Class Of Kripke Frames Research Articles

On semantically labelled syntax trees and the non-existence of certain Sahlqvist formulae

Abstract We elaborate on semantically labelled syntax trees that provide a method of proving the non-existence of modal formulae satisfying certain syntactic properties and defining a given class of frames and use them to show that there are classes of Kripke frames that are definable by both non-Sahlqvist and Sahlqvist formulae, but the latter requires more propositional variables.

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

Second-order propositional modal logic: Expressiveness and completeness results

In this paper we advance the state-of-the-art on the application of second-order propositional modal logic (SOPML) in the representation of individual and group knowledge, as well as temporal and spatial reasoning. The main theoretical contributions of the paper can be summarised as follows. Firstly, we introduce the language of (multi-modal) SOPML and interpret it on a variety of different classes of Kripke frames according to the features of the accessibility relations and of the algebraic structure of the quantification domain of propositions. We provide axiomatisations for some of these classes, and show that SOPML is unaxiomatisable on the remaining classes. Secondly, we introduce novel notions of (bi)simulations and prove that they indeed preserve the interpretation of formulas in (the universal fragment of) SOPML. Then, we apply this formal machinery to study the expressiveness of Second-order Propositional Epistemic Logic (SOPEL) in representing higher-order knowledge, i.e., the knowledge agents have about other agents' knowledge, as well as graph-theoretic notions (e.g., 3-colorability, Hamiltonian paths, etc.). The final outcome is a rich formalism to represent and reason about relevant concepts in artificial intelligence, while still having a model checking problem that is no more computationally expensive than that of the less expressive quantified boolean logic.

Kripke semantics for provability logic GLP

A well-known polymodal provability logic GLP due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of GLP is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for GLP . First, we isolate a certain subsystem J of GLP that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for GLP are defined as the limits of chains of finite expansions of models for J . The techniques involves unions of n -elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic.

Theoremhood-preserving Maps Characterizing Cut Elimination for Modal Provability Logics

Propositional modal provability logics like G and Grz have arithmetical interpretations where □φ can be read as ‘formula φ is provable in Peano Arithmetic’. These logics are decidable but are characterized by classes of Kripke frames which are not first‐order definable. By abstracting the aspects common to their characteristic axioms we define the notion of a formula generation map F(P) in one propositional variable. We then focus our attention on the properly displayable subset of all (first‐order definable) Sahlqvist modal logics. For any logic L from this subset, we consider the (provability) logic LF obtained by the addition of an axiom based upon a formula generation map F(P) so that LF = L + F(P). The class of such logics includes G and Grz. By appropriately modifying the right introduction rules for □, we give (not necessarily cut‐free) display calculi for every such logic. We define the pseudo‐displayable subset of these logics as those whose display calculi enjoy cut‐elimination for sequents of the form ⊤ ⊢ φ for any formula φ. We then show that for any provability logic LF having a conservative tense extension, there is a map f on formulae such that LF is pseudo‐displayable if and only if f maps theorems of LF to theorems of the underlying logic L and vice versa. By using a standard renaming technique we can guarantee that there is a polynomial‐time translation from LF into L. All proofs are purely syntactic and show the versatility of display calculi since similar results using traditional Gentzen calculi are not possible for as broad a range of logics and require further conditions. Our maps generalize previously known maps from G into K4. An application of our results gives an O(n.log n)3) translation from the (‘second order’) provability logic Grz into a decidable subset of first‐order logic. Since each of our logics L is a Sahlqvist logic, it is first‐order definable, and hence each L has a translation into first‐order logic. Our results therefore show that all pseudo‐displayable logics LF are ‘essentially first‐order’ even though their characteristic axiom may not be first‐order definable.

Knowledge in multiagent systems

The semantic framework for the modal logic of knowledge due to Halpern and Moses provides a way to ascribe knowlegde to agents in distributed and multiagent systems. In this paper we study two special cases of this framework:full systemsandhypercubes. Both model static situtations in which no agents has any information about another agent's state. Full systems and hypercubes are an appropriate model for the initial configurations of many systems of interest. We establish a correspondence between full systems and hypercube systems and certain classes of Kripke frames. We show that these classes of systems correspond to the same logic. Moreover, this logic is also the same as that generated by the larger class ofweakly directed frames. We provide a sound and complete axiomatization, S5WDnof this logic, and study its computational complexity. Finally, we show that under certain natural assumptions, in a model where knowledge evolves over time, S5WDncharacteristics the properties of knowledge not just at the initial configuration, but also at all later configurations. In this particular, this holds forhomogeneous broadcast systems,which capture settings in which agents are intially ignorant of each others local states, operate synchronously, have perfect recall, and can communicate only by broadcasting.

Characterizations of Negative Definability in Modal Logic

Negative definability ([18]) is an alternative way of defining classes of Kripke frames via a modal language, one that enables us, for instance, to define the class of irreflexive frames. Besides a list of closure conditions for negatively definable classes, the paper contains two main theorems. First, a characterization is given of negatively definable classes of (rooted) finite transitive Kripke frames and of such classes defined using both traditional (positive) and negative definitions. Second, we characterize the negatively definable classes of rooted general frames.

Expressive power and semantic completeness: Boolean connectives in modal logic

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of ‘dense’ Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective).