A Four-Valued Hybrid Logic with Non-dual Modal Operators

14 Hybrid logics

Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators , thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K , a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

We show that every extension of the basic hybrid logic with modal Sahlqvist axioms is complete. As a corollary of our approach, we also obtain the Beth property for a large class of hybrid logics. Finally, we show that the new completeness result cannot be combined with the existing general completeness result for pure axioms.

Hybrid logic is usually viewed as a variant of modal logic in which it is possible to refer to worlds. But when one moves beyond propositional hybrid logic to first or higher-order hybrid logic, it becomes useful to view it as a systematic modal language of rigidification. The key point is this: @ can be used to rigidify not merely formulas, but other types of symbol as well. This idea was first explored in first-order hybrid logic (without function symbols) where @ was used to rigidify the first-order constants. It has since been used in hybrid type-theory: here one only has function symbols, but they are of every finite type, and @ can rigidify any of them. This paper fills the remaining gap: it introduces a first-order hybrid language which handles function symbols, and allows predicate symbols to be rigidified. The basic idea is straightforward, but there is a slight complication: transferring information about rigidity between the level of terms and formulas. We develop a syntax to deal with this, provide an axiomatization, and prove a strong completeness result for a varying domain (actualist) semantics.

We generalize the notion of symmetries of propositional formulas in\nconjunctive normal form to modal formulas. Our framework uses the coinductive\nmodels and, hence, the results apply to a wide class of modal logics including,\nfor example, hybrid logics. Our main result shows that the symmetries of a\nmodal formula preserve entailment.\n

Information flow control is a long-studied approach for establishing non-interference properties of programs. For instance, it can be used to prove that a secret does not interfere with some computation, thereby establishing that the former does not leak through the latter. Despite their potential as a holy grail for security reasoning and their maturity within the literature, information flow type systems have seen limited adoption. In practice, information flow specifications tend to be excessively complex and can easily spiral out of control even for simple programs. Additionally, while non-interference is well-behaved in an idealized setting where information leakage never occurs, most practical programs must violate non-interference in order to fulfill their purpose. Useful information flow type systems in prior work must therefore contend with a definition of non-interference extended with declassification , which often offers weaker modular reasoning properties. We introduce structural information flow , which both illuminates and addresses these issues from a logical viewpoint. In particular, we draw on established insights from the modal logic literature to argue that information flow reasoning arises from hybrid logic , rather than conventional modal logic as previously imagined. We show with a range of examples that structural information flow specifications are straightforward to write and easy to visually parse. Uniquely in the structural setting, we demonstrate that declassification emerges not as an aberration to non-interference, but as a natural and unavoidable consequence of sufficiently general machinery for information flow. This flavor of declassification features excellent local reasoning and enables our approach to account for real-world information flow needs without compromising its theoretical elegance. Finally, we establish non-interference via a logical relations approach, showing off its simplicity in the face of the expressive power captured.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

Pomset logic introduced by Retor\'e is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. As for a sequent calculus formulation, it has not been known for either of these logics, and there are convincing arguments that such a sequent calculus in the usual sense simply does not exist for them. In an on-going work on semantics we discovered a system similar to Pomset logic, where a noncommutative connective is no longer self-dual. Pomset logic appears as a degeneration, when the class of models is restricted. Motivated by these semantic considerations, we define in the current work a semicommutative multiplicative linear logic}, which is multiplicative linear logic extended with two nonisomorphic noncommutative connectives (not to be confused with very different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets and show how this logic degenerates to Pomset logic. However, a more interesting problem than just finding yet another noncommutative logic is to find a sequent calculus for this logic. We introduce decorated sequents, which are sequents equipped with an extra structure of a binary relation of reachability on formulas. We define a decorated sequent calculus for semicommutative logic and prove that it is cut-free, sound and complete. This is adapted to "degenerate" variations, including Pomset logic. Thus, in particular, we give a variant of sequent calculus formulation for Pomset logic, which is one of the key results of the paper.

