Global View On Reactivity:
Switch Graphs and Their Logics
Dov Gabbay
King’s College London, UK
Bar Ilan University, Israel
University of Luxembourg, Luxembourg
S
´
ergio Marcelino
∗
SQIG - Instituto de Telecomunicac¸
˜
oes, Portugal
Departamento de Matem
´
atica, IST, Lisboa, Portugal
Abstract
The notion of reactive graph generalises the one of graph by allowing the base
accessibility relation to change when its edges are traversed. Can we represent
these more general structures using points and arrows? We prove this can be done
by introducing higher order arrows: the switches.
The possibility of expressing the dependency of the future states of the accessi-
bility relation on individual transitions by the use of higher-order relations, that is,
coding meta-relational concepts by means of relations, strongly suggests the use of
modal languages to reason directly about these structures. We introduce a hybrid
modal logic for this purpose and prove its completeness.
1 Introduction
In computer science the word reactivity has been used to denote systems that react to
their environment and are not meant to terminate, as coined by Pnueli and Harel in [25].
In this paper the word has a different meaning, reactive systems are history-dependent
relational structures, where the accessibility relation is determined not only by the point
where one is, but also by the previous transitions. This concept was introduced by Dov
Gabbay in 2004, see [14] and the extended version [15]. We show that the concept of
reactivity by presenting some structures that embody it and some logics to reason about
them. Let us start by explaining how this concept of reactivity was born and outlining
its short life-story.
∗
The author S
´
ergio Marcelino thanks the support of FCT and EU FEDER, via the postdoc scholarship
SFRH/BPD/76513/2011, the PhD grant SFRH/BD/27938/2006, the project FCT PEst-OE/EEI/LA0008/2011
of IT, the FP7-PEOPLE-2012-IRSES GetFun Marie Curie International Research Staff Exchange Scheme
Fellowship within the 7th European Community Framework Programme, as well as the PQDR initiative of
SGIQ.
1
New kind of arrows. In [14], Dov Gabbay introduced the idea of enriching graph-
based structures with arrows of a new type, calling it the double arrows. Double arrows,
instead of connecting points, connect arrows with arrows or other double arrows, see
Figure 1. The idea is that this new kind of arrows can represent the dependence of
Figure 1: An enriched graph.
the state of the targeted arrow (or double arrow) upon the crossing of the arrow in its
origin. In this first presentation the double arrows would simply change the targeted
arrow state. Let us see how it works by playing with the example in Figure 1. We
represent the fact that an arrow is off by drawing its body as a dotted line. Let us see
Figure 2: The effect of crossing edges.
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the effect that crossing some of its edges has. As shown in Figure 2, when we cross
the edges (a, b) (left) or (b, c) (right), the arrows that are in the scope of the double
arrows coming out of them, become off or on if they are on or off respectively (that is,
their state changes). This process is cumulative, after crossing (a, b) we can also cross
(b, c) and the effects are determined by the new state of double arrows, see Figure 3.
These ideas were presented using suggestive motivational cases, for example in Figure
Figure 3: The effect of crossing edges.
4 we see how these new arrows can represent a classical inheritance networks case.
Figure 4: A classical inheritance networks example. We can represent simple excep-
tions: birds do fly, but although penguins are birds they do not fly. And we can also
represent higher order exceptions (exceptions to exceptions): even though the son of
Tweety is a special penguin (so also a penguin) he does fly.
The general idea that a relational structure may vary when one moves through it
and the enriched kind of frames that came along with it fuelled publications in many
areas. Indeed, there are applications of the reactive ideas in such diverse areas as modal
logic, preferential non-monotonic logic, inheritance systems, context-free grammars,
automata theory, deontic logic and contrary to duty, argumentation and other networks,
see papers [5, 20, 12, 16, 23, 21, 22, 17, 28]. For example when one adds these kind
of double arrows to the structure of an automata, one allows it to modify its transi-
tion relation while reading a sentence, that is, one makes it reactive. This alternative
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paradigm competes with non-determinism in the task of obtaining automata with mini-
mal number of states accepting a language. Indeed, the following theorem (Proposition
6.1, [12]) is proven:
If A is deterministic automaton with k
n
states, it has an equivalent reactive
automaton R(A) with kn states.
Another interesting application of this kind of enriched graphs (though not using the
dynamical counterpart) can be found in [28], where Dung’s abstract argumentation
theory is extended incorporating the meta-level argumentation-based reasoning, about
possibly conflicting preferences between arguments.
