DS-2000-05:
Areces, Carlos
(2000)
*Logic Engineering. The Case of Description and Hybrid Logics.*
Doctoral thesis, University of Amsterdam.

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## Abstract

As the title indicates, there are two levels involved in the research

carried out in this thesis: the general issue of understanding (and

promoting) Logic Engineering, together with a detailed study of its

particular instantiation for Description and Hybrid Languages.

For some years now, a trend has been developing in the field of

computational logic: given the wide diversity of applications the

field has advanced into (theorem proving, software and hardware

verification, computational linguistics, knowledge representation,

etc.), a multiplicity of formal languages has been developed, offering

a wealth of alternatives to classical languages. With the advantages

of the diversity of choice, comes its complexity. How do we decide

what the best formalism is for a given reasoning or modeling task? Or

even more, what are the important rules to take into account when

designing yet another formal language? How do we compare, how do we

measure, how do we test? These are the questions that the young field

of Logic Engineering is supposed to investigate and, if possible,

answer.

What we know about Logic Engineering is still not a lot, and as yet

there are no general answers to these questions. Don't expect to find

a list of ``recipes'' of how things should be done here. But much can

be learned from analyzing in detail a particularly interesting case.

This will be the main thrust of the work carried out in the thesis.

Description logics are a family of formal languages used for

structured knowledge representation. They have been designed as a

tool to describe information in terms of concepts and their

interrelation (definitions), together with means to specify that

certain elements of the domain actually fit such definitions

(assertions). In addition, they provide a formal notion of inference

in terms of this structured knowledge. Description logics constitute

the best example we are aware of, of a broad, homogeneous collection

of formal languages with a clearly specified semantics (in terms of

first-order models) devised to deal with particular applications.

They offer an assortment of specialized inference mechanisms to handle

tasks like knowledge classification, structuring, etc. The complexity

of reasoning in the different languages of this family has been widely

investigated, theorem provers effectively deciding some of the most

expressive languages have been implemented (and they are among the

fastest provers for non-classical languages available), and these

languages have been successfully applied in many realistic problems,

even at an industrial level. Connections between description

languages and modal logics have been investigated, but a unifying

logical background theory explaining their expressive power and

logical characteristics was largely missing. This is the role to be

played by hybrid logics.

Hybrid languages are modal languages extended with the ability to

explicitly refer to elements in the domain of a model. They were

first introduced in the mid 1960s, in the field of temporal logic, and

were subsequently developed mainly in a purely theoretical

environment. The work in the field focused on investigating complete

axiomatizations for these languages, characterizing their meta-logical

properties and understanding their semantic and proof-theoretical

behavior.

Hybrid languages provide the exact kind of expressive power required

to match description languages. Having been optimized for

applications, description logics are difficult to handle with

classical model- and proof-theoretical tools, but given the close

match between description and hybrid logics we will be able to apply

these techniques to the hybrid logic counterpart of description logics

instead. Going in the other direction, description logics provide

hybrid logics with extensively tested examples of useful languages,

knowledge management lore, and implementations. In this thesis we will

draw these two complementary fields together and investigate in detail

what each of them has to offer to the other. Given that the two areas

have developed different techniques and evolved in divergent

directions, ``trading'' between them will be especially

fruitful. Description logics can export reasoning methods, complexity

results and application opportunities; while hybrid logics have their

model-theoretical tools, axiomatizations and analyses of expressive

power to offer.

The particular aim of this thesis is, then, to explore and exploit the

connections between description and hybrid logic, their similarities

and differences. The main results we will present specifically

concern this issue. But we hope to take the first steps in

setting and discussing this work in the wider perspective of logic

engineering, and provide a small contribution to the general issue of

better understanding the rules behind the good design of new formal

languages.

The thesis is organized in four parts. In the first, containing

Chapter 1, we discuss different ways of identifying

interesting fragments (and fragments of extensions) of first-order

logic. We argue that traditional methods, like prenex normal form and

finite variable fragments, are not completely satisfactory. We

propose, instead, to capture relevant fragments _via translations_.

The semantics of many formal languages (including

modal, description and hybrid languages) can be given in terms of

classical logics, and as such they can be considered fragments of

classical languages. But now, these fragments come together with an

extremely simple presentation --- modal languages, for example, are

usually introduced as extensions of propositional logic ---

and with novel and powerful proof- and model-theoretical tools (simple

tableaux systems, elegant axiomatizations, fine-grained notions of

equivalence between models, new model-theoretical constructions, etc).

Modal-like logics in general, and description and hybrid logics in

particular, will be presented as examples of useful fragments

identified in such a way.

Part II introduces both description and hybrid logics (in

Chapters 2 and 3 respectively) providing the

necessary background and the basic notions which will be used in the

rest of the thesis. The chapters can be read independently and serve

as introductions to the kinds of methods and results which have been

developed in these areas. They also provide a detailed guide to the

literature. As we make clear in our presentation, description and

hybrid logics are closely related, and their connections are spelled

out in Chapter 4. We start by presenting already known

embeddings of description languages into converse propositional

dynamic logics, and discussing why they provide a less satisfactory

match than the one obtained through hybrid languages. In particular,

we highlight that two ingredients are needed for a successful

embedding: the ability to refer to elements in the domain of a model,

and the ability to make statements about the whole model from a local

point. The first ingredient is needed to account for assertions, the

second to account for definitions. Both are provided, in an elegant

and direct way, by hybrid languages in the form of nominals, the

satisfiability operator and the existential modality. We also clarify

the relation between local and global notions of consequence, the

first being the standard notion of consequence for hybrid (and in

general modal) languages while the second is predominant in the

description logic community.

After providing two-way satisfaction preserving translations between

description and hybrid logics, we explore the transfer of results. We show how

the embedding into hybrid languages provides sharp upper and lower

complexity bounds, separations in terms of expressive power and

characterizations, and meta-logical

properties like interpolation and Beth definability.

Concerning interpolation and Beth

definability, to the best of our knowledge this is the first time that

such results have been investigated in connection with description

languages. Many of these results are obtained from the general

theorems we will prove in Part III. We also discuss how results from

description logics can fill important gaps which have not yet received

attention in the hybrid logic community. Some examples are the known

complexity bounds concerning description logics with counting

operators, or the PSpace results when certain syntactic restrictions

are imposed on the existential modality.

Part III of the thesis contains the core technical work. In

Chapter 5 we show how ideas from description and hybrid

logics can be put to work with benefit even when the subject is purely

modal. In particular, aided by the notions of nominal/individual, we

define well behaved direct resolution methods for modal languages.

This example shows how the additional flexibility provided by the

ability to name states can be used to greatly simplify reasoning

methods. We proceed to build over the basic resolution method and

obtain extensions for description and hybrid languages. In

Chapters 6 and 7 we take a hybrid logic

perspective as we dive into model-theoretical issues. But we have

already demonstrated in Chapter 4 how hybrid logic

results shed their light on description languages.

In Chapter 6 we turn to expressive power. We start by

considering $\Hls(@,\downarrow)$, a very expressive hybrid language.

The two main results concerning this language are

Theorems~\ref{the:charac} and~\ref{general-arrow}. The first theorem

provides a five fold characterization of the first-order formulas

equivalent to the translation of a formula in $\Hls(@,\downarrow)$.

In particular, it identifies this fragment as the set of formulas

which are invariant for generated submodels.

Theorem~\ref{general-arrow} shows that the arrow interpolation

property not only holds in this language, but also for any system

obtained from $\Hls(@,\downarrow)$ by the addition of pure axioms. In

a more general perspective, the results in Chapter 6 show

that $\Hls(@,\downarrow)$ is surprisingly well behaved in

model-theoretical terms. As we discuss in this chapter, it can be

characterized in many different and natural ways, it responds with

ease to both modal and first-order techniques, and possess one of the

strongest versions of the interpolation and Beth properties we are

aware of for modal languages. For these reasons, $\Hls(@,\downarrow)$

can be used as a ``logical laboratory:'' what we learn from it using

the plethora of techniques it offers, can provide us, in many cases,

with intuitions on restrictions and extensions. We see this process

in action throughout the chapter, as we are able to transfer certain

results from $\Hls(@,\downarrow)$ to extensions and sublanguages.

In Chapter 7 we discuss complexity. We start with an

excursion into undecidability and we prove that a small fragment of

$\Hls(\downarrow)$ already has an undecidable local satisfiability

problem. This is a hint that only very

severe restrictions on the $\downarrow$ binder will bring us back into

decidability. We show in Theorem~\ref{the:hl-decnn} that if we

restrict ourselves to sentences of

$\Hls(\pmodop,\umodop,@,\downarrow)$, where $\downarrow$ appears

non-nested, decidability is regained. In Chapter 4 we

have already shown that even this restricted use of binding proves

interesting from a description logic perspective. We then turn to

weaker languages (without binders) which remain closer to standard

description languages. In Theorem~\ref{the:b.k.pspace} we prove that

the addition of nominals and the satisfiability operator to the basic

modal language $\logic{K}$ does not modify its complexity, while it

greatly increases its expressive power. Interestingly, the same is

not true when we extend the basic temporal language $\logic{K}_t$: the

addition of just one nominal increases the complexity of the local

satisfiability problem to \exptime, when the

class of all models is considered. But usually temporal languages are

interpreted on models where the accessibility relation is forced to

adopt a ``time-like'' structure, the two best known cases being strict

linear orders (linear time) and transitive trees (branching time). We

prove in Theorems~\ref{the:t.linear} and~\ref{the:t.trees.pspace} that

over these classes of models, complexity is tamed and again coincides

with the complexity of the basic temporal language.

Part IV contains our conclusions and directions for further research.

Here we highlight some of the lessons we have learned during the

research presented in this thesis. As we said, we cannot hope yet for

general answers concerning logic engineering, but we can proceed by

analogy: the same questions we posed and answered for description and

hybrid logics can be tested on other formal languages, and we have

presented tools and methodologies (bisimulations, model construction

and comparison games, translations, etc.) which are powerful and

versatile enough to be useful in many diverse situations.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2000-05 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2000 |

Subjects: | Computation Language Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2017 |

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