Showing papers on "Modal operator published in 2009"

Logical Step-Indexed Logical Relations

Abstract: We show how to reason about "step-indexed" logical relations in an abstract way, avoiding the tedious, error-prone, and proof-obscuring step-index arithmetic that seems superficially to be an essential element of the method. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadi's logic for parametricity, but also supports recursively defined relations by means of the modal"later" operator from Appel et al.'s "very modal model" paper. We encode in LSLR a logical relation for reasoning(in-)equationally about programs in call-by-value System F extended with recursive types. Using this logical relation, we derive a useful set of rules with which we can prove contextual (in-)equivalences without mentioning step indices.

Modal and Temporal Aspects of Habituality

Abstract: In this paper, we explore the modal characteristics of habituality, and the relation of habituality to imperfectivity. We have already argued in previous work (Boneh and Doron 2008) for the existence of a habituality modal operator Hab which is independent of imperfective aspect. Here we defend this analysis further, in particular in the face of reductionist views such as Ferreira (2005), who treats Hab as reducible to imperfectivity of plural events, and Hacquard (2006), who treats imperfective aspect as reducible to modal operators such as Hab/Prog. The reductionist views of Ferreira and Hacquard seem natural for languages with imparfait-type morphology expressing both continuity and habituality, such as the Romance languages. For us, the existence of this type of morphology shows that it is indeed natural to present habituals as ongoing. Yet we do not believe in the reduction of Hab to imperfectivity, or vice-versa. Rather, we assume that the output of the modal operator Hab is the input to aspectual operators, normally the imperfective aspect, since Hab is stative, but not exclusively. We show that Hab can be the input to the perfective aspect as well. Thus, it is possible to separate habituality from imperfectivity.

A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics

Abstract: Graded modal logic is the formal language obtained from ordinary modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart---especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.

Coalgebraic Hybrid Logic

Abstract: 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.

Bialgebraic methods and modal logic in structural operational semantics

Abstract: Bialgebraic semantics, invented a decade ago by Turi and Plotkin, is an approach to formal reasoning about well-behaved structural operational semantics (SOS). An extension of algebraic and coalgebraic methods, it abstracts from concrete notions of syntax and system behaviour, thus treating various kinds of operational descriptions in a uniform fashion. In this paper, bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel, general approach to proving the compositionality of process equivalences for languages defined by structural operational semantics. To prove compositionality, one provides a notion of behaviour for logical formulas, and defines an SOS-like specification of modal operators which reflects the original SOS specification of the language. This approach can be used to define SOS congruence formats as well as to prove compositionality for specific languages and equivalences.

A Note on the Complexity of the Satisfiability Problem for Graded Modal Logics

Abstract: Graded modal logic is the formal language obtained from ordinary (propositional) modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart--especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.

Extended Full Computation-Tree Logic with Sequence Modal Operator: Representing Hierarchical Tree Structures

Abstract: An extended full computation-tree logic, CTLS*, is introduced as a Kripke semantics with a sequence modal operator. This logic can appropriately represent hierarchical tree structures where sequence modal operators in CTLS* are applied to tree structures. An embedding theorem of CTLS* into CTL* is proved. The validity, satisfiability and model-checking problems of CTLS* are shown to be decidable. An illustrative example of biological taxonomy is presented using CTLS* formulas.

