MODAL LOGIC

For starters, let us quote three slogans with which Patrick Blackburn, Maarten de Rijke and Yde Venema - the authors of one of the most influential textbooks of modal logic - characterize contemporary modal logic (see [15, 11]):

slogan 1: Modal languages are simple yet expressive languages for talking about relational structures.

slogan 2: Modal languages provide an internal, local perspective on relational structures.

slogan 3: Modal languages are not isolated formal systems.

Yet, since our aim is not to give a versatile overview of modal logics - in fact, the discussion on standard modal logics only serves further considerations on hybrid logics - let us concisely comment on all three slogans.

First, since the ground-breaking paper of Kripke ([63]) appeared in which he exposes his possible world semantics for modal logics, it has been quite clear that models for these logics can be seen as relational structures (or directed graphs). More precisely, a Kripke model for a modal logic M is a triple 〈W, R, V〉, where W is a non-empty set called a universe, R is a binary (or n-ary) relation over W called an accessibility relation, and V is a valuation function mapping propositional formulas to subsets of W. Elements of W are called worlds, states, points or nodes (in places where it does not lead to a confusion, we use all these names interchangeably). Primarily, elements of W were considered to be possible worlds which are accessible to each other through R. However, the contemporary modal logic departs from such strict and narrow alethic understanding of relational structures. In fact, we can interpret W and R arbitrarily: as states of a computer linked by the execution of an appropriate program (the case of the multi-modal logic PDL), as states of knowledge of an agent (the case of epistemic logics), or as obligated and permitted states of affairs (the case of deontic logics).

In the chapter, we provide tableau-based decision procedures for the logic M(En). In the existing literature, several approaches for deciding modal/description logics with counting operators can be found. [6] describes a decision procedure for modal logics with counting operators that exploits the translation function from the modal counting language to the hybrid language with the universal modality H(A). Tableau-based decision procedures for analogs of modal logics with counting operators were established in the field of description logics, where counting operators are known under the guise of cardinality constraints. Sound, complete and terminating tableau-calculi for these logics can be found in [65, 60, 31]. These calculi, in general, do not differ in the rules for cardinalities, however, they utilise different blocking mechanisms for ensuring termination, such as pairwise blocking or pattern-based blocking.

We exploit the framework from [84] to synthesise a sound, complete and terminating prefixed tableau calculus for M(En). We also provide a refinement of this calculus consisting in internalizing the semantics of the logic within the language of the logic. We only exploit global counting operators to dispense with meta-linguistic expressions like M, x ⊧ ϕ or R(x, y) to achieve a full internalization.

We also describe a successful implementation of the calculus using the MetTeL2 tableau prover generator [90, 1].

The chapter is structured as follows. In Section 8.1, we introduce a prefixed tableau calculus TpM(En) for the logic MEn. We also prove soundness and completeness of the calculus. Termination of TpM(En) is proven in Section 8.2. Section 8.3 is devoted to possible refinements of particular rules of TpM(En). The internalized version of the calculus named TiM(En) is presented in Section 8.4. We briefly comment on a successful implementation of TiM(En) on MetTeL2 prover. Section 8.6 is an extensive overview of other decision procedures for logics with global counting operators.

In the first place, let's recall and briefly recapitulate some basic facts about hybrid logics that we use in the present chapter. Various issues related to hybrid logics are discussed more thoroughly in Chapter 4. Hybrid logics are powerful extensions of modal logics which allow for referring to particular states of a model without using meta-language. In order to achieve it, the language of standard modal logics is enriched with the countably infinite set of propositional expressions called nominals (we fix the notation nom = {i, j, k, …} to stand for the set of nominals), disjoint from the set of propositional variables prop. Each nominal is true at exactly one world and therefore can serve both as a label and as a formula. Supplying a language with nominals significantly strengthens its expressive power. In the present chapter we consider a suitably modified hybrid logic obtained by adding the satisfaction operators, the universal modality, the difference modality and the inverse modality. As indicated in Chapter 4, the satisfaction operators of the form @i allow for stating that a particular formula holds at a world labelled by i, the universal modality E expresses the fact that there exists a world in a domain, at which a particular formula holds, the difference modality D stands for the fact that a particular formula holds at a world different from the current one and, eventually, the inverse modality allows us to “jump back” to a predecessor-world along the accessibility relation edges.