Changing Kripke structures One can say that the relational semantics of modal
logic already encompasses change. In fact, one can consider (or access to) different
worlds, and the propositional truths, given by the propositional variables valuation,
may change. Also the accessible worlds may change with these transitions. Yet, the
truths at a given world are ‘still’. In a Kripke model both propositional truths and the
accessible worlds are fixed for each world. One can take it a step further and let these
vary. In many situations it makes sense to consider semantics such that when certain
operators are evaluated, the model, where the formula is being evaluated, changes.
Therefore, the interpretation of a formula in the scope of a modal operator is given by
a general condition of the type:
M, x ϕ if M
′
, x
′
ϕ
where x
′
is a point in a new model M
′
. Indeed there are various examples of such
approaches, e.g.:
• in dynamic epistemic logics with agent’s public announcements [33];
• in sabotage logics edges can be deleted [32];
• in memory logics one may keep the information that a certain world was visited,
adding it to the memory of the model [4];
• in Hyper-modalities the meaning of the modal operators depends on where in the
formula they occur [13];
• in product logics one may think that while moving along one direction the valu-
ation of the remaining hyper-plan is changing, for example modelling valuation
change in time if that direction is a time flow [18].
Local view: reactive logics, making Kripke reactive In [15] Kripke structures are
made reactive. Gabbay introduces a semantics based on Kripke frames enriched with
double arrows, where the basic relation changes along the interpretation of a formula
by the action of the double arrows. The dependence is on where one has been before,
that is, neither relation nor propositional variables values change with the clock ticking
but they react when and because we move. Moreover, the changes are sensitive to the
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way we got to the current world. Subsequently it is proven that this semantics strictly
generalises the Kripke semantics, in fact, we may have classes of these frames origi-
nating logics that are not closed under substitution. It is not claimed that the classical
Kripke frames cannot cope with these types of change, the matter is more about how to
incorporate these meta-level notions into the models and which language to consider
in order to reason about them. In [19, 27] is introduced a more abstract notion of reac-
tive Kripke frames. Whereas in [15] the changes in the accessible relation are the ones
produced by the action of the double arrows, in the abstract notion of reactive Kripke
frames these changes are given. In reality, in the usual semantics of modal logic the
only important information to the value of a modal formula is the set of successors
at each moment, i.e the local accessibility relation. Therefore the notion of reactive
Kripke frame boils down to a set of admissible sequences of points, that is, the set of
admissible paths. One can picture the initial accessibility relation by considering the
paths of size two, and its evolution is encoded in the one step prolongment relation
on the bigger paths. The semantics presented in [15] is generalised over these abstract
structures, allowing the valuation to vary along the paths, and the language is enriched
with an extra operator relating paths that have the same endpoint. This is similar to
what it is done in the branching-time logic with quantification over branches in [34].
Finally results of soundness and completeness are presented characterising some prop-
erties of these frames (generalising some familiar properties of ‘static’ Kripke frames:
reflexivity, symmetry and transitivity). Furthermore some results regarding the decid-
ability of the resulting logics are obtained. Let us look to a concrete case and see some
examples of what can be expressed in the considered language.
Example 1.1. Let us consider the situation of a traveller with a budget. The set of his
possible moves depends on whether he has enough money to do them (to pay tolls, oil,
train or flight tickets), furthermore his actual moves also determine his future possibil-
ities. So the paths of the correspondent reactive frame are the sequences of cities he
can visit with a certain budget. The formulas are interpreted over these paths. Let
R
stand for the dynamics operator, that is, corresponding to the accessibility relation, and
P
to the relation identifying the paths with the same endpoint. So,
R
ϕ means that
after the current path we can access to a city such that the resulting path satisfies ϕ.
R
ϕ means that there is a path to the current world satisfying ϕ. Let us consider that
the propositional symbols p
b
and p
w
correspond to the predicate of being able to buy
bread and wine respectively, and m be true if there is still some money left. See Table
1 for examples of what can be said.
Modal language Natural language
¬m → ◻
R
If the traveller has no money left then he cannot move
◇
P
(◻
R
p
b
∧ ◇
R
⊺) There is a path to the current city, after which the traveller is not blocked
and he has enough money to buy bread every city he can access to
(p
w
∧ ◻
R
p
w
) If the traveller can buy wine now, and at any immediate next stop, then
→ ◻
P
p
b
he would always be able to buy bread in the current city regardless of
how he got there
Table 1: Possible statements in the considered modal semantics.
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