Inference, Promotion, and the Dynamics of Awareness

Abstract: Classical epistemic logic describes implicit knowledge of agents about facts and knowledge of other agents, based on semantic information. The latter is produced by acts of observation or communication, that are described well by dynamic epistemic logics. What these logics do not describe, however, is how significant information is also produced by acts of inference – and key axioms of the system merely postulate “deductive closure”. In this paper, we take the view that all information is produced by acts, and hence we also need a dynamic logic of inference steps showing what effort on the part of the agent makes a conclusion explicit knowledge. Strong omniscience properties of agents should be seen not as static idealizations, but as the result of dynamic processes that agents engage in. This raises two questions: (a) how to define suitable information states of agents and matching notions of explicit knowledge, (b) how to define natural processes over these states that generate new explicit knowledge. To this end, we extend earlier epistemic “awareness models” into a dynamic system that includes acts of public observation, but also adding and dropping formulas from the currently ‘entertained’ set, we give a completeness theorem, and we show how this dynamics updates explicit knowledge. Similar ideas have been proposed before, but they were restricted to update with factual propositions; our new dynamic system applies to arbitrary formulas. We also extend our approach to multi-agent scenarios where awareness changes may happen privately. Finally, we mention further directions and related approaches. 1. The problem of omniscience: what is ‘missing in action’ The usual discussions of the problem of omniscience in epistemic logic revolve around the distribution axiom K(φ→ ψ) → (Kφ → Kψ). Is knowledge closed under drawing logical inferences? If it is, so the story goes, then we have idealized our knowing agents too much. But this story is misleading on two accounts. First, with the usual semantics of epistemic logic, the K operator really just describes implicit semantic information of the agent, which definitely has the preceding closure property. The point is rather that closure need not hold for a related, but different intuitive notion, viz. explicit “aware-that” knowledge Exφ, in some suitable sense to be defined. So, what we really need is not “epistemic logic bashing”, but a richer account of agents’ attitudes. Our first question, then, is how to define explicit knowledge. 2 Johan van Benthem and Fernando R. Velazquez-Quesada But there is more. The interesting issue is not whether explicit knowledge has deductive closure. It is rather: “what do agents have to do to make their implicit knowledge explicit?” Consider the premises Ex (φ → ψ),Exφ of the distribution axiom, saying that the agent explicitly knows both φ→ ψ and φ. These do implyKψ, that is, the agent knows ψ implicitly. But crucially, in order to make this information explicit, the agent has to do some work, namely, perform an act of “awareness raising” that leads to Exψ. Stated more syntactically, the usual implication Ex (φ→ ψ)→ (Exφ→ Exψ) contains a gap [ ]: Ex (φ→ ψ)→ (Exφ→ [ ]Exψ) and in that gap, we should place an action. Note that, then, the agent is no longer omniscient, but she is not defective either: with the right repertoire of actions available, she can do awareness raising as needed. This paper explores these ideas. We first introduce simple epistemic awareness models, with a specific interpretation of the syntactic component as the formulas ‘entertained’ by the agent – and recall their standard axiomatization. Then, we explore some proposals for defining explicit knowledge, picking one that we will work with. Next, we define basic dynamic actions that modify our models, and provide examples of their behaviour, alone and in combination. Representing the actions in the language yields a sound and complete logic that clarifies our initial issues. We also develop some formal properties and raise some open problems. Then we move to the multi-agent case, developing tools for private and even unconscious versions of our actions, that were public so far. Finally, we relate our proposal to earlier ones, and end with conclusions and further directions, in particular, toward agents with beliefs that are modified by default inferences. 2. A static system for different agent attitudes We assume that the reader is familiar with classical epistemic logic (EL for short). We have already stated our motivation for working with this: even though this system fails for its intended interpretation of ‘full-blooded knowledge’, like so many logical systems, it has turned out quite adequate for other, perhaps originally non-intended interpretations. In particular, it deals well with implicit knowledge of the semantic range kind (cf. van Benthem and Martinez (2008)). Now we extend the base language of EL. We add an operator Eφ saying that the agent “is aware of φ” (Fagin and Halpern, 1988) or, in less psychological terms, that she “entertains φ” as a matter of Inference, Promotion, and the Dynamics of Awareness 3 attention. Notice that this does not imply any attitude pro or con: the agent may believe φ, but also reject it. Stated in other, but related terms, “awareness of” does not imply “awareness that”. Definition 2.1 (Language L). Let P be a set of atomic propositions. Formulas φ of the epistemic awareness language L are given by φ ::= p | Eφ | ¬φ | φ ∧ ψ | Kφ with p ∈ P. Other Boolean connectives ∨,→,↔, as well the existential modal operator (K) are defined as usual. We will read formulas Eφ as “the agent entertains φ”, and formulas Kφ as “the agent knows φ implicitly”. The language is interpreted in epistemic models assigning to each agent in each world a set of formulas, representing the information she entertains. Definition 2.2 (Semantic model). An epistemic awareness model is a tuple M = 〈W,R,E, V 〉 where − 〈W,R, V 〉 is a standard epistemic model: a set or worlds W , an accessibility relation R ⊆ (W×W ), and a valuation V : P→ ℘(W ). − E : W → ℘(L) is the “entertain” function giving the formulas that the agent ‘has in mind’. E(w) is the entertained set at w. As usual, a pointed model (M,w) also has a distinguished world w. The semantic interpretation of formulas in L is entirely as expected: Definition 2.3. Let (M,w) be a pointed semantic model with M = 〈W,R,E, V 〉. Atomic propositions and boolean connectives are interpreted as usual; for Eφ and Kφ we have: (M,w) |= Eφ iff φ ∈ E(w) (M,w) |= Kφ iff for all u ∈W , Rwu implies (M,u) |= φ. On these models we can impose standard epistemic assumptions about the accessibility relation, such as reflexivity, transitivity, and symmetry. But these are orthogonal to our main concerns in this paper. It is easy to visualize how our mixed models work: Example 1.