As we mention in Section 4.2 of the preceding chapter, some hybrid logics additionally contain a different sort of expressions, the state variables, which allow quantifying over worlds, and additional operators like the ↓ operator or ∃. However, these logics are proven to be undecidable ([3]) so, in principle, they cannot be subjected to a terminating tableau-based decision procedure. We therefore hold our word from Chapter 4.3 and confine ourselves only to the foregoing decidable hybrid logic.

This part introduces and explains some preliminary notions used throughout the book. The reader is assumed to have some basic knowledge of modal logics, however to a large extent this part of the book may be considered as self-contained. The main issues discussed are: expressivity, decidability and computational complexity of standard and non-standard hybrid logics.

The present chapter is more extensive than the previous one since the reported results are mostly new. They previously occurred in two papers by the author of the book: [102] and [104]

MOTIVATION

Ordinary modal logics do not distinguish between worlds satisfying the same formulas. It means that whenever a modal formula φ is satisfied by a model M, it is also satisfied by a model M′ which collapses identical states from M′ to only one state. If we reason about concrete domains represented by modal models and we want to express the fact that a particular formula (representing, e.g., a particular state of affairs) occurs in a domain, we need to augment the ordinary modal language with some additional operators. Introducing graded modalities of the form ◊>n, ◊<n, ◊=n allowed to express the cardinality of a set of a particular world's successors that satisfy a certain formula φ. However, graded modalities do not enable making global numerical restrictions. Only introducing global counting operators of the form E>n, E<n, and E=n fills this gap. The syntax and semantics of a language enriched with global counting modalities is presented in Section 5.2. In Section 5.3, it is shown that taking into account modal logics with global counting operators in the book devoted to hybrid logics is not a whim and that it is fully justified by the expressive power of such logics.

A few words of comment should be devoted to the problem of coding numerical subscripts in counting operators. Subscript n coded in unary can be regarded as a sequence of n consecutive 1s. The size of this sequence equals n. Subscript n coded in binary is a sequence of consecutive bits which can take values from the set {0, 1}, whose binary value equals n.

This chapter provides a concise survey on hybrid logics, including their expressive power, axiomatization, decidability and computational complexity. It also sketches a historical overview of the development of these logics.

MOTIVATION

Suppose that a modal logic M is defined over the signature 〈prop, R, R−〉, where prop is a denumerable set of propositional variables {p1, p2, …} and R− is the inverse of the relation R. In fact, we can consider M as a temporal logic, so R is a strict total order without endpoints. We can express the thought “in the future it will be cold” by a formula:

Fφ,

where φ stands for “it is cold”. As mentioned in Section 2.1, modal formulas do not distinguish between worlds satisfying formulas under the scope of a modal operator. Consequently, we do not know whether the F sends us three days or three years ahead. By means of the standard modal logic (or standard temporal logic), we cannot express the thought “on John's birthday it will be cold” or “on September 15th it will be cold” since these sentences contain direct pointers that distinguish particular worlds in a model (“John's birthday”, “September 15th”) and, as such, have some kind of global flavour. If we introduced an additional sort of expressions each of which would label exactly one point in a model, we would be able to express all the thoughts quoted above.

Hybrid logic introduces such expressions and calls them nominals. Each of them indeed holds at exactly one world in a model (although many of them can hold at the same world). Additionally, it augments the basic modal language with the so-called satisfaction operators @i that enable us to retrieve all information from a world labelled by i.

Hybrid logics are powerful extensions of modal logics, which, roughly speaking, allow for referring to the semantics of the logic by means of syntactic tools. Augmenting standard modal logics with expressions of an additional sort - nominals, and the so-called satisfaction operators caused a spectacular boost to their expressive power. The satisfaction relation, accessibility statements and equality assertions, hitherto only expressible in meta-language, became available within the object language of the logic. A natural question arose: how do these highly expressive logics behave in terms of decidability? How much must we sacrifice to the altar of this increased expressive power? However, whereas the sole origins of hybrid logics can be traced back to the 1960s and the works of Arthur Prior, some additional “computational” properties of these logics have become a subject of systematic research relatively recently. There has been a lot done since then, though. From the mid-1990s to the present times, a systematic theory of the computational complexity of hybrid logics has been developed by many researchers: Patrick Blackburn, Jerry Seligman, Carlos Areces, Maarten Marx, Thomas Schneider - just to mention a few. Up to now almost all the facts from this area of investigation have been revealed. Theorems placing particular hybrid logics in concrete complexity classes have been established for most of them, defined over most of the well-known frame classes. It seems that not much is left to be done.

