Instantaneous Reaction-Time in Dynamic-Consistency Checking of Conditional Simple Temporal Networks

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Instantaneous Reaction-Time in Dynamic-Consistency Checking of Conditional Simple Temporal Networks

Massimo Cairo, Carlo Comin, Romeo Rizzi

To cite this version:

Massimo Cairo, Carlo Comin, Romeo Rizzi. Instantaneous Reaction-Time in Dynamic-Consistency Checking of Conditional Simple Temporal Networks. TIME 2016 23rd International Symposium on Temporal Representation and Reasoning, Oct 2016, Kongens Lyngby, Denmark. pp.80-89,

�10.1109/TIME.2016.16�. �hal-01577460�

Instantaneous Reaction-Time in

Dynamic-Consistency Checking of Conditional Simple Temporal Networks

Massimo Cairo Department of Mathematics University of Trento, Trento, Italy

(massimo.cairo@unitn.it)

Carlo Comin Department of Mathematics University of Trento, Trento, Italy;

LIGM (UMR 8049)

Universit´e Paris-Est, Marne-la-Vall´ee, France (carlo.comin@unitn.it)

Romeo Rizzi

Department of Computer Science University of Verona, Verona, Italy

(romeo.rizzi@univr.it)

Abstract—Conditional Simple Temporal Network (CSTN) is a constraint-based graph-formalism for conditional temporal planning. Three notions of consistency arise for CSTNs and CSTPs: weak, strong, anddynamic. Dynamic-Consistency (DC) is the most interesting notion, but it is also the most challenging. In order to address the DC-Checking problem, in [5] we introduced ε-DC (a refined, more realistic, notion of DC), and provided an algorithmic solution to it. Next, given that DC implies ε- DC for some sufficiently small ε > 0, and that for every ε >0 it holds thatε-DC implies DC, we offered a sharp lower bounding analysis on the critical value of the reaction-time εˆ under which the two notions coincide. This delivered the first (pseudo) singly-exponential time algorithm for the DC-Checking of CSTNs. However, the ε-DC notion is interesting per se, and theε-DC-Checking algorithm in [5] rests on the assumption that the reaction-time satisfiesε >0; leaving unsolved the question of what happens whenε= 0. In this work, we introduce and study π-DC, a sound notion of DC with aninstantaneousreaction-time (i.e., one in which the planner can react to any observation at the same instant of time in which the observation is made). Firstly, we demonstrate by a counter-example thatπ-DC is not equivalent to 0-DC, and that0-DC is actually inadequate for modeling DC with an instantaneous reaction-time. This shows that the main results obtained in our previous work do not apply directly, as they were formulated, to the case ofε= 0. Motivated by this observation, as a second contribution, our previous tools are extended in order to handle π-DC, and the notion of ps-tree is introduced, also pointing out a relationship betweenπ-DC and HyTN-Consistency.

Thirdly, a simplereductionfrom π-DC to DC is identified. This allows us to design and to analyze the first sound-and-complete π-DC-Checking procedure. Remarkably, the time complexity of the proposed algorithm remains (pseudo) singly-exponential in the number of propositional letters.

Index Terms—Conditional Simple Temporal Networks, Dynamic-Consistency, Instantaneous Reaction-Time, Hyper Tem- poral Networks, Singly-Exponential Time.

I. INTRODUCTION ANDMOTIVATION

Intemporal planningandtemporal scheduling,Simple Tem- poral Networks (STNs)[6] are directed weighted graphs, where

This work was partially supported byDepartment of Computer Science, University of Verona, Verona, Italy, under PhD grant “Computational Mathe- matics and Biology”; and, for the 2nd author, on a co-tut´elle agreement with LIGM, Universit´e Paris-Est in Marne-la-Vall´ee, Paris, France.

nodes are events to be scheduled in time and arcs represent temporal distance constraints between pairs of events.

This work is focused on theConditional Simple Temporal Problem (CSTP)[12] and its graph-based counterpart Condi- tional Simple Temporal Network (CSTN) [10], a constraint- based model for conditional temporal planning. The CSTN formalism extends STNs in that: (1) some of the nodes are called observation events and to each of them is associated a propositional letter, to be disclosed only at execution time;

(2) labels (i.e. conjunctions over the literals) are attached to all nodes and constraints, to indicate the scenarios in which each of them is required. The planning agent (planner) must schedule all the required nodes, while respecting all the re- quired temporal constraints among them. This extended frame- work allows for the off-line construction of conditional plans that are guaranteed to satisfy complex temporal constraints.

Importantly, this can be achieved even while allowing the decisions about the precise timing of actions to be postponed until execution time, in a least-commitment manner, thereby adding flexibility and making it possible to adapt the plan dynamically, during execution, in response to the observations that are made [12].

Then, three notions of consistency arise for CSTNs: weak, strong, and dynamic. Dynamic-Consistency (DC) is the most interesting one, as it requires the existence of conditional plans where decisions about the precise timing of actions are postponed until execution time, but anyhow guaranteeing that all of the relevant constraints will be ultimately satisfied. Still, it is the most challenging one and it was conjectured to be hard to assess by Tsamardinos, Vidal and Pollack [12].

In our previous work [5], it was unveiled that HyTNs and MPGs are a natural underlying combinatorial model for the DC-Checking of CSTNs. Indeed, STNs have been recently generalized into Hyper Temporal Networks (HyTNs)[3], [4], by considering weighted directed hypergraphs, where each hy- perarc models a disjunctive temporal constraint called hyper- constraint. In [3], [4], the computational equivalence between checking the consistency of HyTNs and determining the win-

ning regions in Mean Payoff Games (MPGs) was also pointed out. The approach was shown to be viable and robust thanks to some extensive experimental evaluations [4]. MPGs [1], [7], [13] are a family of two-player infinite games played on finite graphs, with direct and important applications in model- checking and formal verification [8], and they are known for having theoretical interest in computational complexity for their special place among the few (natural) problems lying inNP∩coNP.

All this combined, in [5] it was provided the first (pseudo) singly-exponential time algorithm for the DC-Checking prob- lem, also producing a dynamic execution strategy whenever the input CSTN is DC. For this, it was introduced ε-DC (a refined, more realistic, notion of DC), and provided the first algorithmic solution to it. Next, given that DC implies ε-DC for some sufficiently small ε > 0, and that for every ε > 0 it holds that ε-DC implies DC, it was offered a sharp lower bounding analysis on the critical value of the reaction-time ˆ

ε under which the two notions coincide. This delivered the first (pseudo) singly-exponential time algorithm for the DC- Checking of CSTN. However, the ε-DC notion is interesting per se, and the ε-DC-Checking algorithm in [5] rests on the assumption that the reaction-time satisfies ε > 0; leaving unsolved the question of what happens when ε= 0.

Contribution: In this work we introduce and study π- DC, a sound notion of DC with aninstantaneousreaction-time (i.e., one in which the planner can react to any observation at the same instant of time in which the observation is made).

Firstly, we provide a counter-example showing that π-DC is not just the ε= 0special case ofε-DC. This implies that the algorithmic results obtained in [5] do not apply directly to the study of those situation where the planner is allowed to react instantaneously. Motivated by this observation, as a second contribution, we extend the previous formulation to capture a sound notion of DC with an instantaneous reaction-time, i.e., π-DC. Basically, it turns out thatπ-DC needs to consider an additional internal ordering among all the observation nodes that occur at the same time instant. Next, the notion of ps- tree is introduced to reflect the ordered structure of π-DC, also pointing out a relationship between π-DC and HyTN- Consistency. Thirdly, a simple reduction from π-DC to DC is identified. This allows us to design and to analyze the first sound-and-completeπ-DC-Checking procedure. The time complexity of the proposed algorithm remains (pseudo) singly- exponential in the number|P| of propositional letters.

