Resolution Based Theorem Proving for Temporal
Logics of Knowledge and Belief with InteractionsFinal Report Summary
Background/Context
The project involved the development of proof methods for combined temporal
and modal logics which are useful for specifying and verifying complex systems such
as multiagent systems, security protocols and knowledge games. In
particular, the focus of the project was on
developing proof methods for the combination of propositional lineartime
temporal logic with the modal logic S5 to represent knowledge, and allowing
interaction between the modal and temporal components.
Work on temporal logic and verification is widespread, both within the
UK and internationally. The work of this project is well known
internationally, and members of the research team have been centrally
involved in the development of the field of temporal logic during the
period of this project. For example, Clare Dixon was local organisation cochair in 2002 of
the TIME International Symposium of Temporal Representation and
Reasoning, the only regular international event concerning temporal
reasoning. Clare was one of four editors of a Special Issue on Temporal
Representation and Reasoning of the Annals of Mathematics and Artificial
Intelligence journal in 2001 [2] and is
currently coediting a special issue of the Journal of Logic and
Computation related to time in Computer Science, Artificial Intelligence and Databases.
Key Advances and Supporting Methodology
In the following sections, we outline the key advances made in areas
relating to the project.
The Development of Resolution Methods for Temporal Logics of
Knowledge with Interactions
Work was focused on interactions concerning synchrony and perfect recall, and
synchrony and no learning (see for example [17]), which can be
axiomatised by the usual axioms of
temporal logic, the modal logics of knowledge and the axioms
KOφ ⇒ OKφ
or
OKφ ⇒ KOφ
respectively. It was noticed that these
axioms correspond closely with particular clauses of
the normal form for temporal logics of
knowledge. Rather than developing special resolution rules to account for
the axioms we applied the axioms directly to clauses in the normal
form. Rather than allow a normal form that referred to both temporal and
modal operators in a single clause, renaming was carried out to keep the
temporal and modal components separate.
This led to resolution methods for temporal logics of knowledge with
synchrony and perfect recall or synchrony and no learning and associated
soundness and completeness
results [3,5,24].
Using these results, we considered how to generate resolution proof methods for
interactions of the form
KO^{m}φ ⇒ O^{n}Kφ
or
O^{m}Kφ ⇒ KO^{n}φ
[23] (where
O^{i}
denotes i next operators).
The Development of Tableaux Methods for Temporal Logics of
Knowledge with Interactions
Having devised resolution methods for temporal logics of knowledge with
interactions relating to synchrony
it seemed natural to investigate
other proof methods. Tableaux based methods involve attempting to construct
a structure from which a model for the input formula can be generated. If
no such structure can be constructed then the input formula is
unsatisfiable otherwise the input formula is satisfiable. The grid like
structure of these logics complicate the development of such
methods. Related to the work on the resolution methods for these logics, we
translated the input formulae to a normal form similar to that used
in [8]. The axioms relating to the required interaction are
then applied directly to formulae in the normal form. The tableau
construction then follows that for tableau for temporal logics of knowledge
without interaction. The correctness proofs involve showing that from the
generated structure we really can build a model of the required
form [9,10].
The Development of Resolution Methods for Combinations of
Temporal and Modal Logics
Dealing with combinations of logics is an area of interest for many
researchers. In [1] the issues relating to combining logics
are discussed. This paper arose from the contributions during a panel
session on this topic at the annual UK Automated Reasoning Workshop.
Prior to the commencement of the project we developed resolution methods for
the fusion of propositional lineartime temporal
logics with the modal logic S5 for knowledge [8]. This
concentrated on the single modal case. This
was extended to the multimodal case in [4].
We have also developed resolution based proof methods for different
combinations of temporal and modal logics.
the fusion between the branchingtime
temporal logic CTL and the modal logic KD45 representing
belief [6,7]. From the
development of resolution methods for these particular combinations
we have devised resolution rules for combinations of any normal modal logic
with propositionallinear time temporal logics [23]. In an alternative
approach to this problem we used a translation to a fragment of first order
classical logic for the modal part whilst retaining the temporal clauses in their
original form to provide
a resolution based decision procedure for the fusion of any normal modal logic
with propositionallinear time temporal logic [18].
Efficient Temporal Reasoning
In [11] we present the temporal
resolution method and associated
correctness proofs for propositional lineartime temporal logic. Whilst
this method is complete, more work needed to be done to try apply the resolution
rules in a more efficient way and develop relevant strategies to guide the
proof.
Initially a set of support strategy was defined for normal
form clauses without eventuality operators [13]. This was
essentially an extension of the set of support strategy for classical
propositional logic. Its extension to the full logic (i.e. allowing
eventuality clauses) was not obvious.
The temporal resolution method defined in [11] included a complex
temporal resolution rule. To apply this rule, an external, nonresolution
based procedure call is required to
derive the sets of clauses to which this rule may be applied. To take the
place of this procedure call we developed algorithms to generate formulae
with which to apply the complex temporal resolution rule by repeatedly
applying the (less complex) step resolution rule. This algorithm has been described
in [14, 15, 16]. These algorithms have the advantage that
they allow the reuse of previously
derived clauses for subsequent applications of the temporal resolution
rule to the same eventuality. This algorithm has been used to define a set
of support strategy for the full propositional lineartime temporal
logic [14].
Implementation
A prototype system in Prolog has been developed for the basic system of
temporal logics of knowledge (without interactions) and allowing the
synchrony and no learning interaction [23].
Similarly a prototype using the algorithms to apply the temporal resolution
rule has been developed in Prolog [14].
These have been used to evaluate the proof methods experimentally on
examples.
Whilst both these systems have been useful for initial experiments we
expect that in the longer term extending the TRP++
program [21], an implementation of temporal resolution for propositional
lineartime temporal logics in C++ developed as part of the EPSRC funded
project GR/L87491, would be a more robust solution.
Applications
KARO is an agent specification language using a complex
combination of dynamic logic and modal logics with several nonstandard
operators. In collaboration with the KARO developers we considered proof
methods for KARO by defining a core of this complex language
and using one of two methods [22,19,20]. The first
involves a translation into a
combination of the branchingtime temporal logic (CTL) with knowledge or
belief. The second requires direct translation into firstorder classical
logic. Whilst the core of KARO doesn't allow interactions, interactions
analogous to synchrony and perfect recall and synchrony and no learning are
required for the full KARO logic.
Recently we have been attempting to specify and verify properties of
security protocols using combinations of temporal and modal
logics. In [12] we specified a simple
security protocol using temporal logics of knowledge and were able to prove
properties of the protocol. A similar
study of the Needham Schroeder Protocol is being carried out.
Research Impact
The work carried out in
this project provided significant advances in a number of areas
of temporal logic research, namely
 The development of sound, complete and terminating resolution based
proof procedures for temporal logics of knowledge with the interactions
synchrony and perfect recall or synchrony and no
learning [3,5,24].
 The development of sound, complete and terminating tableaux based
proof procedures for temporal logics of knowledge with the interactions
synchrony and perfect recall or synchrony and no
learning [9,10].
 The development of sound, complete and terminating resolution based
proof procedures for propositional linear time logics combined with
normal modal logics [18, 23] and the branchingtime
temporal logic CTL combined with the modal logic KD45 representing
belief [6,7].
 Efficient algorithms for applying the temporal resolution rule for
propositional lineartime temporal logics and the definition of a set of
support for temporal resolution [14, 15,16]
 An investigation into the agent based specification language
KARO and development of proof methods for a core of its
features [19,20,22].
 Prototype implementations of resolution for temporal logics of
knowledge with interactions [23], new algorithms for applying the
temporal resolution rule and set of support strategy for temporal
resolution [14].
References