Modal‐type orthomodular logic

Abstract: In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripkestyle semantic founded on Baer*-semigroups as in [22] (© 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)

Characterizing Fuzzy Modal Semantics by fuzzy multimodal systems with crisp accessibility relations

Abstract: In (1) the authors considered finitely-valued modal logics with Kripke style semantics where both propositions and the accessibility relation are valued over a finite residuated lattice. Un- fortunately, the necessity operator does not satisfy in general the nor- mality axiom (K). In this paper we focus on the case of finite chains, and we consider a different approach based on introducing a multi- modal logic where the previous necessity operator is replaced with a family, parametrized by truth values different from zero, of necessity operators each one semantically defined using the crisp accessibility relation given by the corresponding cut of the finitely-valued origi- nal accessibility relation. This multimodal logic is somehow more appealing than the original modal one because axiom (K) holds for each necessity operator. In this paper we axiomatize this multimodal logic and we prove that, in the case the starting residuated lattice is a finite BL chain, the modal and the multimodal languages have the same expressive power iff this algebra is an MV chain.

Nominals for everyone

Abstract: It has been recognised that the expressivity of description logics benefits from the introduction of non-standard modal operators beyond existential and number restrictions. Such operators support notions such as uncertainty, defaults, agency, obligation, or evidence, whose semantics often lies outside the realm of relational structures. Coalgebraic hybrid logic serves as a unified setting for logics that combine non-standard modal operators and nominals, which allow reasoning about individuals. In this framework, we prove a generic EXPTIME upper bound for concept satisfiability over general TBoxes, which instantiates to novel upper bounds for many individual logics including probabilistic logic with nominals.

Using modal logics to express and check global graph properties

Abstract: Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL∗ with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

Quantified Multimodal Logics in Simple Type Theory

Abstract: We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple experiments, using existing higher-order theorem provers, to demonstrate that the embedding allows automated proofs of statements in these logics, as well as meta properties of them.

Some new observations on 'because (of)'

Abstract: Because (of ) is ambiguous between a 'reason' and a 'plain cause' interpretation. Presenting a semantic analysis within the framework of Discourse Representation Theory, I argue that because (of ) always denotes a causal relation between causing facts and caused entites of various sorts and that its interpretational variance is dependent on the ontological nature of the caused entity. Finally, I point to a difference between sentential-complement because and nominal-complement because of with regard to their interaction with modals. Whereas both because and because of may outscope e.g. deontic necessity modals, only because may outscope epistemic modal operators.

A note on brute vs. institutional facts: Modal logic of equivalence up to a signature

Abstract: The paper investigates the famous Searlean distinction between "brute" and "institutional" concepts from a logical point of view. We show how the partitioning of the non-logical alphabet—e.g., into "brute" and "institutional" atoms—gives rise to interesting modal properties. A modal logic, called UpTo-logic, is introduced and investigated which formalizes the notion of (propositional) logical equivalence up to a given signature.

