Relation-changing modal logics are extensions of the basic modal logic that allow changes to the accessibility relation of a model during the evaluation of a formula. In particular, they are equipped with dynamic modalities that are able to delete, add, and swap edges in the model, both locally and globally. We provide translations from these logics to hybrid logic along with an implementation. In general, these logics are undecidable, but we use our translations to identify decidable fragments. We also compare the expressive power of relation-changing modal logics with hybrid logics.
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $\exists y. \forall x. (\Box_x P(x,y) \supset\Diamond_y P(y,x))$. Like First order modal logic, TML is also "notoriously" undecidable, in the sense that even very simple fragments are undecidable. In this paper, we show the decidability of one interesting fragment, that of two variable TML. This is in contrast to two-variable First order modal logic, which is undecidable.
Lindström theorems characterize logics in terms of model-theoretic conditions such as Compactness and the Löwenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindström theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we …
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard translation of modal logic to first-order logic. This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two. The aim of this talk is to review recent work concerning these fragments and their popular extensions. When presenting the material spe…
Modal logic is a paradigm for several useful and applicable formal systems in computer science. It generally retains the low complexity of classical propositional logic, but notable exceptions exist in the domains of description, temporal, and spatial logic, where the most expressive formalisms have a very high complexity or are even undecidable. In search of computationally well-behaved fragments, clausal forms and other sub-propositional restrictions of temporal and description logics have been recently studied. This renewed interest on sub-propositional logics, which mainly focus on the com…
In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implication-negation fragment) of the nonsense logics B3, introduced by Bochvar, and H3, introduced by Halldén, are presented. These calculi are obtained by restricting in an appropriate way the application of the rules of a sequent calculus for classical propositional logic CPL. The nice symmetry between the provisos in the rules reveal the semantical relationship between these logics. The Soundness and Completeness theorems for both calculi are obtained, as well as the respective Cut elimination th…
We study the expressive power of the two-variable fragment of order-invariant first-order logic. This logic departs from first-order logic in two ways: first, formulas are only allowed to quantify over two variables. Second, formulas can use an additional binary relation, which is interpreted in the structures under scrutiny as a linear order, provided that the truth value of a sentence over a finite structure never depends on which linear order is chosen on its domain. We prove that on classes of structures of bounded degree, any property expressible in this logic is definable in first-order …
In this paper we introduce the convex fragment of Łukasiewicz Logic and discuss its possible applications in different learning schemes. Indeed, the provided theoretical results are highly general, because they can be exploited in any learning framework involving logical constraints. The method is of particular interest since the fragment guarantees to deal with convex constraints, which are shown to be equivalent to a set of linear constraints. Within this framework, we are able to formulate learning with kernel machines as well as collective classification as a quadratic programming problem.
We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain which can be compared wrt. equality. As the satisfiability problem for this logic is undecidable in general, in a previous work, we have introduced a family of local fragments that restrict quantification to neighbourhoods of a given reference point. We provide here the precise complexity characterisation of the satisfiability problem for the existential fragments of this local logic depending on the number of data values carried by each element and the radius of the…
We study the logic FO(~), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(~) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(~) with respect to the number of variables and the quantifier rank, and prove a dichotomy betwee…
MV-algebras are an algebraic semantics for Lukasiewicz logic and MV-algebras generated by a finite chain are Heyting algebras where the Godel implication can be written in terms of De Morgan and Moisil's modal operators. In our work, a fragment of trivalent Lukasiewicz logic is studied. The propositional and first-order logic is presented. The maximal consistent theories are studied as Monteiro's maximal deductive systems of the Lindenbaum-Tarski algebra, in both cases. Consequently, the adequacy theorem with respect to the suitable algebraic structures is proven.
We investigate the complexity of modal satisfiability for certain combinations of modal logics. In particular we examine four examples of multimodal logics with dependencies and demonstrate that even if we restrict our inputs to diamond-free formulas (in negation normal form), these logics still have a high complexity. This result illustrates that having D as one or more of the combined logics, as well as the interdependencies among logics can be important sources of complexity even in the absence of diamonds and even when at the same time in our formulas we allow only one propositional variab…
STIT (sees to it that) semantics is one of the most prominent tools in modal logic of agency, widely used among both philosophers and responsible AI scholars. STIT logic surveys the properties of agents seeing to it that some state of affairs holds without specifying concrete actions by which that state of affairs is guaranteed. In comparison with other multi-agent modal logics, the main advantage of STIT theories is expressive power. STIT logic allows to study not only statements about agents abilities to perform certain actions (as it is in variations of Coalition Logic or Propositional Dyna…
We investigate the combination of fragments of classical logic as a way of conservatively extending a given Boolean logic by the addition of new connectives, and we precisely characterize the circumstances in which such a combination produces the corresponding fragment of classical logic over the signature containing connectives from both fragments given as input. If the thereby produced combined fragment is only incompletely characterized by the components given as input, this means that connectives from one component need to interact with connectives from the other component, giving rise to …
We study fragments of dependence logic defined either by restricting the number k of universal quantifiers or the width of dependence atoms in formulas. We find the sublogics of existential second-order logic corresponding to these fragments of dependence logic. We also show that these both ways of defining fragments of dependence logic give rise to a hierarchy in expressive power with respect to k.
