Epistemic Actions and Ontic Actions:
a Unified Logical Framework
Andreas Herzig and Tiago De Lima?
IRIT, Université Paul Sabatier, Toulouse, France
Abstract. We present a reasoning about actions framework based on a
sum of epistemic logic S5 and propositional dynamic logic PDL together
with a ‘no forgetting’ principle, also called perfect recall. We show that
in our framework an action may be decomposed into a purely ontic action followed by a purely epistemic action. We also show that the latter
is completely definable in terms of simple observations, i.e., ‘test that’
actions and that they are equivalent to public announcements of public
announcement logic PAL as studied by Plaza, van Benthem and others.
Finally, since these actions respect ‘no learning’ principle we show that
a unified reduction method based on regression, as studied by Reiter,
applies.
1
Introduction
We present a general framework allowing to reason about actions in the case
where the agent has no complete information about the state of the world.
In this case the agent must be able to perform not only physical, STRIPSlike actions (that we call here ontic actions), but also epistemic actions that
allow to acquire information about the state of the world. Such actions include
observations (learning that a proposition is true) and tests (sensing actions).
We are interested in mono-agent environments where all action laws are
known and events are public. The domain described below, taken from [1], is
an example.
Example 1 (Princess). The domain consists of 2 doors. Behind one of the doors
there is a tiger and behind the other one there is a princess. The agent does not
know where tiger and princess are. The available actions are the following.
– listen(i): listen to what happens behind door i, which results in hearing the
tiger roaring if there is one behind the door;
– open(i): open door i, which results in marrying the princess or being eaten
by the tiger, depending on what is behind the door.
The agent’s goal is to marry the princess and to stay alive. Notice that listen(i)
is an epistemic action and open(i) is an ontic action. A possible plan to achieve
that is the sequence of actions: listen(1); if K tiger (1)thenopen(2)elseopen(1). u
t
?
Supported by the Programme Alßan, the European Union Programme of High Level
Scholarships for Latin America, scholarship number E04D041703BR.
2
Andreas Herzig and Tiago De Lima
Our aim in this paper is to design the simplest framework able to deal with
such examples. This framework will be in terms of an epistemic dynamic logic
EDL. It is the sum of S5 logic for modelling knowledge and propositional dynamic logic (without ‘∗’) for modelling actions, together with a perfect recall
(‘no forgetting’) axiom. The latter says that if the agent knows the effects of an
action, then after execution of that action the agent knows the effects.
After introducing syntax and semantics of EDL in Section 2 we show, in
Section 3, that in EDL every action can be decomposed into an ontic action
followed by an epistemic action. We then show that epistemic actions can be
reduced to sequences of observations in Section 4. It turns out that the latter
are nothing but announcements of public announcement logic PAL (as studied
by Plaza [2], Gerbrandy [3], Baltag [4], van Benthem [5] and van Ditmarsch [6]).
In Section 5 we briefly recall the dynamic logic regression method. In Section 6
we show that under some reasonable hypothesis the reduction axiom based proof
method for PAL can be combined with Reiter’s regression into a proof method
for our general framework. Finally, in Section 7 we present some discussions and
the conclusion of the paper.
2
Preliminaries
Definition 1. Let P be a countably infinite set of propositional letters. The
language of epistemic dynamic logic, LEDL , is defined by the following BNF.
ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | K ϕ | [α]ϕ
where p ranges over P and α belongs to the action language defined as follows.
Let A be a countably infinite set of (abstract) actions. The action language L(A)
is defined by the following BNF.
α ::= skip | a | if ϕ then α else α | α; α | α ∪ α
where a ranges over A.
The formula K ϕ is read ‘agent knows that ϕ’ and the formula [α]ϕ is read
‘ϕ holds after every execution of α’. The construction α1 ; α2 is the sequential
composition of α1 and α2 , and the construction α1 ∪ α2 is the nondeterministic
choice between α1 and α2 . To simplify notation, we use the common abbreviations for ∨, →, ↔, hαi (we recall that hαiϕ = ¬[α]¬ϕ). In the sequel, two
fragments of LEDL are important. The first one contains no modal operators.
These formulae are called boolean formulae and we note this fragment LPL . The
second one contains LPL and also formulae with the operator K . These formulae
are called epistemic formulae and we note this fragment LS5 .
