, effective Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to a large class of GNFO sentences within very restricted logics.
This version of the paper contains streamlined and corrected versions of results concerning entailment of a conjunctive query from a set of ground facts and a theory consisting of GNFO sentences of a special form ("dependencies").
We consider an extension of the unary negation fragment of first-order logic in which arbitrarily many binary symbols may be required to be interpreted as equivalence relations. We show that this extension has the finite model property. More specifically, we show that every satisfiable formula has a model of at most doubly exponential size. We argue that the satisfiability (= finite satisfiability) problem for this logic is TwoExpTime-complete. We also transfer our results to a restricted variant of the guarded negation fragment with equivalence relations.
We develop a doubly-exponential decision procedure for the satisfiability problem of guarded separation logic -- a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the above connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly-ex…
This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclu…
University of Bremen（不来梅大学）；University of Liverpool（利物浦大学）
We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the gu…
We show that the finite satisfiability problem for the unary negation fragment with arbitrary number of transitive relations is decidable and 2-ExpTime-complete. Our result actually holds for a more general setting in which one can require that some binary symbols are interpreted as arbitrary transitive relations, some as partial orders and some as equivalences. We also consider finite satisfiability of various extensions of our primary logic, in particular capturing the concepts of nominals and role hierarchies known from description logic. As the unary negation fragment can express unions of…
University of Hildesheim（希尔德斯海姆大学）；Leipzig University（莱比锡大学）；University of Liverpool（利物浦大学）
Finding a logical formula that separates positive and negative examples given in the form of labeled data items is fundamental in applications such as concept learning, reverse engineering of database queries, generating referring expressions, and entity comparison in knowledge graphs. In this paper, we investigate the existence of a separating formula for data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and the following important fragments thereof: the description logic $\mathcal{ALCI}$, the guarded fragment,…
In [1], systems of weakening of intuitionistic negation logic called Z_n and CZ_n were developed in the spirit of da Costa's approach(c.f. [2]) by preserving, differently from da Costa, its fundamental properties: antitonicity, inversion and additivity for distributive lattices. However, according to [3], those systems turned out to be not paraconsistent but extensions of intuitionistic logic. Taking into account of this result, we shall here make some observations on the modified systems of Z_n and CZ_n, that are paraconsistent as well.
The semijoin algebra is the variant of the relational algebra obtained by replacing the join operator by the semijoin operator. We discuss some interesting connections between the semijoin algebra and the guarded fragment of first-order logic. We also provide an Ehrenfeucht-Fraisse game, characterizing the discerning power of the semijoin algebra. This game gives a method for showing that certain queries are not expressible in the semijoin algebra.
We translate the argumentation networks ${\cal A}=(S, R)$ into a theory $D$ of intuitionistic logic, retaining $S$ as the domain and using intuitionistic negation to model the attack $R$ in ${\cal A}$: the attack $xRy$ is translated to $x\to\neg y$. The intuitionistic models of $D$ characterise the complete extensions of ${\cal A}$. The reduction of argumentation networks to intuitionistic logic yields, in addition to a representation theorem, some additional benefits: it allows us to give semantics to higher level attacks, where an attack "$xRy$" can itself attack another attack "$uRv$"; one …
Universidade da Coruña（拉科鲁尼亚大学）；University of Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）；Universität Potsdam（波茨坦大学）；Universidad Politécnica de Madrid（马德里理工大学）
A common feature in Answer Set Programming is the use of a second negation, stronger than default negation and sometimes called explicit, strong or classical negation. This explicit negation is normally used in front of atoms, rather than allowing its use as a regular operator. In this paper we consider the arbitrary combination of explicit negation with nested expressions, as those defined by Lifschitz, Tang and Turner. We extend the concept of reduct for this new syntax and then prove that it can be captured by an extension of Equilibrium Logic with this second negation. We study some proper…
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for h…
Reactive synthesis is an automatic way to translate a human intention expressed in some logic into a system of some kind. This thesis has two parts, devoted to logic and to systems. In Part I, we develop two new approaches to CTL* synthesis. The first approach consists of two extensions of the SMT-based bounded synthesis: one follows bottom-up CTL* model checking, another one follows the automata framework. The second approach reduces CTL* synthesis to LTL synthesis. The reduction turns any LTL synthesiser into a CTL* synthesiser. The two approaches were implemented and are available online. I…
In this paper we consider Epistemic Logic Programs, which extend Answer Set Programming (ASP) with "epistemic operators" and "epistemic negation", and a recent approach to the semantics of such programs in terms of World Views. We propose some observations on the existence and number of world views. We show how to exploit an extended ASP semantics in order to: (i) provide a characterization of world views, different from existing ones; (ii) query world views and query the whole set of world views.