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…
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…
Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas. We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that: (i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear …
We study the satisfiability problem for the 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 nor the tree model property, to show decidability we introduce 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…
Using a recently introduced algebraic framework for the classification of fragments of first-order logic, we study the complexity of the satisfiability problem for several ordered fragments of first-order logic, which are obtained from the ordered logic and the fluted logic by modifying some of their syntactical restrictions.
In this paper we introduce a term calculus ${\cal B}$ which adds to the affine $λ$-calculus with pairing a new construct allowing for a restricted form of contraction. We obtain a Curry-Howard correspondence between ${\cal B}$ and the sub-structural logical system which we call "minimal Łukasiewicz logic", also known in the literature as the logic of hoops (a generalisation of MV-algebras). This logic lies strictly in between affine minimal logic and standard minimal logic. We prove that ${\cal B}$ is strongly normalising and has the Church-Rosser property. We also give examples of terms in ${…
We investigate the relation between the theory of the iterations in the sense of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for several logics. These logics are obtained from the restriction of set quantification in monadic second order logic to certain subsets like, e.g., finite sets, chains, and finite unions of chains. We show that these theories of the Shelah-Stupp iteration can be reduced to corresponding theories of the base structure. This fails for Muchnik's iteration.
During the last decades, a lot of effort was put into identifying decidable fragments of first-order logic. Such efforts gave birth, among the others, to the two-variable fragment and the guarded fragment, depending on the type of restriction imposed on formulae from the language. Despite the success of the mentioned logics in areas like formal verification and knowledge representation, such first-order fragments are too weak to express even the simplest statistical constraints, required for modelling of influence networks or in statistical reasoning. In this work we investigate the extensions…
This paper explores the computational complexity of various natural one-variable fragments of first-order modal logics with the addition of counting quantifiers, over both constant and varying domains. The addition of counting quantifiers provides us a rich language with which to succinctly express statements about the quantity of objects satisfying a given first-order property, using a single variable. Optimal NExpTime upper-bounds are provided for the satisfiability problems of the one-variable fragment of the minimal first-order modal logic QK, over both constant and expanding/decreasing do…
We consider a specific class of tree structures that can represent basic structures in linguistics and computer science such as XML documents, parse trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We present axiomatizations of the monadic second-order logic (MSO), monadic transitive closure logic (FO(TC1)) and monadic least fixed-point logic (FO(LFP1)) theories of this class of structures. These logics can express important properties such as reachability. Using model-theoretic techniques, we show by a uniform argument that these axiomatizations are complete, i.e., eac…
Horn description logics are syntactically defined fragments of standard description logics that fall within the Horn fragment of first-order logic and for which ontology-mediated query answering is in PTime for data complexity. They were independently introduced in modal logic to capture the intersection of Horn first-order logic with modal logic. In this paper, we introduce model comparison games for the basic Horn description logic hornALC (corresponding to the basic Horn modal logic) and use them to obtain an Ehrenfeucht-Fraïssé type definability result and a van Benthem style expressive co…
We investigate properties of monadic purely negational fragment of Intuitionistic Control Logic (ICL). This logic arises from Intuitionistic Propositional Logic (IPL) by extending language of IPL by additional new constant for falsum. Having two different falsum constants enables to define two forms of negation. We analyse implicational relations between negational monadic formulae and present a poset of non equivalent formulae of this fragment of ICL.
