KR'2008 paper

1. Proceedings, Eleventh International Conference on Principles of Knowledge Representation and Reasoning (2008)
Action Theory Contraction and Minimal Change
Ivan Jos´ Varzinczak
e
IRIT – Universit´ de Toulouse
e Meraka Institute
Toulouse, France CSIR, Pretoria, South Africa
ivan.varzinczak@irit.fr ivan.varzinczak@meraka.org.za
Abstract In the second example, the executability of the action under
concern is questioned in the light of new information show-
This work is about changing action domain descriptions
in dynamic logic. We here revisit the semantics of ac-
ing a context that was not known to preclude its execution.
tion theory contraction, giving more robust operators Such cases of theory change are very important in logical
that express minimal change based on a notion of dis- descriptions of dynamic domains: it may always happen that
tance between models. We then define syntactical con- one discovers that an action actually has a behavior that is
traction operators and establish their correctness w.r.t. different from that one has always believed it had.
our semantics. Finally we show that our operators sat-
isfy the PDL-counterpart of the standard postulates for Up to now, theory change has been studied mainly for
theory change adopted in the literature. knowledge bases in classical logics, both in terms of revision
and update. Since (Fuhrmann 1989), only in a few recent
works it has been considered in the realm of modal logics,
Introduction and Motivation viz. in epistemic logic (Hansson 1999) and in dynamic log-
Let an intelligent agent be designed to perform rationally ics (Herzig, Perrussel, and Varzinczak 2006), and in action
in a dynamic world, and suppose she should reason about languages (Eiter et al. 2005). Some other works (Shapiro
the dynamics of an automatic coffee machine. Suppose that et al. 2000; Jin and Thielscher 2005) have investigated re-
the agent believes that a coffee is a hot beverage. Now vision of beliefs about facts of the world. In our scenario,
suppose that some day she gets a coffee and observes it is this would concern for example the truth of token in a given
cold. In such a case, the agent must change her beliefs state: the agent believes she has a token, but is wrong about
about the relationship between the propositions “I have a that and might subsequently be forced to revise her beliefs
coffee” and “I have a hot beverage”. This example is an about the current state of affairs. Such belief revision opera-
instance of the problem of changing propositional belief tions do not modify the agent’s beliefs about the action laws.
bases and is largely addressed in the literature about belief In opposition to that, here we are interested exactly in such
change (G¨ rdenfors 1988) and belief update (Katsuno and
a modifications.
Mendelzon 1992).
Next, let our agent believe that whenever buying a coffee Logical Preliminaries
on the machine, she gets a hot beverage. This means that in Our base formalism is Propositional Dynamic Logic (PDL)
every state of the world that follows the execution of buying, without the ∗ operator (Harel, Tiuryn, and Kozen 2000).
the agent possesses a hot beverage. Some day, it may hap-
pen that the machine is running out of cups, and then after Action Theories in Dynamic Logic
buying, the coffee runs through the shelf and so the agent Let Act = {a1 , a2 , . . .} be the set of atomic actions of a
holds no hot beverage. given domain. An example of atomic action is buy. To each
Imagine now the agent believes that if she has a token, action a there is associated a modal operator [a]. Prop =
then it is always possible to buy coffee. However, during a {p1 , p2 , . . .} denotes the set of propositional constants, also
blackout, even with a token the agent does not manage to called fluents or atoms. Examples of those are token (“the
order her coffee on the machine. agent has a token”) and coffee (“the agent holds a coffee”).
The last two examples illustrate situations where chang- The set of all literals is Lit = {ℓ1 , ℓ2 , . . .}, where each ℓi
ing the beliefs about the behavior of the action of buying is either p or ¬p, for some p ∈ Prop. If ℓ = ¬p, then we
coffee is mandatory. In the first one, buying coffee, once identify ¬ℓ with p. By |ℓ| we denote the atom in ℓ.
believed to be deterministic, has now to be seen as nonde- We use ϕ, ψ, . . . to denote Boolean formulas, an example
terministic, or alternatively to have a different outcome in a of which is coffee → hot. Fml is the set of all Boolean for-
more specific context (e.g. if there is no cup in the machine). mulas. A propositional valuation v is a maximally consistent
Copyright c 2008, Association for the Advancement of Artificial set of literals. We denote by v ϕ the fact that v satisfies ϕ.
Intelligence (www.aaai.org). All rights reserved. By val(ϕ) we denote the set of all valuations satisfying ϕ.
651

2. |= is the classical consequence relation. Cn(ϕ) denotes all
CPL
For the sake of clarity, we will here abstract from the
logical consequences of ϕ in classical propositional logic. frame problem (McCarthy and Hayes 1969) and the rami-
With IP(ϕ) we denote the set of prime implicants (Quine fication problem (Finger 1987), and assume that the agent’s
1952) of ϕ. By π we denote a prime implicant, and atm(π) theory contains all frame axioms (cf. (Herzig, Perrussel, and
is the set of atoms occurring in π. For given ℓ and π, ℓ ∈ π Varzinczak 2006) for a contraction approach within a solu-
abbreviates ‘ℓ is a literal of π’. tion to the frame problem). The action theory of our example
We will use Φ, Ψ, . . . to denote complex formulas (formu- will be:
las with modal operators). An example of a complex for- 
coffee → hot, token → buy ⊤,

mula is ¬coffee → [buy]coffee. a is the dual operator of
 
¬coffee → [buy]coffee, token → [buy]¬token,
 
[a] ( a Φ =def ¬[a]¬Φ). T=
 ¬token → [buy]⊥, ¬token → [buy]¬token, 
coffee → [buy]coffee, hot → [buy]hot
 