II. BACKGROUND

This section provides some background and preliminary no- tations. To begin, if G= (V, A)is a directed weighted graph, every arc a ∈A is a triplet (u, v, w_a) where u=t(a)∈ V is the tail of a, v = h(a) ∈ V is the head of a, and w_a =w(u, v)∈Zis the (integer)weightof a.

Let us now recall Simple Temporal Networks (STNs) [6].

Definition 1 (STNs). An STN [6] is a weighted directed graph whose nodes are events that must be placed on the real

time line and whose arcs, calledstandard arcs, express binary constraints on the allocations of their end-points in time.

An STN G = (V, A) is called consistent if it admits a feasible schedule, i.e., a schedule φ : V 7→ R such that φ(v)≤φ(u) +w(u, v)for all arcs (u, v, w(u, v))∈A.

A. Conditional Simple Temporal Networks

Let us recall the CSTN model from [5], [10], [12].

LetP be a set of propositional letters (boolean variables), a label is any (possibly empty) conjunction of letters, or negations of letters, drawn fromP. Theempty labelis denoted byλ. The set of all these labels is denoted byP^∗. Two labels,

`1and`2, are calledconsistent, denoted byCon(`1, `2), when

`1∧`2 is satisfiable. A label`1 subsumesa label`2, denoted by Sub(`1, `2), whenever `1 ⇒`2 holds. We are now in the position to recall the formal definition of CSTNs.

Definition 2(CSTNs [10], [12]). AConditional Simple Tem- poral Network (CSTN)is a tuplehV, A, L,O,OV, Piwhere:

• V is a finite set of events;P ={p1, . . . , p_|P|} is a finite set of propositional letters;

• A is a set of labelled temporal constraints (LTCs) each having the form hv−u≤w(u, v), `i, whereu, v ∈ V, w(u, v)∈Z, and `∈P^∗ is satisfiable;

• L:V →P^∗ assigns a label to each event inV;

• OV ⊆V is a finite set of observation events;O:P → OV is a bijection that associates a unique observation event O(p) =O_p to each proposition p∈P;

• The followingreasonability assumptions must hold:

(WD1) for any LTC hv−u≤w, `i ∈ A the label ` is satisfiable and subsumes bothL(u)andL(v); intuitively, whenever a constraint hv−u≤wiis required, then its endpointsuand v must be scheduled (sooner or later);

(WD2) for eachp∈P and eachu∈V such that either por¬pappears inL(u), we require:Sub(L(u), L(Op)), andhOp−u≤0, L(u)i ∈A; intuitively, whenever a label L(u), for some u∈V, contains some p∈P, and u is eventually scheduled, thenOp=O(p)must be scheduled before or at the same time ofu.

(WD3) for each labelled constraint hv−u≤w, `iand p∈ P, for which either por ¬pappears in `, it holds thatSub(`, L(O_p)); intuitively, assuming that a required constraint contains some p∈P, then O_p =O(p)must be scheduled (sooner or later).

In all of the following definitions we shall implicitly refer to some CSTNΓ =hV, A, L,O,OV, Pi.

Definition 3 (Scenario). A scenarioover a subset U ⊆P of boolean variables is a truth assignments:U → {0,1}, i.e., sis a function that assigns a truth value to each proposition p ∈ U. When U ( P and s : U → {0,1}, then s is said to be a partial scenario; otherwise, when U = P, thens is said to be a(complete)scenario. The set comprising all of the complete scenarios over P is denoted byΣ_P. Ifs∈Σ_P is a scenario and ` ∈ P<sup>∗</sup> is a label, then s(`) ∈ {0,1} denotes the truth value of` induced bysin the natural way.

Notice that any scenarios∈ΣP can be described by means of the label`s,l1∧· · ·∧l_|P|such that, for every1≤i≤ |P|, the literal li ∈ {pi,¬pi}satisfies s(li) = 1.

Example 1. Consider the set of boolean variablesP ={p, q}.

The scenarios:P → {0,1}defined ass(p) = 1ands(q) = 0 can be compactly described by the label `_s=p∧ ¬q.

