Actions over a constructive semantics for ALC

Actions over a constructive semantics for ALC

Loris Bozzato, Mauro Ferrari, and Paola Villa Dipartimento di Informatica e Comunicazione

Universit`a degli Studi dell’Insubria Via Mazzini 5, 21100, Varese, Italy

Abstract. Following the approaches and motivations given in recent works about action languages over description logics, we propose an ac- tion formalism based on a constructive semantics forALC.

1 Introduction

In [3] we have introduced theinformation terms semanticsfor the basic descrip- tion logic ALC. This semantics is related to the BHK interpretation, the well known constructive explanation of logical connectives (see, e.g., [14]). Specifi- cally, in our semantics the truth of a formula in a given interpretation is explained by a mathematical object that we callinformation term. For the time being, the latter can be visualized as a sort of proof term inhabiting a type/formula. One of the main features of information terms semantics is that it supports a natural notion of state which has already been used to define the semantics of CooML (Constructive Object Oriented Modeling Language) [7, 12], a modeling language based on a constructive first-order logic.

In recent papers [2, 5, 6] different approaches to the definition of action lan- guages over description logics have been addressed and different solutions have been proposed. In this paper we describe an approach based on the notion of state given by information terms. In our context an action is an expression P ⇒ Qwhere P andQ are sets ofALC formulas. An action can be informally understood as follows: if in a given state the formulas inP (preconditions) are true, then the action can be applied and in the resulting state the formulas inQ (postconditions) will be true. In this paper we address the problems to determine executability of an action, to build the state obtained by an action application and to check its consistency. As discussed in [7, 8], these problems are strictly connected with snapshot generation in CooML and with model generation in SAT [10] and in Answer Set Programming [11].

For expository reasons, in this paper we restrict our attention to the simple case where the formulas occurring inP ⇒ Qare literals over the terminological alphabet. However, our approach can be extended to more complex actions.

2 ALC language and semantics

We begin introducing the language L for ALC [1, 13], based on the following denumerable sets: the setNRofrole names, the setNCofconcept namesand the setNIofindividual names. AconceptH is an expression of the kind:

H ::= C | ¬H | HuH | HtH | ∃R.H | ∀R.H

whereC∈NCandR∈NR. LetVarbe a denumerable set ofindividual variables;

theformulas K ofLare defined according to the following grammar:

K ::= ⊥ |(s, t) :R| (s, t) :¬R|t:H | ∀H

where s, t∈NI∪Var,R∈NRandH is a concept. Anatomic formulaof Lis a formula of the kind ⊥, (s, t) :R or t:C where R∈ NRandC ∈ NC; a negated formula is a formula of the kind (s, t) : ¬R or t: ¬H. A formula is closed if it does not contain variables. We write ¬((s, t) : R), ¬((s, t) : ¬R), ¬(t : H) as abbreviations for (s, t) :¬R, (s, t) : R, t : ¬H respectively; A vB stands for

∀(¬AtB).

LetN be a finite subset ofNI. By L_N we denote the set of formulasK ofL such that all the individual names occurring inK belong toN.

A substitution (over N) is a function σ : Var → N; we extend σ to L_N so that for every c ∈ N, σ(c) = c. Given a set of formulas Γ of L_N and a substitutionσoverN,σΓ denotes the set of closed formulas obtained replacing every variablexoccurring inΓ withσ(x).

Amodel (interpretation) MforL is a pair (D^M,·^M), whereD^M is a non- empty set (the domain of M) and ·^M is a valuationmap such that: for every c ∈ NI, c^M ∈ D^M; for every C ∈ NC, C^M ⊆ D^M; for every R ∈ NR, R^M ⊆ D^M× D^M. A non atomic concept H is interpreted by a subset H^M of D^M as usual:

- (¬A)^M=D^M\A^M, (AuB)^M=A^M∩B^M, (AtB)^M=A^M∪B^M - (∃R.A)^M={d∈ D^M | ∃d⁰∈ D^M s.t. (d, d⁰)∈R^M andd⁰∈A^M} - (∀R.A)^M={d∈ D^M | ∀d⁰∈ D^M, (d, d⁰)∈R^M impliesd⁰ ∈A^M}

We say that a model Mof L_N is reachable if, for everyd ∈ D^M, there exists c∈NIsuch thatc^M=d.

Anassignment on a modelMis a mapθ:Var→ D^M. Ift∈NI∪Var,t^M,θ is the element ofD^M denoting tin Mw.r.t.θ, namely: t^M,θ =θ(t) ift∈Var;

t^M,θ=t^Mift∈NI. A formulaKisvalidinMw.r.t.θ, and we writeM, θ|=K, ifK6=⊥and one of the following conditions holds:

- M, θ|=t:H ifft^M,θ∈H^M

- M, θ|= (s, t) :R iff (s^M,θ, t^M,θ)∈R^M - M, θ|= (s, t) :¬R iff (s^M,θ, t^M,θ)6∈R^M - M, θ|=∀H iffH^M=D^M

We writeM |=K iffM, θ|=K for every assignmentθ. Note thatM |=∀H iff M |=x:H, with xany variable. IfΓ is a set of formulas,M |=Γ means that M |=K for everyK∈Γ. We say thatK is alogical consequenceofΓ, and we writeΓ |=K, iff, for everyMand everyθ, M, θ|=Γ impliesM, θ|=K.

3 Information terms semantics

We introduceinformation terms that will be the base structure of our construc- tive semantics. Given a finite subsetN ofNIand a closed formulaK ofL_N, we define the set of information termsitN(K) by induction onK as follows.

itN(K) = {tt}, ifK is an atomic or negated formula

itN(c:A₁uA₂) = {(α, β)|α∈itN(c:A₁) andβ∈itN(c:A₂)} itN(c:A1tA2) = {(k, α)|k∈ {1,2}andα∈itN(c:Ak)} itN(c:∃R.A) = {(d, α)|d∈ N andα∈itN(d:A)} itN(c:∀R.A) = itN(∀A) = {φ:N →S

d∈NitN(d:A)|φ(d)∈itN(d:A)} Information terms for a formulaKprovide possible justifications for the validity ofKin a classical model. Formally, letMbe a model forL,Ka closed formula ofL_N andη∈itN(K). We define therealizability relation MhηiKas follows:

MhttiK iffM |=K, whereK is an atomic or negated formula Mh(α, β)ic:A1uA2 iffMhαic:A1and Mhβic:A2

Mh(k, α)ic:A1tA2 iffMhαic:Ak

Mh(d, α)ic:∃R.AiffM |= (c, d) :R andMhαid:A Mhφic:∀R.Aiff M |=c:∀R.Aand, for everyd∈ N,

M |= (c, d) :RimpliesMhφ(d)id:A Mhφi ∀AiffM |=∀Aand, for everyd∈ N,Mhφ(d)id:A

IfΓ ={K1, . . . , K_n}is a finite set of closed formulas ofLN,itN(Γ) denotes the set of n-tuples η = (η₁, . . . , η_n) such that, for every 1≤j ≤n, η_j ∈ itN(K_j);

MhηiΓ iff, for every 1≤j ≤n,Mhη_jiK_j.

We remark thatMhηiKimpliesM |=K, hence the constructive semantics is compatible with the usual classical one. The converse in general does not hold and stronger conditions are required:

Proposition 1. LetK be a closed formula ofLand letMbe a finite model for L. If M |=K, there exists a finite subset N of NI and η ∈ itN(K) such that

MhηiK. ut

We point out that in our setting negation has a classical meaning, thus negated formulas are not constructively explained by an information term. However, in- formation terms semantics can be extended to treat various kinds of constructive negation as those discussed in [9].

In the following we will indicate with the term theory a set T of closed formulas ofL_N consisting of a TBox and an ABox. The TBox is a set of formulas of the kindK=∀A, representing the constraints of our system. The ABox is a set of concept and role assertions that represent our knowledge about the current state of the system. Astate of the system is anyγ∈itN(T).

Example 1 (The alert system). In this example we model a simple home alert system. The system has two kinds of sensors, namely to detect fire and flood events: whenever a sensor activates and signals the occurrence of one of these events, a corresponding alert goes off. When a sensor stops signaling an event, the alarm must stop.

The theory that models our system consists of the TBoxTas, representing the constraints of the model:

(Ax1) :∀(¬CurrentAlertt(∃hasReason.CurrentSignaluAlert)) (Ax2) :∀(¬CurrentSignalt(Activeu(FiretFlood)))

that can be restated as:

(Ax1) :CurrentAlertv ∃hasReason.CurrentSignaluAlert (Ax2) :CurrentSignalvActiveu(FiretFlood)

and an ABoxAas0:

alert1:Alert fire s1:Fire flood s1:Flood alert2:Alert fire s2:Fire flood s2:Flood

asserting that our system has two sensors for every kind. Moreover, in the initial state none of the signals is active.

LetAs0=Tas∪Aas⁰and letWbe the set of the individual names occurring in Aas⁰. As an example, an element ofit^W(Ax1) is a functionφmapping each c∈Wto an element

δ∈it^W(c:¬CurrentAlertt(∃hasReason.CurrentSignaluAlert)) where,δis either (1,tt) (meaning thatcis not a current alert) or (2,((d,tt),tt)) (i.e.,c is a current alert and his reason is the current signald).

Now, we have to select an initial state of the system consistent with our knowledge contained inAas⁰. To this aim letγ1∈it^W(Ax1) andγ2∈it^W(Ax2) the functions mapping every c∈Win (1,tt). The initial state of our system is the information term γ₀ ∈itW(As0) associatingγ1 to Ax1, γ2 to Ax2 and tt to every formula of the ABoxAas0. It is easy to define a modelMofAs0such that Mhγ₀iAs0. We remark that γ₀ is the only state that can justify our theory only assuming the information contained inAas0. We also notice thatγ₀ assumes a sort of closed world assumption about the current state, in the sense that what is not true in the current state is assumed as false. 3 According to the above definitions an information term is a structured data whose correct reading is provided by the related formula. According to this interpretation we call piece of information over L_N a pair hαiF, whereF is a closed formula ofL_N andα∈itN(F). Given a theoryToverL_N, we introduce two notions of consistency:

- A state (information term) γ ∈ itN(T) isconsistent if there exists a model Msuch thatMhγiT.

- Tisstate consistent if there existsγ∈itN(T) such thatγ is consistent.

Now, we will see how, introducing the notion of information content, we can reduce the problem to check consistency of a state to the problem to check classical consistency of a set of atomic and negated formulas.

Given a finite subsetN ofNI, letR_N ={(s, t) :R | R∈NRands, t∈ N }.

This set intuitively represents the set of all the possible role assertions that we can express overN. Given a finite subsetRof R_N, we define theinformation content (w.r.t.R) of a piece of information as follows:

- icR(httic:H) ={c:H}, ifc:H is an atomic or negated formula - icR(h(α, β)ic:A₁uA₂) =icR(hαic:A₁)∪icR(hβic:A₂)

- icR(h(k, α)ic:A₁tA₂) =icR(hαic:A_k) (k= 1 ork= 2) - icR(h(d, α)ic:∃R.A) =icR(hαid:A)∪ {(c, d) :R}

- icR(hφic:∀R.A) =S

(c,d):R∈RicR(hφ(d)id:A)∪ {(c, d) :R |(c, d) :R∈ R}

- icR(hφi∀A) =S

d∈NicR(hφ(d)id:A)

If Γ = {K1, . . . , Kn} is a set of closed formulas of LN and γ ∈ itN(Γ), icR(hγiΓ) =S

K_i∈Γ(hγiiKi).

We remark that the information content of a piece of information is a set of atomic and negated formulas. The following relation holds between a piece of information and its information content:

Theorem 1. Let N be a finite subset of NI, let K be a closed formula of LN, letR ⊆ RN and letM be a reachable model ofLN such thatM |= (c, d) :R iff (c, d) :R∈ R. ThenMhαiK iff M |=icR(hαiK). ut According with this theorem, icR(hαiK) intuitively represents the minimum¹ amount of information needed to get evidence forKaccording to the information term α, assuming the roles in R. We remark that, by Theorem 1, checking consistency of a stateγofTis equivalent to check consistency oficR(hγiT) for the givenR.

Example 2 (Information content). Let us consider the theoryAs0 and the in- formation termsγ₁ and γ₂ defined in Example 1. Let R=∅, corresponding to the initial state of our system where no role is specified. Then the information content of the initial state of our system is:

icR(hγ1iAx1) ={a:¬CurrentAlert|a∈W}

icR(hγ2iAx2) ={a:¬CurrentSignal|a∈W}

Moreover, since every formula in theAas⁰ is atomic, its information content is Aas⁰ itself. So we have that:

icR(hγ₀iAs0) ={a:¬CurrentAlert|a∈W} ∪ {a:¬CurrentSignal|a∈W} ∪Aas⁰

Now, letM0be the model having the set of individual namesWas domain and mapping every individual name in itself, every concept so to satisfy Aas0 and every role in the empty set.M0satisfies the hypothesis of Theorem 1 and hence

the stateγ₀ is consistent. 3

1 Minimality ofic^R(hαiK) can be formalised in model-theoretic terms (see [7]).

4 Actions

We call literal a formula either of the kindt:C, t:¬C, (s, t) :R or (s, t) :¬R where C ∈ NC and R ∈ NR. Given a set of literals L, we denote with L = {¬K | K ∈ L} where we assume that ¬¬H = H if H is either a role or a concept name.

Anaction overLN is an expression of the kindP ⇒ QwhereP and Qare sets of literals overLN and every individual variable occurring inQalso occurs in P. Informally an action can be understood as follows: if in a given state the formulas inP(preconditions) are true, then the action can be applied and in the resulting state the formulas inQ(postconditions) will be true. Given an action α≡ P ⇒ Qwe denote withPre(α) the preconditions ofαand withPost(α) the postconditions of α.

Now, letTbe a theory with ABoxAoverLN and letγ∈itN(T). The action α≡ P ⇒ Qisactive in the state γ w.r.t. a substitution σ ifσP ⊆ icR(hγiT).

An active action can be applied and its application in stateγhas two effects:

- We get a new ABoxA⁰= (A \σQ)∪σQ(ABox update);

- We get the setOut(α) = ((icR(hγiT)∪ R)\σQ)∪σQ(action output).

An action application changes both the ABox of the theory and the state of the system. Obviously, given a consistent state of our system, an action appli- cation could lead the system to an inconsistent state. According to Theorem 1 to guarantee that action application is consistent we must prove that the action output is consistent and that we can construct an information term for Tfrom the formulas in the action output. Before discussing the consistency issues let us define the actions governing the behaviour of our alert system.

Example 3 (Actions of the alert system). For every fire and flood sensor, we define the following actions (here defined forfire s1) that model the beginning and the end of a signal from the sensor:

SignalF ireS1() :∅ ⇒ {fire s1:Active}

U nSignalF ireS1() :∅ ⇒ {fire s1:¬Active}

We remark that the above actions have an empty set of preconditions, thus they are active in every state and they simply update the knowledge base. The system reacts to these signals with the following actions:

StartF ireAlert(x) :{x:Fire, x:Active} ⇒

{alert1:CurrentAlert, x:CurrentSignal, (alert1,x):hasReason}

StartF loodAlert(x) :{x:Flood, x:Active} ⇒

{alert2:CurrentAlert, x:CurrentSignal, (alert2,x):hasReason}

StopAlert(x, y) :{x:¬Active,(y, x) :hasReason} ⇒ {y:¬CurrentAlert, x:¬CurrentSignal, (y, x) :¬hasReason}

We write actions in a function-like form to emphasise the free variables and how they are instantiated. Note that the above three actions are not active in the initial stateγ₀.

Now, let us consider the situation where a fire is detected byfire s1. This raises the actionSignalF ireS1(). LetAs₁=Tas∪Aas1whereAas1=Aas0∪ {fire s1:Active} and let γ₁ be the information term for As₁ associating γ₁ and γ₂ of Example 1 to Ax₁ andAx₂ respectively, and associating ttto every formula inAas1. It is easy to check that icR(hγ₁iAs₁) is exactly the result of applying this action to the stateγ₀.

In the stateγ₁ the action StartF ireAlert(fire s1) can be activated, and the result of its application is the output

Out=Aas0∪ {a:¬CurrentAlert|a∈(W\ {alert1})} ∪ {a:¬CurrentSignal|a∈(W\ {fire s1})} ∪

{fire s1:Active,alert1:CurrentAlert,fire s1:CurrentSignal, (alert1,fire s1):hasReason}

Now, letAas2=Aas1∪Post(StartF ireAlert(fire s1)), and letAs₂=Tas∪ Aas2. Let M₂ be a model ofOut, it is easy to check thatM₂ is also a model

ofAs₂. 3

5 An algorithm to build up information terms

We remark that we have concluded the above example without giving the state corresponding to the action application. To build up the output state of an action we will use the algorithmGenIt(X, F) of Figure 1. This algorithm takes as input a set X of closed atomic and negated formulas of L_N and a closed formulaF ofL_N and generates as output a (possibly empty) set of information terms initN(F). GenItinvokes the function OpenIt of Figure 2 to compute the information terms for compound and open formulas.OpenIttaken as input the set X and a formula F and returns a (possibly empty) set of pairs (α, c) where α∈itN(F) andc ∈ N (intuitively, if the pair (α, c) is built, αjustifies the formulaF with respect to the individual name c).

It is easy to prove, by induction on the structure of the formula F, the following result:

Theorem 2. Let X be a set of closed atomic and negated formulas ofLN, let R={(t, t⁰) :R|(t, t⁰) :R∈X} and letF be a closed formula ofLN. Then, for

every τ∈GenIt(X, F),icR(hτiF)⊆X. ut

if(F is either (c, d) :R or (c, d) :¬R)then if(F∈X)then return{tt};

else return∅;

else if(F=∀A)then begin letZ =OpenIt(X, x:A);

letΓ ={c|(α, c)∈Z)};

if(Γ 6=N)then return∅;

else return{φ|φ(c) =αwith (α, c)∈Z};

end

else return{α|(α, c)∈OpenIt(X, F)};

Fig. 1.TheGenItalgorithm

Example 4 (State generation). The execution ofGenIt(Out, Ax₁) provides the following information term γ⁰₁ ∈ it^W(Ax1), where we enclose between square brackets the pairs (c, γ₁⁰(c)) belonging to the function:

[ (alert1,(2,((fire_s1, tt), tt))), (alert2,(1,tt)), (fire_s1,(1,tt)), (fire_s2,(1,tt)), (flood_s1,(1,tt)), (flood_s2,(1,tt)) ]

while the execution of GenIt(Out, Ax₂) provides the information term γ₂⁰ ∈ itW(Ax₂):

[ (alert1,(1,tt)), (alert2,(1,tt)), (fire_s1,(2,(tt,(1,tt)))), (fire_s2,(1,tt)), (flood_s1,(1,tt)), (flood_s2,(1,tt)) ]

ConsiderAas2as defined in the previous example. Ifγ₂is the information term associatingγ₁⁰ toAx1,γ₂⁰ toAx2andttto every formula ofAas2and ifM2is a model ofOut, then it follows thatM2hγ₂iAs2. Hence the action application leads to a consistent state and thusAs2 is state consistent.

If we considerR={(alert1,fire s1):hasReason}, which is the set of role formulas asserted byAas2, then the information content ofAs₂ for the axioms in Tasis defined as:

icR(hγ₁⁰iAx1) ={alert1:Alert,fire s1:CurrentSignal, (alert1,fire s1):hasReason} ∪

{a:¬CurrentAlert|a∈W\ {alert1}}

icR(hγ₂⁰iAx2) ={fire s1:Active,fire s1:Fire} ∪ {a:¬CurrentSignal|a∈W\ {fire s1}}

So we have that icR(hγ₂iAs2) = icR(hγ⁰₁iAx1)∪icR(hγ₂⁰iAx2)∪Aas2. As in Example 2, if M2 satisfies the hypotheses, then Theorem 1 holds and M2 hγ₂iAs₂ iffM₂|=icR(hγ₂iAs₂).

Note that, if fire s1 stops being active and U nSignalF ireS1() is exe- cuted, in the new state it does not hold thatfire s1:Activeand so the action StopAlert(fire s1,alert1) can be fired. If that is the case, it is easy to verify that the system returns in the initial state of our example.