Automated reasoning techniques for hybrid logics

Abstract: Hybrid logics augment classical modal logics with machinery for describing and reasoning about identity, which is crucial in many settings. Although modal logics we would today call ``hybrid'' can be traced back to the work of Prior in the 1960's, their systematic study only began in the late 1990's. Part of their interest comes from the fact they fill an important expressivity gap in modal logics. In fact, they are sometimes referred to as ``modal logics with equality''. One of the unifying themes of this thesis is the satisfiability problem for the arguably best-known hybrid logic, H(@,dwn), and some of its sublogics. Satisfiability is the basic problem in automated reasoning. In the case of hybrid logics it has been studied fundamentally using the tableaux method. In this thesis we attempt to complete the picture by investigating satisfiability for hybrid logics using first-order resolution (via translations) and variations of a resolution calculus that operates directly on hybrid formulas. We present firstly several satisfiability-preserving, linear-time translations from H(@,dwn) to first-order logic. These are conceived in a way such that they tend to reduce the search space of a resolution-based theorem prover for first-order logic. We then move our attention to resolution-based calculi that work directly on hybrid formulas. In particular, we will consider the so-called direct resolution calculus. Inspired by first-order logic resolution, we turn this calculus into a calculus of ordered resolution with selection functions and prove that it possesses the reduction property for counterexamples from which it follows its completeness and that it is compatible with the well-known standard redundancy criterion. We also show that certain refinement of this calculus constitutes a decision procedure for H(@), a decidable fragment of H(@,dwn). In the last part of this thesis we investigate certain normal forms for hybrid logics and other extended modal logics. We are interested in normal forms where certain modalities can be guaranteed not to occur under the scope of other modal operators. We will see that these kind of transformations can be exploited in a pre-processing step in order to reduce the number of inferences required by a modal prover. In an attempt to formulate these results in a way that encompasses also other extended modal logics, we arrived at a formulation of modal semantics in terms of a novel type of models that are coinductively defined. Many extended modal logics (such as hybrid logics) can be defined in terms of classes of coinductive models. This way, results that had to be proved separately for each different language (but whose proofs were known to be mere routine) now can be proved in a general way.

Lukasiewicz's 4-valued logic and normal modal logics.

Abstract: In this paper, we investigate the Lukasiewicz’s 4-valued modal logic based on the Aristotele’s modal syllogistic. We present a new interpretation of the set of algebraic truth values by introducing the truth and knowledge orderings similar to those in Belnap’s 4-valued bilattice but by replacing the original Belnap’s negation with the lattice pseudo-complement instead. Based on it, we develop a formal modal Boolean algebra for Lukasiewicz’s system. We show that this modal algebra corresponds to the standard normal modal logic and develop an autoreferential Kripke-style semantics for it, where Lukasiewicz/Aristotele’s ”necessity” operator is an existential additive instead of an universal (standard) multiplicative modal operator. Moreover, we show that the standard (modern) necessity modal operator based on S4 accessibility relation (the truth partial order in Lukasiewicz/Belnap’s bilattice) is an identity and consequently is not useful in Lukasiewicz’s system.

Assessing the modality particles of the Yi group in fuzzy possible-worlds semantics

Abstract: Of late, evidentiality has received great attention in formal semantics. In this paper I develop ‘evidentiality-informed’ truth conditions for modal operators such as must and may. With language data drawn from Luoping Nase (a Tibeto-Burman language spoken in the P.R. of China and belonging to the Yi Nationality), I illustrate that epistemic modals clash with clauses articulating first-hand information. I then demonstrate that existing models such as Kratzer’s graded possible-worlds semantics fail to provide accurate truth conditions for modals tagging clauses with first-hand information. As a remedy I propose a fuzzy version of possible-worlds semantics with various grades of belief and knowledge. In addition to preserving the expressive power of graded possible-worlds semantics, the fuzzy model will be shown to supply appropriate truth conditions for epistemic modals appended to evidential clauses (i.e. clauses expressing first-hand information).

Franchement et personnellement : deux attitudes énonciatives, deux moments de l'énonciation

Abstract: The aim of this paper is to propose a comparative analysis of the two French sentence adverbs franchement and personnellement, within a polyphonic framework (see Anscombre-Ducrot 1983). Our claim is that each of these adverbs qualifies the utterance in which itoccurs in a different way : while the first one modifies the act of saying, the second works asa modal operator, more specifically, as a domain adverb of the speaker’s assumption carriedby the proposition. As we shall see, both syntactic and semantic lexical properties of these adverbs provide two different types of polyphonic structures, that is to say, franchement andpersonnellement both work as polyphonic markers that express different kinds of relations between the speaker of an utterance and the discursive roles it convokes : these relations are exactly what we are intending to account for here.

