This paper provides an algebraic background for the formal derivation of greedy-like algorithms. Such derivations have previously been done in various frameworks including relation algebra. We propose Kleene algebra as a particularly simple alternative. Instead of converse and residuation we use modal operators that are definable in a wide class of algebras, based on domain/codomain or image/pr...

One difference between natural and formal languages is that natural languages are ambiguous in many respects, one of them being the ambiguity of sentences with more than one logic operator. In this paper we focus on sentences containing negation and modal operators and their interpretation by 4and 5-years old. In many cases, a sentence with multiple operators is ambiguous and grammars of human ...

Modal logic is a family of logics for reasoning about modalities. For example, given a sentence φ, one may wish to express statements such as “φ is necessarily true” or “φ is possibly true”. Modalities are expressed using modal operators. In the most basic modal logic, called propositional modal logic, one may use the operators 2 and 3 to formalize the above two statements as 2φ and 3φ. In this...

This paper has two major goals. First, it provides a detailed empirical description of how the epistemic and metaphysical readings of the “non-root modal + Perf(ect) have” construction in English (e.g., John might have won the game yesterday) are expressed in Mandarin Chinese. The two readings both involve non-root modals for the past, i.e., non-root modals with either a past temporal perspecti...

We define a nontrivial variety of boolean algebras with operators such that every member of the variety is atomless. This shows that not every variety of boolean algebras with operators is generated by its atomic members, and thus establishes a strong incompleteness result in (multi-)modal logic.

Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several first-order monotonic modal correspondents, where each first-order correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for mono...

Modal μ-calculus is a modal logic with fixed-point operators and well-known in mathematics and computer scince. For example, many verification properties of a system are expressed by formulas of modal μcalculus in computer science. However some verification properties of a system can not be expressed by a formula of modal μ-calculus as we will show later, and quantifiers of first-order logic ar...

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibili...

Distributive lattices with operators (DLO) are a natural generalization of the notion of Boolean algebras with operators. An operator in a bounded distributive lattice A is a function f : An → A which preserves ∧ (or ∨) in each coordinate. In the last few years these classes of algebras have been actively investigated since they appear as algebraic counterpart of many logics. Some important con...

We consider a hierarchy of modal event calculi to represent and reason about partially ordered events. These calculi are based on the model of time and change of Kowalski and Sergot’s Event Calculus (EC ): given a set of event occurrences, EC allows the derivation of the maximal validity intervals (MVIs) over which properties initiated or terminated by those events hold. The formalisms we analy...