if(F =t:AwithA∈NCorA=¬H)then if(t∈ N)then

if(t:A∈X)then return{(tt, t)};

else return∅;

else return{(tt, c)|c:A∈X};

if(F =t:AuB)then

return{((α, β), c)|(α, c)∈OpenIt(X, t:A) and (β, c)∈OpenIt(X, t:B)};

if(F =t:AtB)then

return{((1, α), c))|(α, c)∈OpenIt(X, t:A)} ∪ {((2, β), c)|(β, c)∈OpenIt(X, t:B)};

if(F =t:∃R.A)then

if(t∈ N)thenletD={t};

elseletD={c|(c, d) :R∈X}

return{((d, α), c)|c∈Dand (c, d) :R∈X and (α, d)∈OpenIt(X, x:A)};

if(F =t:∀R.A)then begin if(t∈ N)then begin

letD={t};

letZ ={d|(t, d) :R∈X and (α, d)∈OpenIt(X, x:A)};

else begin

letD={c|(c, d) :R∈X};

letZ ={d|(α, d)∈OpenIt(X, x:A)};

endletΦ=∅;

for all(c∈D)do begin letC={d|(c, d) :R∈X};

if(C⊆Z)then Φ=Φ∪

(φ, c)|φ(d) =

_{α ifd∈Cand (a, d)∈Z} anyη⁺ofit^N(d:A) otherwise

end;

returnΦ;

end

Fig. 2.TheOpenItfunction

Moreover, it is possible to show that, for example, sinceGenIt(Out, Ax1) = {γ₁⁰} andicR(hγ₁⁰iAx1)⊆Out, then Theorem 2 holds. 3 We remark that in generalGenItgenerates more than one state. In this case the action could lead the system to different states and a non deterministic choice has to be done. Moreover, given a state generated by GenIt, one has to show that this state is consistent. As for the latter point, the usual considerations about checking consistency hold (see, e.g., [7]). In relation with this problem we plan to study connections of our semantics with SAT [10] and ASP [11].

An important point of our approach is that we can use GenIt as a first consistency check for an action application. Indeed, if X is the output of an action application over icR(hγiT) and GenIt(X, F) is empty for someF ∈T then T is not state consistent. This usually means that the action does not

provide enough information to justify the system constraints as shown in the following example.

Example 5 (Action consistency check).Suppose that, in the development of our system, we write the following (wrong) version ofStartF ireAlert(x):

StartF ireAlert2(x) :{x:Fire, x:Active} ⇒

{alert1:CurrentAlert,(alert1,x):hasReason}

As can be noted, we forgot to setx:CurrentSignalinPost(StartF ireAlert2(x)).

This action is “wrong” in the sense that its execution leads to an inconsistent action output: if that is the case, GenIt will find this inconsistency as it is unable to generate a set of information terms for an axiom.

For example, consider the set Out⁰ obtained applying the above action to the stateγ₁ of Example 3 (namely,Out⁰=Out\ {fire s1:CurrentSignal}).

The inconsistency of Out⁰ is verified with the execution of GenIt(Out⁰, Ax₁):

this execution leads to the following recursive executions of OpenIt on the subformulas ofAx₁

OpenIt(Out⁰, x⁰:CurrentSignal) =∅

OpenIt(Out⁰, x:∃hasReason.CurrentSignal) =∅ OpenIt(Out⁰, x:∃hasReason.CurrentSignaluAlert) =∅

OpenIt(Out⁰, x:¬CurrentAlertt(∃hasReason.CurrentSignaluAlert)) =H with H = {((1,tt), a) | a ∈ W\ {alert1}}. Since no information term in H can be associated to alert1, GenIt(Out⁰, Ax1) = ∅. This means that the corresponding theory is not state consistent. We also remark that the execution ofGenIt(Out⁰, x⁰:CurrentSignal) traces the reason of state inconsistency. 3 We conclude this section noting thatGenItis exponential in the size ofN: this complexity is due to the case ofF =∀A, where the algorithm must generate all the possible functions φ such that φ: N → S

d∈NitN(d: A). The number of such functions is obviously exponential in the number of elements ofN, hence the complexity ofGenIt.

6 Conclusion and future works

In this paper we have presented an action formalism based on the information terms semantics. We have shown how our semantics supports a natural notion of state and how an action language can be defined on the top of this notion. The problem to determine the consistency of an action is reduced to the problem to study the information terms generated by the application of GenIt. We have shown, by means of an example, how GenIt can be used, in some cases, to debug inconsistent actions. For lack of space we have not treated in details the problem to study the consistency of an action when the output of GenIt is not empty. However, we remark that this problem is similar to the problem to check snapshot consistency in CooML [7]. As for the future works we plan to

investigate this question also considering its relations with model generation in SAT [10] and ASP [11] and with the planning problem. Moreover, we plan to study the projection problem (see, e.g., [2]), that needs to be restated in our setting. We are also working on an implementation of our action language.

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