Definition 4 (Schedule). A schedule for a subset of events U ⊆V is a functionφ:U →Rthat assigns a real number to each event inU. The set of all schedules overUis denotedΦU. Definition 5(Scenario-Restriction). Lets∈ΣPbe a scenario.

TherestrictionofV,OV, andAw.r.t.sare defined as follows:

• V_s⁺,{v∈V |s(L(v)) = 1}; OV_s⁺,OV ∩V_s⁺;

• A⁺_s ,{hu, v, wi | ∃`hv−u≤w, `i ∈A, s(`) = 1}.

The restriction ofΓw.r.t.s∈ΣP is the STNΓ⁺_s ,hV_s⁺, A⁺_si.

Finally, it is worth to introduce the notationV_s⁺_1,s2,V_s⁺₁∩V_s⁺₂. Definition 6 (Execution-Strategy). An Execution-Strategy (ES) for Γ is a mapping σ: ΣP → Φ_V+

s such that, for any scenarios∈ΣP, the domain of the scheduleσ(s)isV_s⁺. The set of ESs ofΓis denoted bySΓ. Theexecution timeof an event v∈V_s⁺ in the scheduleσ(s)∈Φ_V+

s is denoted by[σ(s)]v. Definition 7 (History). Let σ∈ S_Γ be any ES, lets∈Σ_P be any scenario and let τ ∈R. Thehistory Hst(τ, s, σ)ofτ in the scenario sunder strategy σ is defined as:Hst(τ, s, σ), p, s(p)

∈P× {0,1} | Op∈V_s⁺, [σ(s)]_Op< τ .

The history can be compactly encoded as the conjunction of the literals corresponding to the observations comprising it, that is, by means of a label.

Definition 8(Viable Execution-Strategy). We say thatσ∈ S_Γ is a viable ES if, for each scenario s ∈ Σ_P, the schedule σ(s)∈Φ_V+

s is feasible for the STNΓ⁺_s.

Definition 9 (Dynamic-Consistency). An ESσ∈ SΓ is called dynamic if, for any s1, s2 ∈ ΣP and any v ∈ V_s⁺_1,s2, the following implication holds onτ ,[σ(s₁)]_v:

Con(Hst(τ, s₁, σ), s₂)⇒[σ(s₂)]_v=τ.

We say that Γisdynamically-consistent (DC)if it admitsσ∈ S_Γwhich is both viable and dynamic. The problem of checking whether a given CSTN is DC is named DC-Checking.

We provide next the definition ofdifference set ∆(s1;s2).

Definition 10 (Difference-Set). Let s1, s2 ∈ ΣP be any two scenarios. The set of observation events in OV_s⁺₁ at whichs1

and s₂ differ is denoted by∆(s₁;s₂). Formally,

∆(s1;s2),

Op∈ OV_s⁺₁ |s1(p)6=s2(p) .

The various definitions of history and dynamic consistency that are used by different authors [5], [11], [12] are equivalent.

B. Hyper Temporal Networks

This subsection surveys the Hyper Temporal Network (HyTN) model, which is a strict generalization of STNs. The reader is referred to [3], [4] for an in-depth treatise on HyTNs.

Definition 11(Hypergraph). AhypergraphHis a pair(V,A), whereV is the set of nodes, andAis the set of hyperarcs. Each hyperarc A = (tA, HA, wA) ∈ A has a distinguished node tA called the tailof A, and a nonempty setHA⊆V \ {tA} containing theheadsofA; to each headv∈HAis associated a weightwA(v)∈Z.

Provided that |A|,|H_A∪ {t_A}|, thesize of a hypergraph H = (V,A) is defined as m_A , P

A∈A|A|; it is used as a measure for the encoding length of H. If |A| = 2, then A= (u, v, w)can be regarded as astandard arc. In this way, hypergraphs generalize graphs.

A HyTN is a weighted hypergraph H = (V,A) where a node represents an event to be scheduled, and a hyperarc represents a set of temporal distanceconstraints between the tailand theheads.

In the HyTN framework the consistency problem is the following decision problem.

Definition 12 (HyTN-Consistency). Given some HyTN H= (V,A), decide whether there is a schedule φ:V →R such that:

φ(tA)≥ min

v∈HA

{φ(v)−wA(v)}, ∀A∈ A any such a scheduleφ:V →Ris called feasible.

A HyTN is called consistent whenever it admits at least one feasible schedule. The problem of checking whether a given HyTN is consistent is named HyTN-Consistency.

Theorem 1. [3] There exists anO((|V|+|A|)m_AW)pseudo- polynomial time algorithm for checking HyTN-Consistency;

moreover, when the input HyTNH= (V,A)is consistent, the algorithm returns as output a feasible schedule φ :V → R ofH; Here,W ,max_A∈A,v∈HA|wA(v)|.

C. ε-Dynamic-Consistency

In CSTNs, decisions about the precise timing of actions are postponed until execution time, when informations meanwhile gathered at the observation nodes can be taken into account.

However, the planner is allowed to factor in an outcome, and differentiate its strategy according to it, onlystrictlyafter the outcome has been observed (whence the strict inequality in Definition 7). Notice that this definition does not take into account the reaction-time, which, in most applications, is non- negligible. In order to deliver algorithms that can also deal with the reaction-time ε > 0 of the planner, we introduced in [5] a refined notion of DC.

Definition 13 (ε-Dynamic-Consistency). Given any CSTN hV, A, L,O,OV, Piand any real numberε∈(0,+∞), an ES σ∈ SΓ is ε-dynamic if it satisfies all of the Hε-constraints, namely, for any two scenarios s1, s2 ∈ ΣP and any event u∈V_s⁺_1,s2, the ESσsatisfies the following constraint, which is denoted byHε(s1;s2;u):

[σ(s1)]u≥min

{[σ(s2)]u}∪{[σ(s1)]v+ε|v∈∆(s1;s2)}

We say that a CSTN Γ is ε-dynamically-consistent (ε-DC)if it admits σ∈ S_Γ which is both viable andε-dynamic.

v1

v2

us1

us2

s₁

s2 A,−ε

A,−ε

A,0

Fig. 1: AnHε(s1;s2;u)constraint, modeled as a hyperarc.

As shown in [5],ε-DC can be modeled in terms of HyTN- Consistency. Fig. 1 depicts an illustration of anHε(s1;s2;u) constraint, modeled as an hyperarc.

Also, in [5] we proved that DC coincides with ε-DC,ˆ provided thatεˆ,|ΣP|⁻¹|V|⁻¹.

Theorem 2. Letεˆ,|ΣP|⁻¹|V|⁻¹. Then,Γis DC if and only if Γ isε-DC. Moreover, ifˆ Γ is ε-DC for some ε >0, then Γ is ε⁰-DC for everyε⁰∈(0, ε].

Then, the main result offered in [5] is a (pseudo) singly- exponential time DC-checking procedure (based on HyTNs).

Theorem 3. There exists an O |ΣP|³|A|²|V| +

|ΣP|⁴|A||V|²|P| + |ΣP|⁵|V|³|P|

W time algorithm for checking DC on any input CSTN Γ = hV, A, L,O,OV, Pi.

In particular, given any dynamically-consistent CSTN Γ, the algorithm returns a viable and dynamic ES for Γ.

Here, W ,maxa∈A|wa|.

III. DCWITHINSTANTANEOUSREACTION-TIME

Theorem 2 points out the equivalence between ε-DC and DC, that arises for a sufficiently small ε > 0. However, Definition 13 makes sense even if ε = 0, so a natural question is what happens to the above mentioned relationship between DC and ε-DC when ε = 0. In this section we first show that0-DC doesn’t imply DC, and, moreover, that0-DC is in itself too weak to capture an adequate notion of DC with an instantaneous reaction-time. In light of this we will introduce a stronger notion, which is namedordered-Dynamic- Consistency (π-DC); this will turn out to be a suitable notion of DC with an instantaneous reaction-time.