 1

B. Bennett, C. Dixon, M. Fisher, E. Franconi, I. Horrocks, U. Hustadt, and
M. de Rijke.
Combinations of Modal Logics .
Artificial Intelligence Review, 17(1):120, March 2002.
 2

C. Dixon, M. Finger, M. Fisher, and M. Reynolds, editors.
Special Issue on Temporal Representation and Reasoning,
volume 30 of Annals of Mathematics and Artificial Intelligence.
Kluwer, 2001.
 3

C. Dixon and M. Fisher.
Clausal Resolution for Logics of Time and Knowledge with Synchrony
and Perfect Recall .
In H. Wansing and F. Wolter, editors, Proceedings of ICTL00 the
Third International Conference on Temporal Logic, Leipzig, Germany, October
2000.
 4

C. Dixon and M. Fisher.
ResolutionBased Proof for MultiModal Temporal Logics of
Knowledge .
In S. Goodwin and A. Trudel, editors, Proceedings of TIME00 the
Seventh International Workshop on Temporal Representation and Reasoning,
Cape Breton, Nova Scotia, Canada, July 2000. IEEE Press.
 5

C. Dixon and M. Fisher.
Clausal Resolution for Logics of Time and Knowledge with the
Interaction Synchrony and Perfect Recall .
(Under journal review) , 2002.
 6

C. Dixon, M. Fisher, and A. Bolotov.
Resolution in a Logic of Rational Agency .
In 14th European Conference on Artificial Intelligence (ECAI
2000), pages 358362, Berlin, Germany, August 2000. IOS Press.
 7