This language permits description of effect and executability laws. For example, the formula tiger (1) → [open(1)]¬alive means that if there is a tiger behind
door 1, then the agent will die after opening it. The formula hopen(1)i> ↔ alive
means that action open(1) is executable if and only if the agent is alive. Also conditional plans are possible such as listen(1); if K tiger (1) then open(1) else open(2).
Epistemic Actions and Ontic Actions: a Unified Logical Framework
3
Definition 2. An EDL-model is a tuple of the form M = hW, R, T, V i where
–
–
–
–
–
W is a non-empty set of possible worlds;
R ⊆ ℘(W × W ) is a reflexive, symmetric and transitive relation.
T : L(A) → ℘(W × W ) is a set of transitions, noted Tα .
V : P → ℘(W ) associates an interpretation to each p ∈ P , noted V (p).
for all α ∈ L(A) : Tα ◦ R ⊆ R ◦ Tα .
For convenience, we define R(w) = {w0 : (w, w0 ) ∈ R} and Tα (w) = {w0 :
(w, w0 ) ∈ Tα }. The relation R models the epistemic state of the agent: R(w) is
the set of possible worlds the agent considers to be possible at w. The transitions
Tα model actions: Tα (w) is the set of worlds resulting from the execution of α
in w.
The interaction constraint makes that the worlds the agent considers possible
after execution of α in w, i.e. (Tα ◦ R)(w), are a subset of the outcomes of α that
the agent considers possible at w, i.e. (R ◦ Tα )(w). Such a constraint is called
perfect recall in [7] and no forgetting in [1].
Definition 3. The satisfaction relation is defined as follows.
M, w
M, w
M, w
M, w
M, w
p
¬ϕ
ϕ1 ∧ ϕ2
Kϕ
[α]ϕ
iff w ∈ V (p)
iff M, w 6 ϕ
iff M, w ϕ1 and M, w ϕ2
iff for all w0 ∈ R(w), M, w0 ϕ
iff for all w0 ∈ Tα (w), M, w0 ϕ
Definition 4. An EDL-standard model is a model such that
Tskip
= {(w, w) : w ∈ W }
Tif ϕ then α1 else α2 = {(w, w0 ) :M, w ϕ implies (w, w0 ) ∈ Tα1 and
M, w 6 ϕ implies (w, w0 ) ∈ Tα2 }
Tα1 ;α2
= Tα1 ◦ Tα2
Tα1 ∪α2
= Tα1 ∪ Tα2
Definition 5. A formula ϕ is true in M , noted M ϕ, if and only if M, w ϕ
for all w ∈ W . A formula ϕ is valid, noted |= ϕ, if and only if M ϕ for all
standard models M . A formula ϕ is a valid consequence of a set of formulae Σ,
noted Σ |= ϕ, if and only if M ψ for all ψ ∈ Σ implies M ϕ for all standard
models M .
No specific conditions for Ta are given. It means that their behaviour must
be defined with formulae in LEDL . For instance, for the Example 1 we must write
the set of formulae ∆ that allows to infer action laws such as
tiger (i) → [listen(i)]K tiger (i)
¬tiger (i) → [listen(i)]K ¬tiger (i)
hlisten(i)i> ↔ alive
princess(i) → [open(i)]married
tiger (i) → [open(i)]¬alive
hopen(i)i> ↔ alive .
4
Andreas Herzig and Tiago De Lima
A solution to this planning problem is an action (or plan) π such that
∆ |= (alive ∧ ¬married ) → hπi(alive ∧ married ) .
One can prove that it holds for π = listen(1); ifK tiger (1)thenopen(2)elseopen(1).
It follows from standard results in modal logic on Sahlqvist formulae [8] that EDL
has a sound and complete proof system made up of the standard Modus Ponens
and Necessitation inference rules for K and every [α], plus the following axiom
schemes.
K(K ).
K (ϕ1 → ϕ2 ) → (K ϕ1 → K ϕ2 )
T(K ).
Kϕ → ϕ
4(K ).
K ϕ → KK ϕ
5(K ).
¬K ϕ → K ¬K ϕ
K(α).
[α](ϕ1 → ϕ2 ) → ([α]ϕ1 → [α]ϕ2 )
Def(skip). [skip]ϕ ↔ ϕ
Def(if). [if ϕ1 then α1 else α2 ]ϕ2 ↔ ((ϕ1 → [α1 ]ϕ2 ) ∧ (¬ϕ1 → [α2 ]ϕ2 ))
Def(;).