This book stands at the intersection of two topics: the decidability and computational complexity of hybrid logics, and the deductive systems designed for them. As pointed out above, there are not many blanks to be filled left in the first field. Nonetheless, if we relax our definition of hybrid logics a bit, which captures them as modal logics involving an additional sort of expressions, a wide area for investigation opens.

This part is devoted to concrete deductive systems for hybrid logics. The originally established tableau algorithms involving the unrestricted blocking mechanism are presented and serve as a starting point for a comparative study of existing decision procedures for hybrid logics.

Since a major part of the book is devoted to concrete deductive systems for hybrid logics, providing a formal definition of a deductive system is in place. We use this notion in a similar way to Indrzejczak in [53], therefore, we repeat his very general definition of a deductive system.

Definition 3.1 (Deductive system ([53])). Every deductive system may be characterized on two elementary levels of description:

· the calculus,

· the realization.

By a calculus we understand a non-empty set of schemata of rules of the form:

X1, …,Xn/Y1, …,Ym, n ≥ 0, m ≥ 1,

with a possible list of side conditions. Symbols Xi, denote some data structures being transformed into data structures Yi.

By a realization we usually understand the set of instructions of how to apply rules to perform a derivation/construct a proof.

In the book, we describe instantiations of two basic types of deductive systems: tableau calculi and sequent calculi. We dedicate to each of these types a short, introductory note. We skip all historical remarks as hardly relevant for our considerations. We also waive the idea of a profound exposition of Hilbert-style calculi, even though several such calculi are presented in this book. We assume that the notion of Hilbert-style calculus is rather familiar to most readers and we consider its thorough examination as lying outside the scope of the book. Considerations in this section are based on [53, 52].

SEQUENT CALCULI

Originally invented by Gentzen in the 1930s, sequent calculi are finite sets of primitive rules of the form:

S1, …,Sn/Sn + 1,

where Si is a sequent schema.

By a sequent we understand an ordered pair of finite sequences of formulas with a symbol ⇒ between these sequences. Every sequent is therefore of the form:

φ1, …,φn ⇒ ψ1, …,ψmn, m ≥ 0.

This chapter is aimed to provide a concise overview of basic notions used in description logics. Description and modal (and also hybrid) logics have a lot in common. In fact, the former can be considered as a notational variant of the latter. For that reason, references to works from the description logic field occur throughout the whole book. A brief explanation of a description logic terminology we refer to in our considerations is therefore in order.

First, let's recall that the basic modal logic is defined over a sig-nature 〈 prop, R 〉, where prop is a denumerable set of propositional variables p1, p2, … and R is a binary relation over the universe of a model. Description logics use the notion of atomic concepts rather than propositional formulas. Let C0 denote a set of all atomic concepts of (an abstract) description logic D. Again, in description logics we talk about roles rather than about accessibility relations. Let R denote a set of all roles of D. We have all at hand to define a syntax of D.

Definition6.1 (Syntax of D). Let C0 be the set of atomic concepts and R be the set of roles in D. We define the set C of all concepts of the language of the description logic D as follows:

C := ⊤ | A | ⌝C | C ⊓ D | C ⊓ D | ∃R.C | ∀R. C | n ≥ R.C | n ≤ R.C,

where A ∈ C0, C, D ∈ C, R ∈ R and n ∈ ℕ.

Before we draw a parallel between concrete modal and description notions, let's define a model for D which is called an interpretation.

Definition 6.2 (Semantics of D).

RESULTS

The original results of this book split into two parts. This division reflects the division of the book itself. We can say that the first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, our efforts focused on hybrid logics referred to as non-standard in this book. By non-standard hybrid logics, we understand modal logics with global counting operators (M(En)) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise:

1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators (MK(En)), which can be easily extended onto other frame classes,

2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames (MK(En), MT(En), MD(En), MB(En), MK4(En)) with numerical subscripts coded in binary. Establishing the exponential-size model property for logics defined over the classes of Euclidean and equivalential frames (MK5(En), MS5(En)). In the second case, we currently lack complexity bounds which are tight. We only conjecture that the satisfiability problem for the modal logic with global counting operators over Euclidean and equivalential frames is NP-complete. Nevertheless, the computational complexity of non-standard hybrid logics turned out to be rather fixed and less dependant on the frame class it is defined over than standard hybrid logics.

Results of the second type consist of designing concrete deductive (tableau and sequent) systems for standard and non-standard hybrid logics. More precisely, they include:

1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics.

2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics.