Analysis and Mechanisation of Decidable FirstOrder
Temporal Logics
Research Team
Principal Investigators:
CoInvestigators:
 Anatoli Degtyarev,
Department of Computer Science, King's College,
London WC2R 2LS
 Clare Dixon,
Department of Computer
Science, University of Liverpool
 Dov Gabbay,
Department of Computer Science, King's College,
London WC2R 2LS
 Ullrich Hustadt,
Department of Computer
Science, University of Liverpool
 Ian Hodkinson,
Department of Computing, Imperial College,
London SW7 2BZ
 Frank Wolter
Department of Computer
Science, University of Liverpool
Research Staff:
 Ulle Endriss,
Department of Computer Science, King's College,
London WC2R 2LS
 David Gabelaia,
Department of Computer Science, King's College,
London WC2R 2LS
 Boris Konev,
Department of Computer
Science, University of Liverpool
 Roman Kontchakov,
Department of Computer Science, King's College,
London WC2R 2LS
Aims and Objectives
The longterm aim of our work is to develop sophisticated temporal
verification tools that can be widely applied.
In this project we will investigate FOTL in depth, and develop, implement,
and analyse tools mechanising the axiomatisable and decidable fragments of
FOTL identified. We believe this approach represents a sound,
necessary, and feasible step towards our ultimate goal.
Our principal objectives for this proposed research programme are:
 to develop practical proof algorithms, based on decidable
(extensions of) monodic FOTL, using both tableau and
resolution techniques;
 to carry out a detailed analysis of logical and
computational properties of monodic FOTL;
 to extend current results concerning axiomatisable and decidable
classes beyond the monodic case;
 to implement some of the tableau/resolution systems developed in
(1); and
 to evaluate the systems developed, and to apply them to a range of
verification case studies.
Background
It is widely believed in computer science (CS) and artificial
intelligence (AI) that if firstorder temporal logic (FOTL) were not
so impregnable, it could provide elegant solutions to fundamental
longstanding problems in many practically important fields. Areas
where FOTL may be exploited, and where propositional temporal logic
(PTL) has already made significant impact, include:
 specification and verification of reactive (e.g. distributed or
concurrent) systems [20]  FOTL
allows extension of these techniques to both data dependent
systems and hybrid systems [15];
 synthesis of programs from temporal specifications
[24,18]  FOTL again allows the
synthesis technique to be extended to more complex
specifications;
 semantics of executable temporal logic
[11,12]  FOTL provides not
only more expressive tools for formalising the behaviour of
executable temporal logics, but can itself be used as a more
powerful programming notation;
 modelchecking [17,8]  FOTL provides
the possibility of extending model checking techniques to non
finitestate systems, and to systems containing multiple
concurrent processes;
 knowledge representation and reasoning
[9,5,29]  FOTL
allows the extension of techniques for reasoning about
knowledge to more dynamic and powerful classes.
In addition to current applications of temporal logic that would be
enhanced by the use of firstorder techniques, there are several novel
applications that may become feasible via such extensions. For
example, query languages for temporal databases are often based on
(variants of) FOTL [7]. In turn, FOTL could
provide a means for temporal constraint checking [6] and
for verifying properties of transaction protocols (or business models)
in ecommerce [2,25].
As the above enhanced and potential application areas are themselves
increasingly important in both mainstream CS and AI, the development
of appropriate reasoning systems for FOTL will ultimately contribute
to many strands within these disciplines.
Unfortunately, while FOTL is clearly useful, it is a very expressive
language with extremely high computational complexity. Many varieties
of FOTL are not even recursively enumerable
[1,3,13,22,26],
and so do not admit finite proof methods at all. As a consequence,
indepth investigations of FOTL have been rare, with attention largely
focused on developing practical tools based only on PTL. The few
either recursively enumerable or decidable fragments of FOTL that have
been found either use nonstandard timelines [3],
weaker versions of validity [1], minimal interaction
between quantifiers and temporal
operators [21,6,25] or very restricted
firstorder features [23,22]. In each of these
cases, either the logic is so restricted that it represents only a
small extension of PTL, or the logical calculus described is very
difficult to implement in practice.
However, within the past year, this situation has started to
change. In a seminal paper, Hodkinson, Wolter, and
Zakharyaschev [16] introduced a new natural monodic
fragment of FOTL (where formulas beginning with a temporal operator
contain at most one free variable) and showed that it is quite
expressive and yet computationally manageable.
The whole monodic
fragment can be represented as a finite axiomatic system
[30], and so in principle can be supported by
tableau or resolutiontype reasoning machinery. Moreover, by
restricting the pure firstorder part to certain decidable fragments,
we obtain decidable monodic fragments of FOTL over various
flows of time  for example, the monodic guarded or even loosely
guarded fragments [4,27,14].
These recent research results have opened up
new and exciting opportunities for identifying, extending and
mechanising relatively expressive subsets of FOTL
In particular, certain similarities between monodic FOTL and
effective multidimensional knowledge representation formalisms
described in [28] suggest that the
monodic FOTL can be considerably extended. Although
[16,30] establish strict boundaries beyond which
fragments of FOTL again become nonenumerable, there is still scope
for enriching the expressive power of the monodic fragment. For
example, we may allow applications of local temporal operators,
such as `nexttime', to formulas with two or more free variables 
formulas of this form are not only particularly welcome in temporal
databases, but also cover the decidable fragments of FOTL developed by
Pliuskevicius [23].
As regards mechanisation, recent papers
[19] present tableaubased satisfiability
checking algorithms for description logics with temporal and epistemic
operators. Again, similarities between such logics and monodic FOTL
suggest that tableaubased reasoning systems can also be constructed
for decidable monodic fragments of FOTL. Moreover, [19]
shows that this can be carried out in a modular way by
combining existing tableau systems for PTL and the classical
firstorder components. In addition, we intend to develop
an alternative approach to modular proofsearch in FOTL based on the
resolution method, which is more amenable
to automated reasoning in classical firstorder logic than
tableau. As most of the known decidable fragments of firstorder
classical logic are supported by resolution decision
procedures [10], a modular style
resolutionbased decision method for FOTL immediately gains access to
all the benefits, both theoretical and practical, of resolutionbased
decision methods for fragments of classical firstorder logic.
The development of two different approaches to mechanisation will not
only generate further analysis of the monodic fragment, but will
provide alternative tools for practical applications.
Thus, the aim of this project is to analyse, in depth, monodic FOTL
and its decidable extensions, and to develop, implement and apply
methods of automated deduction for the constructed temporal
formalisms. Thus, this project will combine work on logical and
computational properties of wellbehaved fragments of FOTL (Gabbay,
Hodkinson, Kurucz, Zakharyaschev) with work on the theory,
implementation and application of temporal verification methods
(Degtyarev, Dixon, Fisher). It will extend and enhance both areas. The
new work on monodic fragments of FOTL has opened up a range of
important opportunities and it is our aim to follow these up as soon
as possible, providing significant advances in the area of temporal
verification.
The work comprising this project constitutes a natural and feasible
extension to our ongoing work and brings together two of the world's
foremost research teams. In addition, this project complements the
ongoing EPSRC project on ``Mechanising FirstOrder Temporal
Logic'' (GR/M46631), being carried out between Liverpool (Fisher,
Quigley) and Edinburgh (Smaill, Bundy), which is tackling the
undecidable/nonenumerable parts of FOTL using inductive proof
techniques and proof planning.
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