[α1 ; α2 ]ϕ ↔ [α1 ][α2 ]ϕ
Def(∪). [α1 ∪ α2 ]ϕ ↔ [α1 ]ϕ ∧ [α2 ]ϕ
NF.
K [α]ϕ → [α]K ϕ
The first four are the standard axioms of S5 logic, and the following four are
standard axioms of PDL. NF is the ‘no forgetting’ axiom: if the agent knows
that ϕ holds after α, then he indeed knows ϕ after α.
EDL is decidable and expressive enough for our purpose. However, for practical applications, it is not as suitable as desired. The complexity of the validity
problem in EDL is PSPACE-hard [1].
In this paper we go beyond EDL. First, we prove that every action can be
decomposed in a purely ontic followed by a purely epistemic action. Next, we
propose a refined analysis of the concept of epistemic actions. This results in a
stronger logic, which in particular augments EDL by a no learning principle.
3
The decomposition theorem
Roughly, purely ontic actions stand for actions that do not involve any perception: the agent only knows that action α has been performed, without learning
about its (possibly nondeterministic or conditional) effects. On the other hand,
purely epistemic actions cannot change facts about the world. Typical examples
are sensing actions (testing whether a proposition is true or not) and observations (learning that a proposition is true). The definitions below make precise
this distinction.
Definition 6. In a model M , o ∈ A is purely ontic if and only if To satisfies
the following properties.
– Epistemic determinism: if w1 , w2 ∈ To (w), then R(w1 ) = R(w2 ).
– No learning: if w0 ∈ (R ◦ To )(w) and To (w) 6= ∅, then w0 ∈ (To ◦ R)(w).
Epistemic Actions and Ontic Actions: a Unified Logical Framework
5
The epistemic determinism constraint corresponds to the fact that the agent
cannot distinguish between nondeterministic outcomes of an action: whether the
coin falls heads or tails, the agent only knows that a coin has been tossed and
that the disjunction holds. The ‘no learning’ constraint corresponds to the fact
that if the agent considers that w0 is a possible outcome of execution of o in w,
then the agent keeps on considering w0 to be a possible world after o.
Definition 7. In a model M , e ∈ A is purely epistemic if and only if it satisfies
the following property.
– Preservation: If w0 ∈ Te (w), then for all p ∈ P , (w ∈ V (p) iff w0 ∈ V (p)).
This constraint corresponds to the fact that epistemic actions do not change
the facts about the world.
Now, we show that these two kinds of actions is all we need: every transition
relation can be decomposed appropriately.
Theorem 1 (Decomposition). Let α ∈ L(A) and let ϕ ∈ LEDL . The formula
ϕ is satisfiable if and only if there exist actions o and e such that: o is purely
ontic, e is purely epistemic, and ϕ[(e; o)/α] (the formula obtained by replacing
e; o for α in ϕ) is satisfiable.
Proof (sketch). From right to left is straightforward. From left to right is established by introducing intermediate worlds that correspond to the outcome of
action o.
t
u
It enables us to make a partition in the set A of abstract actions. From now
on, it is formed by the union of two disjoint sets: Ae of purely epistemic and Ao
of purely ontic actions.
4
How many kinds of epistemic actions are there?
We start our analysis by considering the most basic kind of epistemic action:
observations 1 . It can roughly be understood as an exogenous event that makes
the agent observe that ϕ. The observation that ϕ holds is noted obs ϕ. Then,
the formula [obs ϕ]ψ is read ‘ψ holds after the agent observes that ϕ’. We now
consider a variant of EDL where the only epistemic actions are observations. We
therefore restrict our attention to the following language.
Definition 8. Let o range over Ao . The language of EDLobs LEDLobs , is the
same as given in Definition 1, with the difference that Ae = ∅ and actions α
are now elements of the ontic action language L(Ao ), defined according to the
following BNF.
α ::= skip | o | obs ϕ | if ϕ then α else α | α; α | α ∪ α
1
This action is also named test that in some different approaches.
6
Andreas Herzig and Tiago De Lima
Definition 9. An EDLobs -standard model is an EDL-standard model such that
every Tobs ϕ satisfies the following constraints:
– If w0 ∈ Tobs ϕ (w) then for all p ∈ P , (w ∈ V (p) iff w0 ∈ V (p));
– If M, w 6 ϕ then Tobs ϕ (w) = ∅;
– If M, w ϕ then exists w0 ∈ Tobs ϕ (w) and for all w00 ∈ Tobs ϕ (w), w0 and
w00 satisfy the same formulae;
– If w0 ∈ Tobs ϕ (w) then R(w0 ) = (R ◦ Tobs ϕ )(w).
The first constraint says that observations are purely epistemic actions. The
second constraint says that truth of ϕ is a necessary condition for executability
of obs ϕ. The third condition says that observations are deterministic. And the
fourth condition says that observations are also ontic actions.
It can be shown that the following validities characterise observations (note
that the last one corresponds to the ‘no learning’ principle).
Pre(obs ). Φ → [obs ϕ]Φ for a boolen formula Φ
Exe(obs ). ϕ ↔ hobs ϕi>
Det(obs ). hobs ϕiψ → [obs ϕ]ψ
NL(obs ). [obs ϕ]K ψ → ([obs ϕ]⊥ ∨ K [obs ϕ]ψ)
However, other kinds of epistemic actions exist. For instance, listen(1) (Example 1) is not an observation. It is what we call a test 2 : given a formula ϕ
it returns whether ϕ holds or not. We note this kind of action test ϕ, and the
formula [test ϕ]χ is read ‘χ holds after the agent tests whether ϕ’. For example,
listen(1) can be written ‘test tiger (1)’.
In fact, listen(1) is conditional: the test depends on the context. It is noted,
test ϕ if ψ, where ψ is the condition for the test whether ϕ. The same may also
be applied to observations. Then, the formula [obs ϕ if ψ]χ is read ‘under the
condition ψ, χ holds after the agent observes that ϕ’. For example, listen(1) is
test tiger (1) if alive. We can see these constructions as abbreviations:
def
hobs ϕ if ψiχ = ψ ∧ hobs ϕiχ
htest ϕiχ
def
= hobs ϕiχ ∨ hobs ¬ϕiχ
def
htest ϕ if ψiχ = ψ ∧ htest ϕiχ
We leave to the reader the confirmation that all these definitions match the
intuitions behind the actions introduced above.
We want to have all kinds of epistemic actions in our logic. Tests and conditional observations are definable in terms of unconditional observations. Still
better, in the theorem below, we prove that every purely epistemic action can
be defined in terms of this operator.
Theorem 2 (Observations are general). Suppose that P and A are finite.
Let ϕ ∈ LEDL and let e be a purely epistemic action. The formula ϕ is satisfiable
in finite models if and only if there exists a (complex) observation such that
ϕ[/e] is satisfiable in finite models.
2
This action is also named test if or sense in the literature.
Epistemic Actions and Ontic Actions: a Unified Logical Framework
7
Proof (sketch). From right to left: since is purely epistemic, take Te = T .
From left to right: suppose M, w ϕ. It is shown in [5] that every w ∈ W can
be characterised by a formula δ(w) such that M, w δ(w0 ) iff w and w0 satisfy
the same formulae. Now, it can be shown that Te = TSw∈W (obs δ(w) if γ(w)) , where
W
γ(w) = v∈(Te ◦R◦Te−1 )(w) δ(v) by using the fact that: ∀w, v, v 0 ∈ W , if v ∈
Tobs γ(w) (w) and v 0 ∈ Te (w) then v and v 0 satisfy the same epistemic formulae.
For a detailed proof the reader can refer to [9].
t
u
In other words, in EDL, sequences and nondeterministic compositions of observations suffice to express every kind of purely epistemic action.
In fact, the operator obs turns out to be the same as ‘!’ of the public announcement logic PAL, originally proposed in [2]. Both PAL and EDLobs have
the same set of axioms for the operators ‘!’ and ‘obs ’ respectively. Here, we use a
mono-agent version of this logic. Some recent works [6, 10] show that PAL is very
suitable for modelling multi-agent communication. In addition, it is possible to
incorporate other notions such as belief and common knowledge. PAL is itself
a special case of other communication logics proposed, for instance, in [4, 3, 10].
This suggests that EDLobs can be extended to handle all these notions.
5
Describing deterministic actions
As shown in [11], purely ontic actions that are deterministic can be eliminated
under the following two hypotheses: ∆ does not contain static laws and the agent
has complete information about executability conditions and action effects. A
method for performing this elimination in dynamic logic was proposed in [12].