A PDL-model is a tuple M = W, R where W is a set
of valuations, and R maps action constants a to accessibility
M
relations Ra ⊆ W × W. Given M , |= p (p is true at world Figure 1 below shows a PDL-model for the theory T.
w
M M
w of model M ) if w p; |= [a]Φ if |= ′ Φ for every w′ s.t.
w w
¬t, c, h
(w, w′ ) ∈ Ra ; truth conditions for the other connectives are b b
as usual. By M we will denote a set of PDL-models.
M M
M is a model of Φ (noted |= Φ) if and only if |= Φ for all
w t, c, h b t, ¬c, h
M
w ∈ W. M is a model of a set of formulas Σ (noted |= Σ)
M
if and only if |= Φ for every Φ ∈ Σ. Φ is a consequence of
the global axioms Σ in all PDL-models (noted Σ |= Φ) if
PDL t, ¬c, ¬h
M M
and only if for every M , if |= Σ, then |= Φ.
Figure 1: A model for the coffee machine scenario. b, t, c,
With PDL we can state laws describing the behavior of and h stand for, respectively, buy, token, coffee, and hot.
actions. Following the tradition in the reasoning about ac-
tions community, we here distinguish three types of them. Sometimes it will be useful to consider models whose
possible worlds are all the possible worlds allowed by S :
Static Laws A static law is a formula ϕ ∈ Fml. It is a
formula that characterizes the possible states of the world. Definition 1 Let T = S ∪ E ∪ X be an action theory. Then
An example of static law is coffee → hot: if the agent holds M = W, R is the big model of T if and only if:
a coffee, then she holds a hot beverage. The set of all static • W = val(S ); and
laws of a domain is denoted by S . M M
• Ra = {(w, w′ ) : ∀.ϕ → [a]ψ ∈ Ea , if |= ϕ then |= ′ ψ}.
w w
Effect Laws An effect law for a is of the form ϕ → [a]ψ,
where ϕ, ψ ∈ Fml. Effect laws are formulas relating an ac- Figure 2 below depicts the big model of T.
tion to its effects, which can be conditional. The consequent
¬t, c, h
ψ is the effect that always obtains when a is executed in a
state where the antecedent ϕ holds. If a is a nondeterminis- b b
tic action, then ψ is typically a disjunction. An example of
such a law is token → [buy]hot: whenever the agent has a
t, c, h b t, ¬c, h
token, after buying, she has a hot beverage. If ψ is incon-
sistent we have a special kind of effect law that we call an
inexecutability law. For example, ¬token → [buy]⊥ says
that buy cannot be executed if the agent has no token. The ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
set of effect laws of a domain is denoted by E .
Executability Laws An executability law for a has the form Figure 2: The big model for the coffee machine scenario.
ϕ → a ⊤, with ϕ ∈ Fml. It stipulates the context in which
a is guaranteed to be executable. (In PDL, the operator a Elementary Atoms
is used to express executability, a ⊤ thus reads “a’s execu- Given ϕ ∈ Fml, E(ϕ) denotes the elementary atoms actu-
tion is possible”.) For instance, token → buy ⊤ says that ally occurring in ϕ. For example, E(¬p1 ∧ (¬p1 ∨ p2 )) =
buying can be executed whenever the agent has a token. The {p1 , p2 }. An atom p is essential to ϕ if and only if p ∈ E(ϕ′ )
set of all executability laws of a domain is denoted by X . for all ϕ′ such that |= ϕ ↔ ϕ′ . For instance, p1 is essential
CPL
Given a, Ea (resp. Xa ) will denote the set of only those to ¬p1 ∧(¬p1 ∨p2 ). E!(ϕ) will denote the essential atoms of
effect (resp. executability) laws about a. ϕ. (If ϕ is a tautology or a contradiction, then E!(ϕ) = ∅.)
For ϕ ∈ Fml, ϕ∗ is the set of all ϕ′ ∈ Fml such that
Action Theories T = S ∪ E ∪ X is an action theory. ϕ |= ϕ′ and E(ϕ′ ) ⊆ E!(ϕ). For instance, p1 ∨ p2 ∈
CPL
/
652

3. p1 ∗, as p1 |= p1 ∨ p2 but E(p1 ∨ p2 ) ⊆ E!(p1 ). Clearly,
CPL
Semantics of Contraction
E( ϕ∗) = E!( ϕ∗). Moreover, whenever |= ϕ ↔ ϕ′ ,
CPL When contracting a law Φ we must ensure that Φ becomes
then E!(ϕ) = E!(ϕ′ ) and also ϕ∗ = ϕ′ ∗. invalid in at least one (possibly new) model of the dynamic
domain. Because there can be lots of models to consider,
Theorem 1 (Least atom-set theorem (Parikh 1999)) we start with a set M of models in which Φ is (potentially)
|= ϕ ↔
CPL
ϕ∗, and E(ϕ∗) ⊆ E(ϕ′ ) for every ϕ′ s.t. valid. Thus contracting Φ amounts to make it no longer valid
′
|= ϕ ↔ ϕ .
CPL in this set of models. What are the operations that must be
carried out to achieve that? Throwing models out of M does
Thus for every ϕ ∈ Fml there is a unique least set of
not work, since Φ will keep on being valid in all models of
elementary atoms such that ϕ may equivalently be expressed
the remaining set. Thus we should add new models to M.
using only atoms from that set.1 Hence, Cn(ϕ) = Cn(ϕ∗).
Which models? Well, models in which Φ is not true. But
not any of such models: taking models falsifying Φ that are
Prime Valuations too different from our original models will certainly violate
Given a valuation v, v′ ⊆ v is a subvaluation. For W a set minimal change.
of valuations, a subvaluation v′ satisfies ϕ ∈ Fml modulo W Hence, we shall take some model M ∈ M as basis and
(noted v′ W ϕ) if and only if v ϕ for all v ∈ W such that manipulate it to get a new model M ′ in which Φ is not true.
v′ ⊆ v. A subvaluation v essentially satisfies ϕ modulo W In PDL, the removal of a law Φ from a model M amounts to
!
(v W ϕ) if and only if v W ϕ and {|ℓ| : ℓ ∈ v} ⊆ E!(ϕ). modifying the possible worlds or the accessibility relation in
M so that Φ becomes false. Such an operation gives as re-
−
Definition 2 Let ϕ ∈ Fml and W be a set of valuations. A sult a set MΦ of models, each of which is no longer a model
subvaluation v is a prime subvaluation of ϕ (modulo W) if of Φ. But if there are several candidates, which ones should
! ! we choose? We shall take those that are minimal modifica-
and only if v W ϕ and there is no v′ ⊆ v s.t. v′ W ϕ. tions of the original M . Note that there can be more than
A prime subvaluation of a formula ϕ is one of the weakest one M ′ that is minimal. Because adding just one of these
states of truth in which ϕ is true. (Notice the similarity with new models is enough to invalidate Φ, we take all possible
the syntactical notion of prime implicant (Quine 1952).) combinations M ∪ {M ′ } of expanding our set of models by
one minimal model. The result will be a set of sets of models.
By base(ϕ, W) we denote the set of all prime subvalua- In each set of models there will be one M ′ falsifying Φ.
tions of ϕ modulo W.
Contraction of Executability Laws
Theorem 2 Let ϕ ∈ Fml and W be a set of valuations. Then
for all w ∈ W, w ϕ if and only if w Intuitively, to contract an executability law ϕ → a ⊤ in one
v∈base(ϕ,W) ℓ∈v ℓ.
model, we remove arrows leaving ϕ-worlds. To success the
operation, we have to guarantee that in the resulting model
Closeness Between Models there is at least one ϕ-world with no departing a-arrow.
When contracting a formula from a model, we will perform
Definition 4 Let M = W, R be a PDL-model. Then
a change in its structure. Because there can be several differ- −
ent ways of modifying a model (not all of them minimal), we M ′ = W′ , R′ ∈ Mϕ→ a ⊤ if and only if
need a notion of distance between models to identify those • W′ = W
that are closest to the original one.
As we are going to see in more depth in the sequel, chang- • R′ ⊆ R
M
ing a model amounts to modifying its possible worlds or • If (w, w′ ) ∈ R R′ , then |= ϕ
w
its accessibility relation. Hence, the distance between two M′
PDL-models will depend upon the distance between their • There is w ∈ W′ s.t. |= ϕ → a ⊤
w
sets of worlds and accessibility relations. These here will be
based on the symmetric difference between sets, defined as To get minimal change, we want such an operation to be
˙
X −Y = (X Y ) ∪ (Y X). minimal w.r.t. the original model: we should remove a min-
imum set of arrows sufficient to get the desired result.
Definition 3 Let M = W, R be a model. M ′ = W′ , R′
is as close to M as M ′′ = W′′ , R′′ , noted M ′ M M ′′ , Definition 5 Let M be a PDL-model and ϕ → a ⊤ an
if and only if executability law. Then
−
• either W−W′ ⊆ W−W′′
˙ ˙ contraction(M , ϕ → a ⊤) = min{Mϕ→ a ⊤, M}
• or W−W˙ ′ = W−W′′ and R−R′ ⊆ R−R′′
˙ ˙ ˙
And now we define the sets of possible models resulting
(Notice that other distance notions are also possible, like from the contraction of an executability in a set of models:
e.g. considering the cardinality of symmetric differences.)
Definition 6 Let M be a set of models, and ϕ → a ⊤ an
1
The dual notion (redundant atoms) is also addressed in the lit- executability law. Then M− a ⊤ = {M′ : M′ = M ∪
ϕ→
erature, e.g. in (Herzig and Rifi 1999), with similar purposes. {M ′ }, M ′ ∈ contraction(M , ϕ → a ⊤), M ∈ M}.
653

4. In our example, consider M = {M }, where M is the (¬coffee is relevant to ¬hot). Hence, we can add arrows
model in Figure 2. When the agent discovers that even with from token-worlds to ¬hot ∧ ¬coffee ∧ token-worlds, as well
a token she does not manage to buy a coffee anymore, she as to ¬hot ∧ ¬coffee ∧ ¬token (Figure 4). Pointing the ar-
has to change her models in order to admit models with row to ¬hot ∧ ¬coffee ∧ token would make us lose the ef-
states where token is the case but from which there is no fect ¬token, true after every execution of buy in the original
buy-transition at all. Because having just one of such worlds model. How to preserve this law while allowing for the new
in each new model is enough, taking those resulting models transition to a ¬hot-world?
whose accessibility relations are maximal guarantees mini- ¬t, c, h
mal change. Hence we get M− ′
token→ buy ⊤ = {M ∪ {Mi } :
1 ≤ i ≤ 3}, where each Mi′ is depicted in Figure 3. b b
¬t, c, h M : t, c, h b t, ¬c, h
b
′
M1 : t, c, h b t, ¬c, h
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
Figure 4: Candidate ¬hot-worlds to receive arrows from
token-worlds.
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
When pointing a new arrow leaving a world w it is enough
to preserve old effects only in w (because the remaining
¬t, c, h structure of the model keeps unchanged after adding this
new arrow). The operation we must carry out is observing
b
what is true in w and in the candidate target world w′ : what
′
M2 : changes from w to w′ (w′ w) must be what is obliged to
t, c, h b t, ¬c, h do so; what does not change from w to w′ (w ∩ w′ ) must be
what is either obliged or allowed to do so.
This means that the only things allowed to change w.r.t.
w in the candidate target world are those that are forced to
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
change: they are relevant to ¬ψ or to another effect that ap-
plies in w. Every change outside that is not an intended one.
¬t, c, h Similarly, we want the literals preserved in the target world
to be those that are relevant to ¬ψ or to some other effect
b b that applies in w or that are usually preserved in w. Every
′
M3 : preservation outside those may make us lose some law.
t, c, h t, ¬c, h Here is where prime subvaluations play their role: the
worlds one should aim the new arrow at are those whose
difference w.r.t. w are literals that are relevant, and whose
similarity w.r.t. w are literals we know may not change.
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
Definition 7 Let M = W, R , w, w′ ∈ W, M be such that
Figure 3: Models resulting from contracting the executabil- M ∈ M, and ϕ → [a]ψ an effect law. Then w′ is a relevant
ity law token → buy ⊤ in the model M of Figure 2. target world of w w.r.t. ϕ → [a]ψ for M in M if and only if
M M
• |= ϕ, |= ′ ψ
w w
• for all ℓ ∈ w′ w
Contraction of Effect Laws
– either there is v ∈ base(¬ψ, W) s.t. v ⊆ w′ and ℓ ∈ v
When the agent discovers that there may be cases when after
– or there are ψ ′ ∈ Fml, v′ ∈ base(ψ ′ , W) s.t. v′ ⊆ w′ ,
buying she gets no hot beverage, she must give up the law M
token → [buy]hot in her models. This means that token ∧ ℓ ∈ v′ , and |= i [a]ψ ′ for every Mi ∈ M
w
buy ¬hot shall now be admitted in at least one world of • for all ℓ ∈ w ∩ w′
some of her new models of beliefs. Hence, to contract an – either there is v ∈ base(¬ψ, W) s.t. v ⊆ w′ and ℓ ∈ v
effect law ϕ → [a]ψ from a given model, we have to put
– or there are ψ ′ ∈ Fml, v′ ∈ base(ψ ′ , W) s.t. v′ ⊆ w′ ,
new arrows leaving ϕ-worlds to worlds satisfying ¬ψ. M
In our example, when contracting token → [buy]hot in ℓ ∈ v′ , and |= i [a]ψ ′ for every Mi ∈ M
w
the model of Figure 2, we add arrows from token-worlds Mi
– or there is Mi ∈ M such that |= [a]¬ℓ
to ¬hot-worlds. The challenge in such an operation is in w
guaranteeing minimal change: because coffee → hot, and By RelTgt(w, ϕ → [a]ψ, M , M) we denote the set of all rel-
then ¬hot → ¬coffee, this should also give buy ¬coffee evant target worlds of w w.r.t. ϕ → [a]ψ for M in M.
654

5. We need the set of models M (and here we can suppose ¬t, c, h
it contains all models of the theory we want to change) be-
cause preserving effects depends on what other effects hold b b
in the other models that interest us. One needs to take them ′
into account in the local operation of changing one model:2 M1 : t, c, h b t, ¬c, h
Definition 8 Let M = W, R be a PDL-model and M be b
−
such that M ∈ M. Then M ′ = W′ , R′ ∈ Mϕ→[a]ψ if and
only if ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
′
• W =W
• R ⊆ R′ ¬t, c, h
• (w, w′ ) ∈ R′ R implies w′ ∈ RelTgt(w, ϕ → [a]ψ, M , M) b b
′ M′
• There is w ∈ W s.t. |= ϕ → [a]ψ
w
′
M2 :
t, c, h b t, ¬c, h
As having just one world where the law is no longer
true in each model is enough, taking those resulting mod-
els whose difference w.r.t. the original accessibility relation
b
is minimal guarantees minimal change: ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
Definition 9 Let M be a PDL-model and ϕ → [a]ψ an
effect law. Then ¬t, c, h
−
contraction(M , ϕ → [a]ψ) = min{Mϕ→[a]ψ , M} b b
′
M3 :
Now we can define the possible sets of models resulting t, c, h b t, ¬c, h
from contracting an effect law from a set of models:
b
Definition 10 Let M be a set of models, and ϕ → [a]ψ
an effect law. Then M− ′ ′
ϕ→[a]ψ = {M : M = M ∪ ¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
{M ′ }, M ′ ∈ contraction(M , ϕ → [a]ψ), M ∈ M}.
Figure 5: Models resulting from contracting the effect law
Taking again M = {M }, for M as in Figure 2, after token → [buy]hot in the model M of Figure 2. The new
contracting the effect law token → [buy]hot from M, we get arrows are the dashed ones.
M− ′ ′
token→[buy]hot = {M ∪ {Mi } : 1 ≤ i ≤ 3}, where all Mi s
are as depicted in Figure 5. What we can do is choose which laws we accept to lose and
postpone their change (by the other operators).
If ϕ is not satisfied by M or ψ is true in M , of course we
do not succeed in falsifying ϕ → [a]ψ. In these cases, prior The tradition in the reasoning about actions community
to do that we must change our set of possible states. says that executability laws are, in general, more difficult to
state than effect laws, and hence are more likely to be incor-
Contraction of Static Laws rect. Relying on this, in (Herzig, Perrussel, and Varzinczak
When contracting a static law in a model, we want to admit 2006) no change in the accessibility relation is made, what
at least one possible state falsifying it. Intuitively this means means preserving effect laws and postponing correction of
that we should add new worlds to the original model. This executability laws. We here embrace this solution. It is con-
is quite easy. A delicate issue however is what to do with troversial whether this approach is in line with the intuition
the accessibility relation: should new arrows leave/arrive at or not (see (Varzinczak 2008a) for an alternative). Anyway,
the new world? If no arrow leaves the new added world, with the information we have at hand, this is the safest way
we may lose an executability law. If some arrow leaves it, of contracting static laws.
we may lose an effect law, the same holding if we add an Definition 11 Let M = W, R be a PDL-model. Then
arrow pointing to the new world. If no arrow arrives at the M ′ = W′ , R′ ∈ Mϕ if and only if
−
new world, what about the intuition? Do we want to have an • W ⊆ W′
unreachable state?
• R = R′
All this discussion shows how drastic a change in the M′
static laws may be: it is a change in the underlying struc- • There is w ∈ W′ s.t. |= ϕ
w
ture (possible states) of the world! Changing it may have as The minimal modifications of one model are as expected:
consequence the loss of an effect law or an executability law.
Definition 12 Let M be a model and ϕ a static law. Then
2
We do not need M in the local contraction of executabilities −
− contraction(M , ϕ) = min{Mϕ , M}
Mϕ→ a ⊤ as all effects are preserved along the removal of arrows.
655

6. And we define the sets of models resulting from contract- contexts where ¬ϕ is the case. Second, in order to get min-
ing a static law from one set of models: imality, we must make a executable in some contexts where
ϕ is true, viz. all ϕ-worlds but one. This means that we can
Definition 13 Let M be a set of models, and ϕ a static have several action theories as outcome.
law. Then M− = {M′ : M′ = M ∪ {M ′ }, M ′ ∈
ϕ Algorithm 1 gives a syntactical operator to achieve this.
contraction(M , ϕ), M ∈ M}.
In our example, contracting the static law coffee → hot Algorithm 1 Contraction of an executability law
from M = {M }, with M as in Figure 2, will give us input: T, ϕ → a ⊤
M− ′ ′
coffee→hot = {M ∪ {M1 }, M ∪ {M2 }}, where each Mi
′ output: T − a ⊤ /* a set of theories */
ϕ→
is as depicted in Figure 6. if T |= ϕ → a ⊤ then
PDL
for all π ∈ IP(S ∧ ϕ) do
for all A ⊆ atm(π) do V
ϕA := pi ∈atm(π) pi ∧ pi ∈atm(π) ¬pi
V
¬t, c, h t, c, ¬h
pi ∈A /
pi ∈A
b b if S |= (π ∧ ϕA ) → ⊥ then
CPL
′
M1 : (T Xa ) ∪ {(ϕi ∧ ¬(π ∧ ϕA )) → a ⊤ :
t, c, h b t, ¬c, h T ′ :=
ϕi → a ⊤ ∈ Xa }
T−
ϕ→ a ⊤
:= T −
ϕ→ a ⊤ ∪ {T ′ }
else
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h T−ϕ→ a ⊤
:= {T}
¬t, c, ¬h ¬t, c, h As an example, contracting token → buy ⊤ from our
b b
theory T would give us three theories. One of them is:
 
′ coffee → hot, ¬coffee → [buy]coffee,
M2 :
 
 token → [buy]¬token, ¬token → [buy]⊥, 

 
t, c, h b t, ¬c, h  

¬token → [buy]¬token, coffee → [buy]coffee,
 
′
T1 =
 hot → [buy]hot, 
(token ∧ ¬coffee ∧ hot) → buy ⊤,

 


 

(token ∧ ¬coffee ∧ ¬hot) → buy ⊤
 
¬t, ¬c, ¬h t, ¬c, ¬h ¬t, ¬c, h
Figure 6: Models resulting from contracting the static law Contracting Effect Laws
coffee → hot in the model M of Figure 2. The new added When contracting an effect law ϕ → [a]ψ from a theory T,
coffee ∧ ¬hot-worlds are dashed. intuitively we should change some effect laws that preclude
¬ψ in target worlds. In order to cope with minimality, we
Notice that by not modifying the accessibility relation all must change only those laws that are relevant to ϕ → [a]ψ.
ϕ,ψ
the effect laws are preserved with minimal change. More- Let Ea denote the minimum subset of the effect laws in
ϕ,ψ
over, our approach is also intuitive: when learning that a Ea such that S , Ea |= ϕ → [a]ψ. In the case where the
PDL
new state is now possible, we do not necessarily know all theory is modular (Herzig and Varzinczak 2005) (see fur-
the behavior of the action in the new added state. ther), interpolation guarantees that such a set always exists.
Moreover, note that there can be more than one such a set,
ϕ,ψ ϕ,ψ
Syntactic Operators for Contraction in which case we denote them (Ea )1 , . . . , (Ea )n . Let
We now turn our attention to the definition of a syntacti- − ϕ,ψ
Ea = (Ea )i
cal counterpart of our semantic operators. As (Nebel 1989) 1≤i≤n
says, “[. . . ] finite bases usually represent [. . . ] laws, and −
when we are forced to change the theory we would like to The laws in Ea will serve as guideline to get rid of ϕ → [a]ψ
stay as close as possible to the original [. . . ] base.” Hence, in the theory.
besides the definition of syntactical operators, we should The first thing that we must do is to ensure that action
also guarantee that they perform minimal change. a still has effect ψ (if that was so) in all those contexts in
which ϕ does not hold. This means that we shall weaken the
By T − we denote in the sequel the result of contracting a ϕ,ψ
Φ laws in Ea specializing them to ¬ϕ.
law Φ from the set of laws T.
Second, we need to preserve all old effects in all ϕ-worlds
but one. To achieve that, we specialize the above laws to
Contracting Executability Laws each possible valuation satisfying ϕ but one. In the left ϕ-
For the case of contracting an executability law ϕ → a ⊤ valuation, we must ensure that action a has either its old
from an action theory, the first thing we do is to ensure that effects or ¬ψ as outcome. We achieve that by weakening
−
the action a is still executable (if that was so) in all those the consequent of the laws in Ea .
656

7. Finally, in order to get minimal change, we must ensure careful approach is to change the theory so that all action
that all literals in this ϕ-valuation that are not forced to laws remain the same in the contexts where the contracted
change in ¬ψ-worlds should be preserved. We do this by law is the case. In our example, if when contracting the law
stating an effect law of the form (ϕk ∧ℓ) → [a](ψ∨ℓ), where coffee → hot we are not sure whether buy is still executable
ϕk is the above ϕ-valuation. The reason why this is needed or not, we should weaken our executability laws specializ-
is clear: there can be several ¬ψ-valuations, and as far as we ing them to the context coffee → hot, and then make buy a
want at most one to be reachable from ϕk , we should force priori inexecutable in all ¬(coffee → hot) contexts.
it to be the one whose difference to ϕk is minimal.
Algorithm 3 below formalizes such an operation.
Again, the result will be a set of theories. Algorithm 2
below gives the operator. Algorithm 3 Contraction of a static law
input: T, ϕ
Algorithm 2 Contraction of an effect law output: T − /* a set of theories */
ϕ
input: T, ϕ → [a]ψ if S |= ϕ then
CPL
output: T −ϕ→[a]ψ /* a set of theories */
for all S − ∈ S ⊖ ϕ do
if T |= ϕ → [a]ψ then
PDL ((T S ) ∪ S − ) Xa ∪
for all π ∈ IP(S ∧ ϕ) do ′
T := {(ϕi ∧ ϕ) → a ⊤ : ϕi → a ⊤ ∈ Xa } ∪
for all A ⊆ atm(π) do V {¬ϕ → [a]⊥}
ϕA := pi ∈atm(π) pi ∧ pi ∈atm(π) ¬pi
V
pi ∈A /
pi ∈A
T − := T − ∪ {T ′ }
ϕ ϕ
if S |= (π ∧ ϕA ) → ⊥ then
CPL else
for all π ′ ∈ IP(S ∧ ¬ψ) do T − := {T}
ϕ
T ′ := (T Ea ) ∪
−
−
{(ϕi ∧ ¬(π ∧ ϕA )) → [a]ψi : ϕi → [a]ψi ∈ Ea } ∪ In our running example, contracting the law coffee → hot
{(ϕi ∧ π ∧ ϕA ) → [a](ψi ∨ π ′ ) : ϕi → [a]ψi ∈ Ea }
−
from T produces two theories, one of them is
for all L ⊆ Lit do 
¬(¬token ∧ coffee ∧ ¬hot),

if S |= (π ∧ ϕA ) → ℓ∈L ℓ and S |= (π ′ ∧
V  
(token ∧ coffee → hot) → buy ⊤,
CPL CPL

 

V
ℓ∈L ℓ) → ⊥ then

 

¬coffee → [buy]coffee, token → [buy]¬token,
 
for all ℓ ∈ L do ′
T1 =
if T |= (π ∧ ϕA ∧ ℓ) → [a]¬ℓ or ℓ ∈ π ′  ¬token → [buy]⊥, ¬token → [buy]¬token, 
PDL
coffee → [buy]coffee, hot → [buy]hot,

 

then 
 

T := T ′ ∪ {(π ∧ ϕA ∧ ℓ) → [a](ψ ∨ ℓ)}
′ (coffee ∧ ¬hot) → [buy]⊥
 
T−ϕ→[a]ψ
:= T − ′
ϕ→[a]ψ ∪ {T }
else Observe that the effect laws are not affected by the
T−ϕ→[a]ψ
:= {T} change: as far as we do not state executabilities for the new
world, all the effect laws remain true in it.
For instance, contracting the effect law token → [buy]hot If the knowledge engineer is not happy with the added in-
from T will give us three resulting theories, one of them is executability law (coffee∧¬hot) → [buy]⊥, she can contract
T1′ = it from the theory using Algorithm 2.
coffee → hot, token → buy ⊤, Correctness of the Operators
 
token → [buy]¬token, ¬token → [buy]⊥,

 

 
Here we show that our algorithms are correct w.r.t. our se-
 
¬token → [buy]¬token,

 

 


(coffee ∧ ¬(token ∧ coffee ∧ hot)) → [buy]coffee,

 mantics for action theory contraction. Before doing that, we
(hot ∧ ¬(token ∧ coffee ∧ hot)) → [buy]hot, need a definition.
 
 (¬coffee ∧ ¬(token ∧ coffee ∧ hot)) → [buy]coffee, 
 

 
 Definition 14 (Modularity (Herzig and Varzinczak 2005))
 (token ∧ coffee ∧ hot) → [buy](coffee ∨ ¬hot), 



 An action theory T is modular if and only if for every
 
(token ∧ coffee ∧ hot) → [buy](hot ∨ ¬coffee) ϕ ∈ Fml, if T |= ϕ, then S |= ϕ.
PDL CPL
Contracting Static Laws For an example of a non-modular theory, suppose in our
Finally, in order to contract a static law from a theory, we can action theory T we had stated the law buy ⊤ instead of
use any standard contraction/revision operator ⊖ for classi- token → buy ⊤. Then T |= token and S |= token.
PDL CPL
cal propositional logic to change the set of static laws S . In (Herzig and Varzinczak 2005) algorithms are given to
Because contracting static laws means admitting new pos- check whether T satisfies the principle of modularity and
sible states (cf. the semantics), it may be the case that just also to make T satisfy it, if that is not the case.
modifying S is not enough.
Since we in general do not necessarily know the behav- Theorem 3 T is modular if and only if its big model is a
ior of the actions in a new discovered state of the world, a model of T.
657

8. Modular theories have interesting properties. For exam- Thanks to modularity, our operators also satisfy Katsuno
ple, if T is modular, then its consistency can be checked by and Mendelzon’s (C5) postulate, recovery:
just checking consistency of the set of static laws S alone. Theorem 10 Let T be modular. T ′ ∪ {Φ} |= T, for all
PDL
Deduction of effect laws does not need the executability
T ′ ∈ T−.
Φ
ones and vice versa. Prediction of an effect of a sequence of
actions a1 ; . . . ; an does not need the effect laws for actions Theorem 11 If T is modular, then every T ′ ∈ T − is also
Φ
other than a1 , . . . , an . This also applies to plan validation modular.
when deciding whether a1 ; . . . ; an ϕ is the case. For more Besides satisfying all postulates for contraction, our opera-
results on modularity, see (Herzig and Varzinczak 2007). tors also preserve modularity. This is a nice property, since
The following theorem (see Appendix A for the proof) it means that modularity can be checked/ensured once for all
establishes that the semantic contraction of the law Φ from during the theory’s evolution.
the set of models of the action theory T produces models of
some contracted theory in T − .
Φ
Related Work
To the best of our knowledge, the first work on updating ac-
Theorem 4 Let T be modular, and Φ be a law. For all M′ ∈
M tion theories is that by (Li and Pereira 1996) in a narrative-
M− such that |= T for every M ∈ M, there is T ′ ∈ T −
Φ Φ based action description language (Gelfond and Lifschitz
M′
such that |= T ′ for every M ′ ∈ M′ . 1993). Contrary to us, however, they investigate the problem
of updating the narrative with new observed facts and (pos-
The next theorem establishes the other way round: models sibly) with occurrences of actions that explain those facts.
of theories in T − are all models of the semantical contraction
Φ This amounts to updating a given state/configuration of
of Φ from models of T. (The proof is in Appendix B.) the world (in our terms, what is true in a possible world) and
Theorem 5 Let T be modular, Φ a law, and T ′ ∈ T − . For
Φ
focusing on the models of the narrative in which some ac-
M′ tions took place (in our terms, the models of the action the-
all M ′ such that |= T ′ , there is M′ ∈ M− such that
Φ ory with a particular sequence of action executions). Clearly
M
M ′ ∈ M′ and |= T for every M ∈ M. the models of the action laws remain the same.
Hence our operators are correct w.r.t. the semantics. (Liberatore 2000) proposes an action language in which
one can express a given semantics for belief update, like
Assessment of Postulates for Change (Winslett 1988) and (Katsuno and Mendelzon 1992). Up-
date operations are then expressed as action laws in a theory.
We now analyze our operator’s behavior w.r.t. Katsuno and
The main difference between Liberatore’s work and Li
Mendelzon’s classical contraction postulates. (Due to space
and Pereira’s is that Liberatore’s framework allows for ab-
limitations, proofs are omitted here. They are all available
ductively adding to the action theory new effect propositions
at (Varzinczak 2008a).)
(effect laws, in our terms) that consistently explain the oc-
Theorem 6 T |= T ′ , for all T ′ ∈ T − .
PDL Φ currence of an event.
This result means our operators satisfy the PDL-version of The work by (Eiter et al. 2005) is similar to ours in that
Katsuno and Mendelzon’s (C1) postulate about monotonic- they also propose a framework for updating action laws.
ity. Such a postulate is not satisfied by the operators given They mainly investigate the case where e.g. a new effect law
in (Herzig, Perrussel, and Varzinczak 2006): there, when re- shall be added to the description. This problem is the dual
moving e.g. an executability law ϕ → a ⊤ one may make of contraction and is then closer to revision.
ϕ → [a]⊥ valid in all models of the resulting theory. In Eiter et al.’s approach, action theories are also de-
scribed in a variant of a narrative-based action language.
Theorem 7 If T |= Φ, then |= T ↔ T ′ , for all T ′ ∈ T − .
Φ
PDL PDL Like here, the semantics is in terms of transition systems.
This corresponds to Katsuno and Mendelzon’s (C2) postu- Contrary to us, the minimality condition on the outcome of
late about preservation. Whenever T |= Φ, then the models
PDL the update is in terms of inclusion of sets of laws, which
of the resulting theory are exactly the models of T, because means the approach is more syntax-oriented than ours.
these are the minimal models falsifying Φ. Both their framework and ours can be qualified as
Theorem 8 Let T = S ∪ E ∪ X be consistent, and Φ be constraint-based update, in that the update is carried out rel-
an executability or an effect law such that S |= Φ. If T is ative to a set of laws that one wants to hold in the result.
PDL Here for example, all changes in the action laws are relative
modular, then T ′ |= Φ for every T ′ ∈ T −
PDL Φ to the static laws in S .
Thus, under modularity our operators satisfy the success One difference between our approach and Eiter et al.’s is
postulate (C3). Still under modularity and the assumption that there it is also possible to update a theory relatively to
that the classical contraction operator satisfies Katsuno and e.g. executability laws: when expanding T with a new effect
Mendelzon’s (C4) postulate, our operations also satisfy it: law, one may want to constrain the change so that the action
Theorem 9 Let T1 and T2 be modular. If |= T1 ↔ T2 under concern is guaranteed to be executable in the result.
PDL This may of course require the withdrawal of some static
−
and |= Φ1 ↔ Φ2 , then for each T1′ ∈ (T1 )Φ2 there is
PDL law. Hence, in Eiter et al.’s framework, static laws do not
−
T2′ ∈ (T2 )Φ1 such that |= T1′ ↔ T2′ , and vice-versa.
PDL
have the same status as in ours.
658

9. Concluding Remarks Gelfond, M., and Lifschitz, V. 1993. Representing action
The contributions of the present work are as follows: and change by logic programs. Journal of Logic Program-
ming 17(2/3&4):301–321.
• What is the meaning of removing a law Φ from an action
Hansson, S. 1999. A Textbook of Belief Dynamics: The-
theory T? How to get minimal change, i.e., how to keep
ory Change and Database Updating. Kluwer Academic
as much knowledge about other laws as possible? We
Publishers.
answered these questions with Definitions 6, 10 and 13.
Harel, D.; Tiuryn, J.; and Kozen, D. 2000. Dynamic Logic.
• How to syntactically contract an action theory so that Cambridge, MA: MIT Press.
its result corresponds to the intended semantics? We
answered this question with Algorithms 1–3 and Theo- Herzig, A., and Rifi, O. 1999. Propositional belief
rems 4 and 5. base update and minimal change. Artificial Intelligence
115(1):107–138.
• Is our method closer to update or revision? Does it Herzig, A., and Varzinczak, I. 2005. On the modularity of
comply with the standard postulates for classical theory theories. In Advances in Modal Logic, volume 5. King’s
change and what are the differences w.r.t. that? We an- College Publications. 93–109.
swered these questions with Theorems 6–11.
Herzig, A., and Varzinczak, I. 2007. Metatheory of actions:
We have shown the importance that modularity has in ac- beyond consistency. Artificial Intelligence 171:951–984.
tion theory change. Under modularity, our operators sat-
Herzig, A.; Perrussel, L.; and Varzinczak, I. 2006. Elab-
isfy all Katsuno and Mendelzon’s postulates for contraction.
orating domain descriptions. In Proc. 17th Eur. Conf. on
This shows that our modularity notion is fruitful. Moreover,
Artificial Intelligence (ECAI’06), 397–401. IOS Press.
considering future modifications one should perform on the
theory, since modularity is preserved by our operators, it suf- Jin, Y., and Thielscher, M. 2005. Iterated belief revision,
fices to check/ensure it only once. revised. In Proc. 19th Intl. Joint Conf. on Artificial Intelli-
gence (IJCAI’05), 478–483. Morgan Kaufmann.
Here we presented the case for contraction. We are cur- Katsuno, H., and Mendelzon, A. 1992. On the difference
rently investigating the definition of the revision counterpart between updating a knowledge base and revising it. In Be-
of action theory change. The first results on this issue are lief revision. Cambridge University Press. 183–203.
available in (Varzinczak 2008b).
Li, R., and Pereira, L. 1996. What is believed is what is ex-
Our ongoing research is on how to contract not only laws plained. In Proc. 13th Natl. Conf. on Artificial Intelligence
but any PDL-formula. Definitions 4, 8 and 11 show up to (AAAI’96), 550–555. AAAI Press/MIT Press.
be important for better understanding the case of general Liberatore, P. 2000. A framework for belief update. In
formulas: the modifications to perform in a given model Proc. 7th Eur. Conf. on Logics in Artificial Intelligence
in order to falsify a general formula will also comprise re- (JELIA’2000), 361–375.
moval/addition of arrows and worlds. The definition of a
McCarthy, J., and Hayes, P. 1969. Some philosophi-
more general contraction method will thus benefit from our
cal problems from the standpoint of artificial intelligence.
present constructions.
In Machine Intelligence, volume 4. Edinburgh University
Press. 463–502.
Acknowledgements
Nebel, B. 1989. A knowledge level analysis of belief re-
The author is thankful to Andreas Herzig and Laurent Per- vision. In Proc. Intl. Conf. on Knowledge Representation
russel for interesting discussions on the subject of this work. and Reasoning (KR’89), 301–311. Morgan Kaufmann.
He is also grateful to the anonymous referees for their useful
Parikh, R. 1999. Beliefs, belief revision, and splitting lan-
comments on an earlier version of this paper.
guages. In Logic, Language and Computation, volume 2
This work has been partially supported by a fellowship of CSLI Lecture Notes. CSLI Publications. 266–278.
from the government of the F EDERATIVE R EPUBLIC OF Quine, W. V. O. 1952. The problem of simplifying truth
B RAZIL. Grant: CAPES BEX 1389/01-7. functions. American Mathematical Monthly 59:521–531.
Shapiro, S.; Pagnucco, M.; Lesp´ rance, Y.; and Levesque,
e
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659

10. − ϕ,ψ
Appendix A: Proof of Theorem 4 Now let Ea = 1≤i≤n (Ea )i and let the theory
Lemma 1 T |= T ′ .
PDL
T ′ = (T Ea ) ∪
−
For the proof of this lemma, the reader is invited to −
{(ϕi ∧ ¬(π ∧ ϕA )) → [a]ψi : ϕi → [a]ψi ∈ Ea } ∪
check (Varzinczak 2008a).
{(ϕi ∧ π ∧ ϕA ) → [a](ψi ∨ π ′ ) : ϕi → [a]ψi ∈ Ea } ∪
−
Proof of Theorem 4
 