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable remains free in the quantified formula. The fragment is closed under Boolean operations, but additional restrictions (called uniformity conditions) apply to combinations of atomic formulae with two or more variables. We argue that the notions of one-dimensionality and uniformity together offer a novel perspective on the robust decidability of modal logics. We also…
It is common in various non-classical logics, especially in relevant logics, to characterize negation semantically via the operation known as Routley star. This operation works well within relational semantic frameworks based on prime theories. We study this operation in the context of "information-based" semantics for which it is characteristic that sets of formulas supported by individual information states are theories that do not have to be prime. We will show that, somewhat surprisingly, the incorporation of Routley star into the information-based semantics does not lead to a collapse or …
University of Athens（雅典大学）；University of Szeged（塞格德大学）
Extensional higher-order logic programming has been introduced as a generalization of classical logic programming. An important characteristic of this paradigm is that it preserves all the well-known properties of traditional logic programming. In this paper we consider the semantics of negation in the context of the new paradigm. Using some recent results from non-monotonic fixed-point theory, we demonstrate that every higher-order logic program with negation has a unique minimum infinite-valued model. In this way we obtain the first purely model-theoretic semantics for negation in extensiona…
University of Liverpool（利物浦大学）；University of Bremen（不来梅大学）
We study the complexity of ontology-mediated querying when ontologies are formulated in the guarded fragment of first-order logic (GF). Our general aim is to classify the data complexity on the level of ontologies where query evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several large and relevant fragments of GF that enjoy a dichotomy between PTime and coNP, some of them additionally admitting a form of counting. In fact, almost all on…
The introduction of explicit notions of rejection, or disbelief, into logics for knowledge representation can be justified in a number of ways. Motivations range from the need for versions of negation weaker than classical negation, to the explicit recording of classic belief contraction operations in the area of belief change, and the additional levels of expressivity obtained from an extended version of belief change which includes disbelief contraction. In this paper we present four logics of disbelief which address some or all of these intuitions. Soundness and completeness results are sup…
We consider the family of guarded and unguarded ordered logics, that constitute a recently rediscovered family of decidable fragments of first-order logic (FO), in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. While the complexities of their satisfiability problems are now well-established, their model theory, however, is poorly understood. Our paper aims to provide some insight into it. We start by providing suitable notions of bisimulation for ordered logics. We next employ bisimulations to compare the rela…
We consider certain infectious logics (Sfde, dSfde, K3w, and PWK) and several their non-infectious modifications, including two new logics, reformulate previously constructed natural deduction systems for them (or present such systems from scratch for the case of new logics) in way such that the proof of normalisation theorem becomes possible for these logics. We present such a proof and establish the negation subformula property for the logics in question.
The paper proposes a new type of negation in multi-valued logics, providing a different way to answer the following question: what does it mean that some object language formula does not have a given truth-value. Along the way, the paper provides a general definition of truth and falsity in an arbitrary many-valued logic.
We consider the satisfiability problem for the two-variable fragment of first-order logic over finite unranked trees. We work with signatures consisting of some unary predicates and the binary navigational predicates child, right sibling, and their respective transitive closures. We prove that the satisfiability problem for the logic containing all these predicates is EXPSPACE-complete. Further, we consider the restriction of the class of structures to singular trees, i.e., we assume that at every node precisely one unary predicate holds. We observe that the full logic and even for unordered t…
Logic programming has developed as a rich field, built over a logical substratum whose main constituent is a nonclassical form of negation, sometimes coexisting with classical negation. The field has seen the advent of a number of alternative semantics, with Kripke-Kleene semantics, the well-founded semantics, the stable model semantics, and the answer-set semantics standing out as the most successful. We show that all aforementioned semantics are particular cases of a generic semantics, in a framework where classical negation is the unique form of negation and where the literals in the bodies…
Argumentation has proved a useful tool in defining formal semantics for assumption-based reasoning by viewing a proof as a process in which proponents and opponents attack each others arguments by undercuts (attack to an argument's premise) and rebuts (attack to an argument's conclusion). In this paper, we formulate a variety of notions of attack for extended logic programs from combinations of undercuts and rebuts and define a general hierarchy of argumentation semantics parameterised by the notions of attack chosen by proponent and opponent. We prove the equivalence and subset relationships …
We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logic…
We extend answer set semantics to deal with inconsistent programs (containing classical negation), by finding a ``best'' answer set. Within the context of inconsistent programs, it is natural to have a partial order on rules, representing a preference for satisfying certain rules, possibly at the cost of violating less important ones. We show that such a rule order induces a natural order on extended answer sets, the minimal elements of which we call preferred answer sets. We characterize the expressiveness of the resulting semantics and show that it can simulate negation as failure, disjuncti…
Hilbert's Entscheidungsproblem has given rise to a broad and productive line of research in mathematical logic, where the classification process of decidable classes of first-order sentences represent only one of the remarkable results. According to the criteria used to identify the particular classes of interest, this process was declined into several research programs, of which some of the most deeply investigated are the ones classifying sentences in prenex normal form in base of their prefix vocabulary. Unfortunately, almost all of these approaches did not shed any light on the reasons why…
Logical systems with classical negation and means for sentential or propositional self-reference involve, in some way, paradoxical statements such as the liar. However, the paradox disappears if one replaces classical by an appropriate non-classical negation such as a paraconsistent one (no paradox arises if the liar is both true and false). We consider a non-Fregean logic which is a revised and extended version (Lewitzka 2012) of Epsilon-T-Logic originally introduced by (Straeter 1992) as a logic with a total truth predicate and propositional quantifiers. Self-reference is achieved by means o…