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

The present thesis focuses on the study of the mathematical structures underlying input/output logic. Building on insights from Abstract Algebraic Logic and on the recent introduction of subordination algebras and related structures as a semantic environment for input/output logic, the present thesis establishes a strong and systematic bridge between areas of logic which have been investigated for decades independently of each other, and by doing so, it opens new possibilities of interpretations and areas of applications for these formal frameworks. Motivated by a research program investigating ‘logic at work’ rather than ‘logic in isolation’, input/output logic has been introduced for modelling the interaction between logical inferences and other agency-related notions such as conditional obligations, goals, ideals, preferences, actions, and beliefs, in the context of the formalization of normative systems in philosophical logic and AI, but also in connection and combination with a very diverse range of issues, also relevant to present-day AI, spanning from formal argumentation, to causal reasoning and non-monotonic reasoning. The first contribution of this thesis is a generalization of the framework of input/output logic from classical propositional logic to selfextensional logics, a large class of logics which prominently includes those logical systems the consequence relation of which corresponds to a partial order on the algebras canonically associated with it. The various notions of permission systems, namely, negative permission, positive static permission and dynamic permission, are generalized and studied uniformly in the context of selfextensional logics, both in themselves, and in connection with normative systems. In the same context, the notion of dual negative permission system is introduced and studied. The second contribution of this thesis is a systematic study of the algebraic counterparts of normative, permission, and dual permission systems based on the algebras of the classes canonically associated with (fully) selfextensional logics. In particular, the proof of the main characterization result of the previous chapter is refined and streamlined, and further extended to characterize properties of various notions of permission systems also in relation to normative systems, by means of modal axioms in an expanded slanted signature which includes negative modal operators (intuitively representing prohibitions). Moreover, these characterization results are then applied to resolve different issues: from the logical characterizations of output operators for both normative and permission systems, to the dual characterizations of conditions on subordination algebra (resp. precontact algebras) and on their associated spaces; the algebraization of positive static permissions; the modal characterization of the notion of cross-coherence; the clarification of the relation between certain conditions on positive bi-subordination lattices and Dunn’s axioms for positive modal logic. The third contribution of this thesis generalizes the modal characterizations, achieved in the previous chapter, of finitely many conditions on normative and permission systems, as well as on subordination algebras and related structures, to those conditions corresponding to the infinite class of clopen-analytic inequalities. Conversely, Kracht (first-order) formulas are introduced as the class of those first order conditions on algebras with subordination, precontact, and dual precontact relations which correspond to clopen-analytic axioms; finally, as an application of these results, Celani’s dual characterization results between subordination lattices and subordination spaces is generalized to the environment of algebras with relations based on distributive lattices and the infinite class of inductive Kracht formulas.

We discuss the celebrated Blok-Esakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimplication.

Head-driven phrase structure grammar (HPSG) is one of the most prominent theories employed in deep parsing of natural language. Many linguistic theories are arguably best formalized in extensions of modal or dynamic logic (Keller, Feature logics, infinitary descriptions and grammar, 1993; Kracht, Linguistics Philos 18:401---458, 1995; Moss and Tiede, In: Blackburn, van Benthem, and Wolther (eds.) Handbook of modal logic, 2006), and HPSG seems to be no exception. Adequate extensions of dynamic logic have not been studied in detail, however; the most important aspect is the reference to sets of substructures. In this paper, an adequate extension is identified, and some important results are established: Satisfiability is highly undecidable, and model checking is shown to be in EXPTIME and PSPACE-hard. A fragment with polynomial time model checking procedures is identified; it is shown to cover considerable fragments of HPSG.

1 - Structures, Languages and Translations: the Structural Approach to Feature Logic

We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal µ-calculus extends basic modal logic with an operator for least fixed points. Least and inflationary fixed point operators have been studied and compared in other contexts, particularly in finite model theory, where it is known that the logics IFP and LFP that result from adding such fixed point operators to first order logic have equal expressive power. As we show, the situation in modal logic is quite different, as the modal iteration calculus (MIC) we introduce has much greater expressive power than the µ-calculus. Greater expressive power comes at a cost: the calculus is algorithmically much less manageable.