 (π ∧ ϕA ∧ ℓ) → [a](ψ ∨ ℓ) : ℓ ∈ L, for L ⊆ Lit s.t. 
M
Let M = {M :|= T}, and M′ ∈ M− . We show that S |= (π ′ ∧ ℓ∈L ℓ) → ⊥, and
CPL
Φ
M′
 ℓ ∈ π ′ or T |= (π ∧ ϕA ∧ ℓ) → [a]¬ℓ 
there is T ′ ∈ T − such that |= T ′ for every M ′ ∈ M′ .
Φ
PDL
M′ (Clearly, T ′ is a theory produced by Algorithm 2.)
By definition, each M ′ ∈ M′ is such that either |= T or
M′ M′ In order to show that M ′ is a model of T ′ , it is enough to
|= Φ. Because T − = ∅, there must be T ′ ∈ T − . If |= T,
Φ Φ show that it is a model of the added laws. Given (ϕi ∧ ¬(π ∧
M′
by Lemma 1 |= T ′ and we are done. Let’s then suppose M′
ϕA )) → [a]ψi ∈ T ′ , for every w ∈ W′ , if |= ϕi ∧¬(π∧ϕA ),
′
M w
that |= Φ. We analyze each case. M′ M M M
then |= ϕi , and then |= ϕi . Because |= ϕi → [a]ψi , |= ′ ψi
w w w
Let Φ have the form ϕ → a ⊤ for some ϕ ∈ Fml. Then for all w′ ∈ W such that (w, w′ ) ∈ Ra . We need to show
M ′ = W′ , R′ , where W′ = W, R′ = R Rϕ , with Rϕ = M
M
a a
that R′ (w) = Ra (w). If |= ϕ, then Rϕ,¬ψ = ∅, and then
a a
{(w, w′ ) :|= ϕ and (w, w′ ) ∈ Ra }, for some M ∈ M. w
M
w
′ M′ M′ R′ (w) = Ra (w). If |= ϕ, then either w = u, and from
a w
Let u ∈ W be such that |= ϕ → a ⊤, i.e., |= ϕ and
u u M′ M′
R′ (u) = ∅. |= π ∧ ϕA we conclude |= (ϕi ∧ ¬(π ∧ ϕA )) → [a]ψi ,
u u
a
Because u ϕ, there must be v ∈ base(ϕ, W′ ) such that or w = u, and then we must have Rϕ,¬ψ = ∅, otherwise
a
v ⊆ u. Let π = ℓ∈v ℓ. Clearly π is a prime implicant of there would be Sϕ,¬ψ ⊂ Rϕ,¬ψ such that R−(R ∪ Sϕ,¬ψ ) ⊂
a a
˙ a
S ∧ ϕ. Let also ϕA = ℓ∈uv ℓ, and consider R−(R ∪ Rϕ,¬ψ ), and then M ′′ = W′ , R ∪ Sϕ,¬ψ would be
˙ a a
M ′′
T ′ = (TXa )∪{(ϕi ∧¬(π∧ϕA )) → a ⊤ : ϕi → a ⊤ ∈ Xa } such that |= ϕ → [a]ψ and M ′′ M M ′ , a contradiction
since M ′ is minimal w.r.t. M . Hence R′ (w) = Ra (w),
a
(Clearly, T ′ is a theory produced by Algorithm 1.) M′
and |= ′ ψi for all w′ such that (w, w′ ) ∈ R′ .
w a
It is enough to show that M ′ is a model of the new added
laws. Given (ϕi ∧ ¬(π ∧ ϕA )) → a ⊤ ∈ T ′ , for every Now, given (ϕi ∧ π ∧ ϕA ) → [a](ψi ∨ π ′ ), for every w ∈
′ ′ M′ M′ M
M
w ∈ W′ , if |= ϕi ∧ ¬(π ∧ ϕA ), then |= ϕi , from what it
M
W′ , if |= ϕi ∧π∧ϕA , then |= ϕi , and then |= ϕi . Because,
w w w
w w
M M
M M
follows |= ϕi . Because |= ϕi → a ⊤, there is w′ ∈ W |= ϕi → [a]ψi , we have |= ′ ψi for all w′ ∈ W such that
w
w ′
M
such that w′ ∈ Ra (w). We need to show that (w, w′ ) ∈ (w, w′ ) ∈ Ra , and then |= ′ ψi for every w′ ∈ W′ such that
M M w
R′ . If |= ϕ, then Rϕ = ∅, and (w, w′ ) ∈ R′ . If |= ϕ, M′
a w a a w (w, w′ ) ∈ R′ Rϕ,¬ψ . Now, given (w, w′ ) ∈ Rϕ,¬ψ , |= ′ π ′ ,
a a a w
M′ M′
either w = u, and then from |= π ∧ ϕA we conclude |=
u u
and the result follows.
(ϕi ∧ ¬(π ∧ ϕA )) → a ⊤, or w = u and then we must Now, for each (π∧ϕA ∧ℓ) → [a](ψ∨ℓ), for every w ∈ W′ ,
have (w, w′ ) ∈ R′ , otherwise there is Sϕ ⊂ Rϕ such that
a a a M′ M′ M
R−(R Sϕ ) ⊂ R−(R Rϕ ), and then M ′′ = W′ , R Sϕ is
˙ ˙ if |= π ∧ ϕA ∧ ℓ, then |= ϕ, and then |= ϕ. Because
w w w
a a a
M ′′ M M
such that |= ϕ → a ⊤ and M ′′ M M ′ , a contradiction |= ϕ → [a]ψ, we have |= ′ ψ for every w′ ∈ W such that
w
because M ′ is minimal w.r.t. M . Thus (w, w′ ) ∈ R′ , and
a M′
(w, w′ ) ∈ Ra , and then |= ′ ψ for all w′ ∈ W′ such that
′ ′
M M w
then |= a ⊤. Hence |= T ′ . M′
w (w, w′ ) ∈ R′ Rϕ,¬ψ . It remains to show that |= ′ ℓ for
a a w
Now let Φ be of the form ϕ → [a]ψ, for ϕ, ψ ∈ Fml. ′
every w ∈ W such that (w, w′ ) ∈ Rϕ,¬ψ . Since M ′ is
′
a
Then M ′ = W′ , R′ , where W′ = W, R′ = R ∪ Rϕ,¬ψ , with
a M′
minimal, it is enough to show that |=′ ℓ for every ℓ ∈ Lit
u
Rϕ,¬ψ = {(w, w′ ) : w′ ∈ RelTgt(w, ϕ → [a]ψ, M , M)} M′
a such that |=
u
π ∧ ϕA ∧ ℓ. If ℓ ∈ π ′ , the result follows.
for some M = W, R ∈ M. M′
M′ Otherwise, suppose |=′ ℓ. Then
′ u
Let u ∈ W be such that |= ϕ → [a]ψ. Then there is
u
M′
• either ¬ℓ ∈ π ′ , then π ′ and ℓ are unsatisfiable, and in this
′
′
u ∈ W such that (u, u ) ∈ ′
R′
and |=′ ψ. Because u ϕ,
u a case Algorithm 2 has not put the law (π ∧ ϕA ∧ ℓ) →
there is v ∈ base(ϕ, W′ ) such that v ⊆ u, and as u′ ¬ψ, [a](ψ ∨ ℓ) in T ′ , a contradiction;
there must be v ′ ∈ base(¬ψ, W′ ) such that v ′ ⊆ u′ . Let • or ¬ℓ ∈ u′ v ′ . In this case, there is a valuation u′′ = (u′
π = ℓ∈v ℓ, ϕA = ℓ∈uv ℓ, and π ′ = ℓ∈v′ ℓ. Clearly π {¬ℓ}) ∪ {ℓ} such that u′′ ψ. We must have u′′ ∈ W′ ,
(resp. π ′ ) is a prime implicant of S ∧ ϕ (resp. S ∧ ¬ψ). otherwise there will be L′ = {ℓi : ℓi ∈ u′′ } such that
660