We present an extension of the argumentation semantics for defeasible logic to cover the temporalisation of defeasible logic with permanent and imminent temporal literals

Governatori, G

Riveret, R.

Rotolo, A.

RIVERET, REGIS

GOVERNATORI, GUIDO

ROTOLO, ANTONINO

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Argumentation Semantics for Temporal Defeasible Logic

University of Queensland eSpace

STAIRS 2006
Pavlos Peppas, Anna Perini and Loris Penserini
IOS Press, 2006
Argumentation Semantics for Temporal
Defeasible Logic1
Régis Riveret a,2x, Guido Governatori b, and Antonino Rotolo a
aCIRSFID, University of Bologna
bSchool of ITEE, University of Queensland
Keywords. Defeasible Logic, Argumentation Semantics, Temporal Reasoning
Temporal Defeasible Logic extends Defeasible Logic (DL) [1] to deal with temporal as-
pects. This extension proved useful in modelling temporalised normative positions [3]
and retroactive rules, which permit to obtain conclusions holding at a time instant that
precedes the time of application of the same rules [4]. Time is added in two ways. First, a
temporalised literal is a pair l : t where l is a literal and t is an instant of time belonging to
a discrete totally ordered set of instants of time T = {t1, t2, . . .}. Intuitively, the meaning
of a temporalised literal l : t is that l holds at time t. Second, rules are partitioned in per-
sistent and transient rules according to whether the consequent persists until an interrupt-
ing event occurs or is co-occurrent with the premises. Hence, D= (T ,F,Rp,Rc,) is a
temporal defeasible theory, whereT is the set of instants, F a set of facts, a superiority
relation over rules, and Rp and Rt the sets of persistent and transient rules. Given a rule
r ∈ Rp such as a : t ⇒p b : t ′, if r is applicable, we can derive b holding at t ′ and at any
t ′+ n, until we can block this inference, for example, by deriving ¬b, at a certain time
t ′+m; given an r′ ∈ Rt such as a : t ⇒t b : t, we derive b at any t only if a holds as well
at t. Proof tags of DL must be duplicated: ±∆pl : t and ±∂pl : t mean, respectively, that
l : t is/is not definitely and defeasibly provable persistently; ±∆t l : t and ±∂t l : t mean,
respectively, that l : t is/is not definitely and defeasibly provable transiently.
On the other hand, DL can also be interpreted in terms of interacting arguments,
giving for it an argumentation semantics [2]. Argumentation systems are of particular
interest in AI & Law, where notions like argument and counter-argument are very com-
mon. For example, a recent development of such semantics is represented by argumen-
tation and mediation systems which assist the users in expressing and organising their
arguments, in assessing their impact on controversial legal issues or in building up an
effective interaction in dialectical contexts [5]. So far, the logic has been only formalised
in a proof-theoretic setting in which arguments play no role. Our purpose is to provide
an argumentation semantics for temporal DL. Note that we can dispense with the superi-
1The full version of this paper is available at http://eprint.uq.edu.au/archive/00003954/
01/Stairs06Long.pdf.
2Corresponding Author: Régis Riveret, CIRSFID, University of Bologna, 40121 Bologna, Italy; E-mail:
rriveret@cirsfid.unibo.it. The first and third authors were supported by the European project for Standardized
Transparant Representations in order to Extend Legal Accessibility (ESTRELLA, IST-4-027655); the second
author was supported by the Australia Research Council under Discovery Project No. DP0558854 on “A
Formal Approach to Resource Allocation in Service Oriented Marketplaces”.
ority relation of standard DL, as we can define a modular transformation similar to that
given in [1] that enables to empty the superiority relation.
In line with [2], an argument for a temporalised literal p : t is a proof tree (or mono-
tonic derivation) for that temporalised literal in DL. Nodes are temporalised literals, arcs
connecting nodes correspond to rules. We introduce a new type of connection, repre-
sented by “dashed arcs”, which are meant to connect a permanent literal to its succes-
sors in time. Types of arguments are distinguished according to the rules used: support-
ive arguments are finite arguments in which no defeater is used and strict arguments use
strict rules only, permanent arguments are arguments in which the ending arc either is a
dashed arc or corresponds to a permanent rule while in transient arguments the ending
arc corresponds to a transient rule.
The notion of attack between arguments is thus defined: a set of arguments S attacks
a defeasible argument B if there is an argument A in S that attacks B such that (i) a : ta
is a conclusion of A and (ii) a : ta is not a conclusion of a dashed arc, and (iii) b : tb is
a conclusion of B, and a is on conflict with b (typically a is the complement of b), and
either ta = tb, or it exists a dashed arc in B with premise b : tb′ and conclusion b : tb,
and tb′ < ta ≤ tb. This definition of attack allows us to reuse the standard definition as
given in [2] of arguments being undercut and arguments being acceptable. Based on
these concepts we proceed to define justified arguments, i.e., arguments that resist any
refutation. Accordingly, a literal p : t is defined as transiently/permanently justified if it
is the conclusion of a supportive and transient/permanent argument in the set of justified
arguments JargsD of a theory D. That a literal p : t is justified means that it is provable
(+∂ ). However, DL permits to express when a conclusion is not provable (−∂ ). This
last notion is captured by assigning the status of rejected to arguments. Roughly, an
argument is rejected if it has a rejected sub-argument or it cannot overcome an attack
from a justified argument. More generally, a literal is transiently/permanently rejected if
it is transiently/permanently rejected by JargsD. One of our results is that permanent and
transient defeasible conclusions can be characterised as follows:
THEOREM 1 Given a theory D and its set of justified arguments JargsD,
• D `+∂p(p : t) iff p : t is permanently justified;
• D ` −∂p(p : t) iff p : t is permanently rejected by JargsD;
• D `+∂t(p : t) iff p : t is transiently justified;
• D ` −∂t(p : t) iff p : t is transiently rejected by JargsD.
Hence, Theorem 1 gives an argumentation semantics with ambiguity blocking to perma-
nent and transient defeasible conclusions.
References
[1] G. Antoniou, D. Billington, G. Governatori, and M.J. Maher. Representation results for defea-
sible logic. ACM Transactions on Computational Logic, 2, pages 255–287, 2001.
[2] G. Governatori, M.J. Maher, D. Billington, and G. Antoniou. Argumentation semantics for
defeasible logics. Journal of Logic and Computation, 14, pages 675–702, 2004.
[3] G. Governatori, A. Rotolo, and G. Sartor. Temporalised normative positions in defeasible
logic. In Proc. ICAIL05. ACM, New York, 2005.
[4] G. Governatori, M. Palmirani, R. Riveret, A. Rotolo, and G. Sartor. Norm modifications in
defeasible logic. In M. Moens, editor, Proc. Jurix’05. IOS Press, Amsterdam, 2005.
[5] B. Verheij. Virtual Arguments: On the Design of Argument Assistants for Lawyers and Other
Arguers. T.M.C. Asser Press, The Hague, 2005.


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Argumentation Semantics for Temporal Defeasible Logic

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