Example 2 (CSTNΓ₂). Consider the following CSTNΓ₂= (V₂, A₂, L₂,O₂,OV₂, P₂); see Fig. 2 for an illustration.

– V₂={⊥,>, A, B, C};

– A₂ = {(> − ⊥ ≤ 1, λ),(⊥ − > ≤ −1, λ),(> −A ≤ 0, b∧ ¬c),(> −B≤0, a∧c),(> −C≤0,¬a∧ ¬b),(⊥ −A≤ 0, λ),(A− ⊥ ≤0,¬b),(A− ⊥ ≤0, c),(⊥ −B ≤0, λ),(B−

⊥ ≤ 0,¬a),(A− ⊥ ≤ 0,¬c),(⊥ −C ≤ 0, λ),(C− ⊥ ≤ 0, a),(C− ⊥ ≤0, b)};

– L₂(A) =L₂(B) =L₂(C) =L₂(⊥) =L₂(>) =λ;

– O₂(a) =A,O₂(b) =B,O₂(c) =C;

– OV₂={A, B, C};

– P₂={a, b, c}.

Proposition 1. The CSTN Γ₂ (Example 2, Fig. 2) is0-DC.

Proof. Consider the execution strategyσ₂ : Σ_P2 →Ψ_V2: – [σ₂(s)]_A,s(a∧b∧ ¬c) +s(¬a∧b∧ ¬c);

>

[1]

B

b?

A

a?

C

c?

⊥

[0]

+1 −1

0,¬b 0 0, c 0, b¬c

0,¬a 0

0,¬c 0, ac

0, a

0 0, b 0,¬a¬b

Fig. 2: A CSTNΓ₂ which is0-DC but not DC.

– [σ₂(s)]_B ,s(a∧b∧c) +s(a∧ ¬b∧c);

– [σ₂(s)]_C,s(¬a∧ ¬b∧ ¬c) +s(¬a∧ ¬b∧c);

– [σ₂(s)]_⊥,0 and[σ₂(s)]_> ,1, for everys∈Σ_P. An illustration of σ₂ is offered in Fig 3. Three cubical graphs are depicted in which every node is labelled asvsfor some(v, s)∈V₂×ΣP₂: an edge connectsv_1s1andv_2s2if and only if: (i) v₁ =v₂ and (ii) the Hamming distance between s₁ and s₂ is unitary; each scenario s ∈ Σ_P2 is represented as s=αβγ for α, β, γ ∈ {0,1}, where s(a) =α,s(b) =β, s(c) = γ; moreover, each node v_s = (v, s) ∈ V₂×Σ_P2 is filled in black if[σ₂(s)]v= 0, and in white if[σ₂(s)]v= 1. So all three3-cubes own both black and white nodes, but each of them, in its own dimension, decomposes into two identically colored 2-cubes. Fig. 4 offers another visualization of σ₂ in

A000 A010

A₀₁₁ A₀₀₁

A100 A110

A111

A101

B000 B010

B₀₁₁ B₀₀₁

B100 B₁₁₀ B111

B101

C000 C010

C₀₁₁ C₀₀₁

C100 C₁₁₀ C111

C101

Fig. 3: The ESσ₂ for the CSTN Γ₂.

which every component of the depicted graph corresponds to a restriction STNΓ⁺_2sfor somes∈Σ_P2, wheres_i, s_j ∈Σ_P2 are grouped together whenever Γ⁺_2s

i = Γ⁺_2s

j. It is easy to see from Fig. 4 that σ₂ is viable for Γ₂. In order to check thatσ₂is0-dynamic, look again at Fig. 4, and notice that for everysi, sj ∈Σ₂, wheresi 6=sj, there exists an event node X ∈ {A, B, C} such that [σ₂(si)]X = 0 = [σ₂(sj)]X and si(X)6=sj(X). With this in mind it is easy to check that all of theH0 constraints are thus satisfied byσ₂. Therefore, the CSTNΓ₂ is0-DC.

Proposition 2. The CSTNΓ₂ is not DC.

Proof. Letσbe a viable ES forΓ₂. Then,σmust be the ESσ₂ depicted in Fig. 4, there is no other choice here. Letˆs∈Σ_P2.

>

1

C

1

B

0

A

0

⊥

0

ΣP₂

000; 001

>

1

C

0

B

0

A

1

⊥

0

010; 110

>

1

C

0

B

0

A

0

⊥

0

011; 100

>

1

C

0

B

1

A

0

⊥

0

101; 111

0

0 [1]

0

[1]

0 0

0

0 0

[1]

0

0

0

[1]

0

Fig. 4: The restrictionsΓ⁺_2sfors∈Σ_P2, where the execution times [σ₂(s)]_v∈ {0,1} are depicted in bold face.

Then, it is easy to check from Fig. 4 that: (i) [σ₂(ˆs)]_⊥ = 0, [σ₂(ˆs)]_> = 1, and it holds [σ₂(ˆs)]_X ∈ {0,1} for every X ∈ {A, B, C}; (ii) there exists at least two observation events X ∈ {A, B, C} such that[σ(ˆs)]_X = 0; still, (iii) there is no X ∈ {A, B, C}such that[σ(s)]_X= 0for everys∈Σ_P2, i.e., no observation event is executed first at all possible scenarios.

Therefore, the ESσ₂ is not dynamic.

We now introduce a stronger notion of dynamic consistency;

it is namedordered-Dynamic-Consistency (π-DC), and it takes explicitly into account an additional ordering between the observation events scheduled at the same execution time.

Definition 14 (π-Execution-Strategy). Anordered-Execution- Strategy (π-ES) for Γis a mapping:

σ:s7→([σ(s)]^t,[σ(s)]^π),

where s∈Σ_P,[σ(s)]^t∈Φ_V, and finally, [σ(s)]^π :OV_s⁺ {1, . . . ,|OV_s⁺|}is bijective. The set ofπ-ES ofΓ is denoted by SΓ. For any s ∈ ΣP, the execution time of an event v ∈ V_s⁺ in the schedule [σ(s)]^t ∈ Φ_V+

s is denoted by [σ(s)]^t_v ∈R; the position of an observation O_p ∈ OV_s⁺ in σ(s)is[σ(s)]^π_O

p. We require positions to becoherentw.r.t. ex- ecution times, i.e.,∀(O_p,O_q∈ OV_s⁺)if[σ(s)]^t_O

p<[σ(s)]^t_O

q

then[σ(s)]^π_O

p<[σ(s)]^π_O

q. In addition, it is worth to adopt the notation[σ(s)]^π_v ,|OV|+ 1wheneverv∈V_s⁺\ OV. Definition 15 (π-History). Let σ ∈ SΓ, s ∈ ΣP, and let τ ∈ R and ψ ∈ {1, . . . ,|V|}. The ordered-history π-Hst(τ, ψ, s, σ)ofτ andψin the scenarios, under theπ-ES σ, is defined as:π-Hst(τ, ψ, s, σ),

p, s(p)

∈P× {0,1} | O_p ∈ OV_s⁺,[σ(s)]^t_O

p ≤τ,[σ(s)]^π_O

p< ψ .

We are finally in the position to defineπ-DC.

Definition 16(π-Dynamic-Consistency). Anyσ∈ SΓis called π-dynamic when, for any two scenarioss1, s2∈ΣP and any event v∈V_s⁺_1,s2, ifτ,[σ(s1)]^t_v and ψ,[σ(s1)]^π_v, then:

Con(π-Hst(τ, ψ, s1, σ), s2)⇒[σ(s2)]^t_v=τ,[σ(s2)]^π_v =ψ.

We say thatΓisπ-dynamically-consistent (π-DC)if it admits σ ∈ SΓ which is both viable and π-dynamic. The problem of checking whether a given CSTN is π-DC is named π-DC- Checking.