An Analytic Logic of Aggregation

Abstract: We present a modular approach to the logic of aggregated group preferences based on hybrid modal logic. The modularity of the system is twofold: 1) lifting preference relations between states to complex relations between propositions and 2) lifting individual preferences to group preferences. The preferences may be doxastic or proairetic, generating a logic of aggregated belief or aggregated desire, respectively, using a specific aggregation policy known as `lexicographic re-ordering'. Each agent and each group of agents has an associated modal operator representing their preferences between states. The addition of the existential modality and nominals allows us to produce, first, a Hilbert-style axiomatization of the logic and then a more thorough analysis of inference using a Gentzen-style sequent calculus, in which the role of each operator is revealed.

Modal logic programming revisited

Abstract: We present optimizations for the modal logic programming system MProlog, including the standard form for resolution cycles, optimized sets of rules used as meta-clauses, optimizations for the version of MProlog without existential modal operators, as well as iterative deepening search and tabulation. Our SLD-resolution calculi for MProlog in a number of modal logics are still strongly complete when resolution cycles are in the standard form and optimized sets of rules are used. We also show that the labelling technique used in our direct approach is relatively better than the Skolemization technique used in the translation approaches for modal logic programming.

Implementation of a cut−free sequent calculus for logics with adjoint modalities

Abstract: Sadrzadeh and Dyckhoff describe in [1] a cut-free sequent calculus for logics with adjoint pairs of modal operators. We give here a Prolog implementation of a decision procedure for this calculus and describe the simple mechanism for loop checking used to guarantee termination, which requires slight modification of some of the inference rules of the calculus.

The ontogeny of complex verb phrases

Abstract: This paper investigate the acquisition of V-complement constructions (complex VPs) by English-speaking children ca age 1;8-to-2;9 It suggests that the child acquires these constructions during intensive epistemic or deontic modal negotiations with the adult In the earliest stage, the main-plus-complement construction is spread over adjacent child-adult or adult-child conversational turns ( Ochs et al 1979) The early precursor of the complex VP construction is thus paratactic , with the two clauses falling under separate intonation contours Only later on is the construction condensed into a complex syntactic construction under a single intonation contour, produced by the child alone The early use of these constructions is as direct speech acts , be they epistemic or deontic (Diessel 2005), whereby the semantic focus resides in the complement clause, and the main clause acts as a modal operator But this is true of both the children and their adult interlocutors, and is also characteristic, at the text-frequency level, of adult oral language (Thomson 2001) However, this characterization of complex VPs is semantic rather than syntactic

Optimization of Bounded Model Checking

Abstract: This paper optimizes the encoding of verifying G(p) and G(p→F(q)) which are two important and frequently used modal operators in optimization of encoding for bounded model checking (BMC). Through analysis of the properties of finite state machine (FSM) and LTL (linear-time temporal logic) when verifying these modal operators, it presents a concise recursive formula, which can efficiently translate BMC instances into SAT (satisfiability) instances. The logical properties of these recursion formulas are verified. The experimental comparison between the optimization of BMC and the other two important methods AA_BMC and Timo_BMC for solving these modal operators in BMC shows that the former is superior to the latter in both the scale of instances and the difficulty to solve the problem. Research of this paper is also beneficial to encoding optimization of verifying other modal operators in BMC.

An Epistemic Logic Characterizing Understanding

Abstract: In this paper,we firs give a claim from two researches that many people think to understand a proposition is to know its meaning.Then we point out that in classical epistemic logic,as a modal operator,understanding can be represented by knowledge in a weak sense as understanding a proposition is to know its truth.In the end,we give an epistemic logic characterizing understanding in order to reveal the connections and differences between understanding and knowledge.

Fictions within fictions

Abstract: This paper examines the logic of fictions within fictions I argue that consistently nested consistent fictions must have certain formal characteristics The most important is that they form a tree structure Depending on one’s theory of fictional objects, additional constraints may apply regarding the appearance of a fictional object in two or more fictional universes The background motivation for the paper is to use iterated fiction operators as a tool for making sense of iterated modal operators; I conclude by noting briefly where the results about nested fictions can, and where they cannot, be extended to nested possible worlds

The Research of Dynamic BDI Model

Abstract: BDI (belief, desire, intention) is the theoretic model of agent, this paper will introduce some non-normal modal operators for the study of the BDI semantic, which are unknown–objective -belief, known-objective -belief, realizable-subjective -belief, known- subjective-belief, h-achieved-desire and u-achieved-desire. And there is no logical omniscience problem that usually exists in the normal modal logic, and the BDI model will be evolving.