From now on, we make the assumption that all actions in Ao are deterministic: for every model M and for every o ∈ Ao , To (w) is a function. Then, we add
the following axiom scheme to EDLobs . Let o ∈ Ao .
Det(o). hoiϕ → [o]ϕ
The hypothesis of complete information enables descriptions of actions in the
following format.
Definition 10. Let P be the finite set of propositional letters of the domain, let
Ao be the finite set of abstract ontic actions of the domain, and let o range over
Ao . An action description of o is a tuple D(o) of the form hPoss, Effect + , Effect − ,
Cond + , Cond − i such that
Poss ∈ LS5 is the executability precondition, noted Poss(o);
Effect + ⊆ P is the set of all possible positive effects, noted Effect + (o);
Effect − ⊆ P is the set of all possible negative effects, noted Effect − (o);
Cond + : Effect + → LPL assigns to each p ∈ Effect + (o) a boolean formula
describing its positive precondition, noted Cond + (o, p); and
– Cond − : Effect − → LPL assigns to each p ∈ Effect − (o) a boolean formula
describing its negative precondition, noted Cond − (o, p).
–
–
–
–
8
Andreas Herzig and Tiago De Lima
In addition, Cond + and Cond − must satisfy: |= ¬(Cond + (o, p)∧Cond − (o, p)).
In the sequel, we also assume that this kind of description is given for observations. In this case, we set Poss(obsϕ) = ϕ, and Effect + (obsϕ) = Effect − (obsϕ) =
Cond + = Cond − = ∅.
Now, suppose that an action description of an action a is given. Then the set
of laws of a, ∆(a), can be automatically generated as follows:
1. Add the following executability axiom to ∆(a): Poss(a) ↔ hai>.
2. For every p ∈ Effect + (a), add the following two positive effect axioms to
∆(a): Cond + (a, p) → [a]p and (¬Cond + (a, p) ∧ ¬p) → [a]¬p.
3. For every p ∈ Effect − (a), add the following two negative effect axioms to
∆(a): Cond − (a, p) → [a]¬p and (¬Cond − (a, p) ∧ p) → [a]p.
4. For every p 6∈ Effect + (a), add the following frame axiom to ∆(a): ¬p →
[a]¬p.
5. For every p 6∈ Effect − (a), add the following frame axiom to ∆(a): p → [a]p.
6
Regression in EDLobs
Planning under incomplete knowledge through regression was originally proposed in [11, 13] using situation calculus as the base formalism. Situation calculus
is a dialect of second order logic, but through regression, a semi-decidable procedure for planning is possible. The corresponding mechanism for our logic should
act as follows: given a problem of the form ∆ |= ι → hαiγ, then, under some reasonable assumptions, return an equivalent problem of the form |= ι → reg(hαiγ)
where reg(hαiγ) is a “simplified” version of the original formula, namely, without dynamic operators. Hence, the result is a formula in S5 logic. Validity in S5
is well known to be NP-Complete. Therefore, although a decision procedure for
EDL is already known, regression based planning is still interesting because in
practice it is often more efficient.
DefinitionS11. Let ϕ, ϕ1 , ϕ2 ∈ LEDLobs , let Φ ∈ LS5 , let α1 , α2 ∈ L(Ao ), let
b ∈ (Ao ∪ ϕ∈LS5 obs ϕ) and let p ∈ P . The EDL regression operator reg() is
inductively defined as follows.
1. reg(Φ) = Φ;
2. if p 6∈ Effect + (b) and p 6∈ Effect − (b), then
reg([b]p) = ¬Poss(b) ∨ p;
3. if p 6∈ Effect + (b) and p ∈ Effect − (b), then
reg([b]p) = ¬Poss(b) ∨ (p ∧ ¬Cond − (b, p));
4. if p ∈ Effect + (b) and p 6∈ Effect − (b), then
reg([b]p) = ¬Poss(b) ∨ Cond + (b, p) ∨ p;
5. if p ∈ Effect + (b) and p ∈ Effect − (b), then
reg([b]p) = ¬Poss(b) ∨ Cond + (b, p) ∨ (p ∧ ¬Cond − (b, p));
6. reg([b]¬ϕ) = ¬Poss(b) ∨ ¬ reg([b]ϕ);
7. reg([b](ϕ1 ∧ ϕ2 )) = reg([b]ϕ1 ) ∧ reg([b]ϕ2 );
8. reg([b]K ϕ) = ¬Poss(b) ∨ K reg([b]ϕ);
Epistemic Actions and Ontic Actions: a Unified Logical Framework
9.
10.
11.
12.
9
reg([skip]ϕ) = reg(ϕ);
reg([if ϕ then α1 else α2 ]ψ) = reg((ϕ → [α1 ]ψ) ∧ (¬ϕ → [α2 ]ψ));
reg([α1 ; α2 ]ϕ) = reg([α1 ] reg([α2 ]ϕ)); and
reg([α1 ∪ α2 ]ϕ) = reg([α1 ]ϕ) ∧ reg([α2 ]ϕ);
By using an appropriate reduction ordering one can prove that the rewriting
procedure reg() is well-defined and terminates.
S
Theorem
3 (EDL regression). Let B = Ao ∪ ϕ∈LS5 obs ϕ. Let ∆(B) =
S
b∈B ∆(b) be the action theory corresponding to action descriptions in terms of
Poss, Effect + , Effect − , Cond + and Cond − . If ϕ is an EDLobs formula, then
∆(B) |= ϕ if and only if |=S5 reg(ϕ).
Proof (sketch). Equalities from 1 to 5 for ontic actions are proved to be logical
equivalences under ∆(B) in [12]. The proof for observations is the same. Equalities from 9 to 12 are proved by using the axiom schemes Def(skip), Def(if),
Def(;) and Def(∪). For ontic actions, equalities from 6 to 8 are proved by using
Definition 6 and for observations they are proved by the following validities that
can be deduced from our axioms for obs . Notice that the last one corresponds
to both ‘no forgetting’ and ‘no learning’ principles for observations.
[obs ϕ]p ↔ (ϕ → p)
[obs ϕ]¬ψ ↔ (ϕ → ¬[obs ϕ]ψ)
[obs ϕ](ϕ ∧ χ) ↔ ([obs ϕ]ψ ∧ [obs ϕ]χ)
[obs ϕ]K ψ ↔ (ϕ → K [obs ϕ]ψ)
t
u
Corollary 1. ∆(B) |= ι → [π]γ if and only if |= ι → reg([π]γ).
7
Discussion and conclusion
Starting with early work by Moore [14], there has been a lot of interest in the
reasoning about actions community on the integration of knowledge and sensing actions into the situation calculus since Scherl and Levesque’s [13]. Some
of these approaches use, without proof, the assumption that actions may be decomposed as shown in the decomposition theorem presented here. In addition, in
such approaches, epistemic actions are characterised by a set sensedFluents().
Suppose sensedFluents() = {p1 , . . . , pn }. Then corresponds to our complex
action obs (p1 ∧ . . . ∧ pn ) ∪ obs (p1 ∧ . . . ∧ ¬pn ) ∪ obs (¬p1 ∧ . . . ∧ ¬pn ) of nondeterministically observing a boolean combination of p1 , . . . , pn . Thus such complex
actions can be reduced to our simpler, basic observations.
Going beyond these approaches, we allow here for more complex observations
of non-boolean, epistemic propositions, such as obs (p ∧ ¬K p). Such observations
are particularly interesting in the multi-agent case that, as said before, is a
straightforward extension of EDLobs . We note that contrarily to boolean observations, epistemic observations are not necessarily successful: [obs (p ∧ ¬K p)](p ∧
¬K p) is not valid (and is even unsatisfiable).
10
Andreas Herzig and Tiago De Lima
Up to now, research in reasoning about actions was quite disconnected from
work on public announcement logic PAL that had been done in the ‘Dutch tradition’ by van Benthem, Gerbrandy, Baltag, van Ditmarsch, Kooi, and others.
Just as we do, PAL combines epistemic logic with PDL. Contrarily to us (and
contrarily to the AI tradition and terminology), work in PAL has focussed almost exclusively on observations (called public announcements in PAL). From
this perspective, we have here extended PAL by ontic actions, connecting it
with Reiter’s solution to the frame problem. The only similar work is [10], who
have augmented PAL by public assignments (of propositional variables to truth
values). We are currently investigating the precise relationship.
Acknowledgements: We thank the reviewers for their helpful comments.
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