While reasoning in a logic extending a complete Boolean basis is coNP-hard, restricting to conjunctive fragments of modal languages sometimes allows for tractable reasoning even in the presence of greatest fixpoints. One such example is the EL family of description logics; here, efficient reasoning is based on satisfaction checking in suitable small models that characterize formulas in terms of simulations. It is well-known, though, that not every conjunctive modal language has a tractable reasoning problem. Natural questions are then how common such tractable fragments are and how to identify…
We compare the model-theoretic expressiveness of the existential fragment of Separation Logic over unrestricted relational signatures (SLR) -- with only separating conjunction as logical connective and higher-order inductive definitions, traditionally known as the symbolic heap fragment -- with the expressiveness of (Monadic) Second Order Logic ((M)SO). While SLR and MSO are incomparable on structures of unbounded treewidth, it turns out that SLR can be embedded in SO, in general, and that MSO becomes a strict subset of SLR, when the treewidth of the models is bounded by a parameter given as i…
Separation Logic (SL) is a well-known assertion language used in Hoare-style modular proof systems for programs with dynamically allocated data structures. In this paper we investigate the fragment of first-order SL restricted to the Bernays-Schoenfinkel-Ramsey quantifier prefix $\exists^*\forall^*$, where the quantified variables range over the set of memory locations. When this set is uninterpreted (has no associated theory) the fragment is PSPACE-complete, which matches the complexity of the quantifier-free fragment. However, SL becomes undecidable when the quantifier prefix belongs to $\ex…
Formalisms based on temporal logics interpreted over finite strict linear orders, known in the literature as finite traces, have been used for temporal specification in automated planning, process modelling, (runtime) verification and synthesis of programs, as well as in knowledge representation and reasoning. In this paper, we focus on first-order temporal logic on finite traces. We first investigate preservation of equivalences and satisfiability of formulas between finite and infinite traces, by providing a set of semantic and syntactic conditions to guarantee when the distinction between r…
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 stratify intuitionistic first-order logic over $(\forall,\to)$ into fragments determined by the alternation of positive and negative occurrences of quantifiers (Mints hierarchy). We study the decidability and complexity of these fragments. We prove that even the $Δ_2$ level is undecidable and that $Σ_1$ is Expspace-complete. We also prove that the arity-bounded fragment of $Σ_1$ is complete for co-Nexptime.
We study the satisfiability problem for the fluted fragment extended with transitive relations. We show that the logic enjoys the finite model property when only one transitive relation is available. On the other hand we show that the satisfiability problem is undecidable already for the two-variable fragment of the logic in the presence of three transitive relations.
Interval temporal logics provide a general framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders. We classify them in terms of both their relative expressive power and their complexity. We show that there are exactly 44 expressively different decidable fragments, whose complexity ranges from NP to EXPSPACE. In addit…
While finite-variable fragments of the propositional modal logic S5--complete with respect to reflexive, symmetric and transitive frames--are polynomial-time decidable, the restriction to finite-variable formulas for logics of reflexive and transitive frames yields fragments that remain "intractable." The role of the symmetry condition in this context has not been investigated. We show that symmetry either by itself or in combination with reflexivity produces logics that behave just like logics of reflexive and transitive frames, i.e. their finite-variable fragments remain intractable, namely …
We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove NP-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.
Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. It turned out that full propositional Linear Logic is undecidable (Lincoln, Mitchell, Scedrov, and Shankar) and, hence, it is more expressive than (modalized) classical or intuitionistic logic. In this paper we focus on the study of the simplest fragments of Linear Logic, such as the one-literal and constant-only fragments (the latter contains no literals at all). Here we demonstrate that all these extremely simple fragments of Linear Logic (one-literal, $\bot$-only, and even unit-only) are exactly of …
Metric Temporal Logic, $\mtlfull$ is amongst the most studied real-time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints $I$. \oomit{The classical results of Alur and Henzinger showed that $\mtlfull$ is undecidable where as $\mitl$ which uses only non-singular intervals $NS$ is decidable. In a surprizing result, Ouaknine and Worrell showed that the satisfiability of $\mtl$ is decidable over finite pointwise models, albeit with NPR decision complexity, whereas it remains …
Recently, the separated fragment (SF) of first-order logic has been introduced. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. SF properly generalizes both the Bernays-Schönfinkel-Ramsey (BSR) fragment and the relational monadic fragment. In this paper the restrictions on variable occurrences in SF sentences are relaxed such that universally and existentially quantified variables may occur together in the same atom under certain conditions. Still, satisfiability can be decided. This result is established in two ways: firstly, …
We characterise non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by a new notion of simulation called a meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs of states from one model to single states from another. En route to this result we use a more traditional notion of simulations and prove a Hennessy-Milner style theorem for it, using an analogue of modal saturation called meet-compactness.
Aix Marseille Université（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Vienna University of Technology（维也纳工业大学）
Recently, belief change within the framework of fragments of propositional logic has gained increasing attention. Previous works focused on belief contraction and belief revision on the Horn fragment. However, the problem of belief merging within fragments of propositional logic has been neglected so far. This paper presents a general approach to define new merging operators derived from existing ones such that the result of merging remains in the fragment under consideration. Our approach is not limited to the case of Horn fragment but applicable to any fragment of propositional logic charact…
In recent work, comonads and associated structures have been used to analyse a range of important notions in finite model theory, descriptive complexity and combinatorics. We extend this analysis to Hybrid logic, a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. In addition to characterising the various resource-indexed equivalences induced by Hybrid logic and the bounded fragment, and the associated combinatorial decompositions of structures, we also give model-theoretic characterisations of bounded formulas in terms of invariance…