C. Dixon, M. Fisher, and A. Bolotov.
Clausal Resolution in a Logic of Rational Agency .
Artificial Intelligence, 139(1):4789, July 2002.
 8

C. Dixon, M. Fisher, and M. Wooldridge.
Resolution for Temporal Logics of Knowledge.
Journal of Logic and Computation, 8(3):345372, 1998.
 9

C. Dixon, C. Nalon, and M. Fisher.
Tableaux for Logics of Time and Knowledge with Interactions Relating
to Synchrony .
(In preparation) , 2003.
 10

C. Dixon, C. Nalon, and M. Fisher.
Tableaux for Temporal Logics of Knowledge: Synchronous Systems of
Perfect Recall or No Learning .
In M. Reynolds and A. Sattar, editors, Proceedings of the Tenth
International Symposium on Temporal Representation and Reasoning and the
Fourth International Conference on Temporal Logic (TIMEICTL 2003), Cairns,
Australia, July 2003. IEEE Computer Society Press.
 11

M. Fisher, C. Dixon, and M. Peim.
Clausal Temporal Resolution .
ACM Transactions on Computational Logic, 2(1):1256, January
2001.
 12

M. Fisher, W. van der Hoek, M.C. Fernández Gago, and C. Dixon.
Specifying and Verifying Security Protocols .
(Internal technical note) , 2003.
 13

M.C. Fernández Gago.
Efficient control of temporal reasoning.
Transfer Report, Manchester Metropolitan University, 2000.
 14

M.C. Fernández Gago.
Efficient Control of Temporal Reasoning.
PhD thesis, Department of Computer Science, University of Liverpool,
2003.
In preparation.
 15

M.C. Fernández Gago, M. Fisher, and C.Dixon.
Algorithms for Guiding Clausal Temporal Resolution .
In Proceedings of 25th German Conference on Artificial
Intelligence (KI2002), Lecture Notes in Artificial Intelligence, Aachen,
Germany, September 2002. Springer.
 16

M.C. Fernández Gago, M. Fisher, and C. Dixon.
An Algorithm for Guiding Clausal Temporal Resolution .
In Proceedings of 4th International Workshop on Strategies in
Automated Deduction (STRATEGIES 2001) , Siena, Italy, June 2001.
Held in conjunction with IJCAR.
 17

J. Y. Halpern and M. Y. Vardi.
The Complexity of Reasoning about Knowledge and Time. I Lower Bounds.
Journal of Computer and System Sciences, 38:195237, 1989.
 18

U. Hustadt, C. Dixon, R. A. Schmidt, and M. Fisher.
Normal Forms and Proofs in Combined Modal and Temporal Logics .
In Proceedings of the Third International Workshop on Frontiers
of Combining Systems (FroCoS'2000), volume 1794 of LNAI. Springer,
2000.
 19

U. Hustadt, C. Dixon, R. A. Schmidt, M. Fisher, J.J. Meyer, and W. van der
Hoek.
Reasoning about agents in the KARO framework.
In C. Bettini and A. Montanari, editors, Proceedings of the
Eighth International Symposium on Temporal Representation and Reasoning
(TIME01), pages 206213, Cividale del Friuli, Italy, June 2001. IEEE
Press.
 20

U. Hustadt, C. Dixon, R. A. Schmidt, M. Fisher, J.J. Meyer, and W. van der
Hoek.
Verification Within the KARO Agent Theory .
In Selected Papers from the first Goddard Workshop on Formal
Approaches to AgentBased Systems (FAABS). Kluwer Academic Publishers, 2001.
 21

U. Hustadt and B. Konev.
TRP++: A Temporal Resolution Prover .
In Proceedings of 3rd International Workshop on the
Implementation of Logics, Tbilisi, Georgia, October 2002.
 22

Ullrich Hustadt, C. Dixon, R. A. Schmidt, M. Fisher, J.J. C. Meyer, and
W. van der Hoek.
Verification within the KARO agent theory.
In J. L. Rash, C. A. Rouff, W. Truszkowski, D. Gordon, and M. G.
Hinchey, editors, Proceedings of the First International Workshop on
Formal Approaches to AgentBased Systems (FAABS 2000), volume 1871 of
LNAI, pages 3347. Springer, 2001.
 23

C. Nalon.
Resolution for Interacting Modal Logics.
PhD thesis, Department of Computer Science, University of Liverpool,
2003.
In preparation.
 24

C. Nalon, C. Dixon, and M. Fisher.
Resolution for synchrony and no learning: Preliminary report.
In Proceedings of the 8th International Workshop on Logic
Language, Information and Computation (WoLLIC'2001), 2001.