Remark 1. It is easy to see that, due to the strict inequality

“[σ(s)]^π_O

p< ψ” in the definition ofπ-Hst(·)(Definition 15), in a π-dynamicπ-ES, there must be exactly oneO_p⁰ ∈ OV, for somep⁰ ∈P, which is executed at first (w.r.t. both execution time and position) under all possible scenarioss∈Σ_P. Proposition 3. The CSTNΓ₂ is not π-DC.

Proof. The proof goes almost in the same way as that of Proposition 2. In particular, no observation event is executed first (i.e., at time t = 0 and position ψ = 1) in all possible scenarios. Since there is no first-in-time observation event, then, the ESσis not π-dynamic.

We provide next a CSTN which isπ-DC but not DC.

Example 3. Define Γ_π = (V_π, A_π,Oπ,OVπ, P_π)as follows.

V_π = {O_p, X,>}, A_π = {(> − O_p ≤ 1, λ),(O_p− > ≤

−1, λ),(X− O_p ≤0, p),(> −X ≤ 0,¬p)}, O_π(p) = O_p, OVπ={Op},Pπ ={p}. Fig. 5 depicts the CSTNΓπ.

Op

p?

X

1 >

−1

0, p 0,¬p

Fig. 5: The CSTNΓπ.

Proposition 4. The CSTNΓπ isπ-DC, but it is not DC.

Proof. Lets1, s2∈ΣP_π be two scenarios such thats1(p) = 1 and s2(p) = 0. Consider the π-ES σ defined as follows:

[σ(s1)]^t_Op= [σ(s1)]^t_X= 0,[σ(s1)]^t_> = 1; and[σ(s2)]^t_Op = 0, [σ(s2)]^t_X = [σ(s2)]^t_> = 1; finally, [σ(s)]^π_Op = 1, [σ(s)]^π_X = [σ(s)]^π_> = 2, for all s ∈ {s1, s2}. Then, σ is viable and π- dynamic for Γπ. To see that Γπ is not DC, pick any ε >0.

Notice that any viable ES must scheduleX either att= 0or t= 1, depending on the outcome ofOp, which in turn happens att= 0; however, in anyε-dynamic strategy, the planner can’t react to the outcome ofOpbefore timet=ε >0. This implies that Γ_π is not ε-DC. Since εwas chosen arbitrarily (ε >0), thenΓ_π can’t be DC by Theorem 2.

SoΓ_πisε-DC forε= 0but fornoε >0. In summary, the following chain of implications holds on the various DCs:

[ε-DC, ∀ε∈(0,ε]]ˆ ⇔ DC⇒^6⇐ π-DC⇒^6⇐ [ε-DC, forε= 0]

whereεˆ,|ΣP|⁻¹· |V|⁻¹ as in Theorem 2.

A. The ps-tree: a “skeleton” structure forπ-dynamicπ-ESs In this subsection we introduce a labelled tree data structure, named the ps-tree, which turns out to capture the “skeleton”

ordered structure of π-dynamicπ-ESs.

Definition 17 (PS-Tree). Let P be any set of boolean vari- ables. A permutation-scenario tree (ps-tree) πT over P is an outward (non-empty) rooted binary tree such that:

• Each nodeuofπ_T is labelled with a letterp_u∈P;

• All the nodes that lie along a path leading from the root to a leaf are labelled with distinct letters from P.

• Each arc(u, v)ofπT is labelled by someb_(u,v)∈ {0,1};

• The two arcs (u, vl) and (u, vr) exiting a same node u have opposite labels, i.e.,b_(u,vl)6=b_(u,vr).

Fig. 6 depicts an example of a ps-tree.

a

b

c c

d d c

d b

d d

0

0 1

0 1

1 0

1

0 1

Fig. 6: An example of a ps-tree overP ={a, b, c, d}.

Definition 18(πs,si, Coherent-PS-Tree). LetπT be a ps-tree over P, letrbe the root andsbe any leaf. Let(r, v2, . . . , s) be the sequence of the nodes encountered along the path going fromr down tosinπT. Then:

• The sequence of labels π_s = (p_r, p_v2, . . . , p_s) is a per- mutation of the subset of letters {p_r, p_v2, . . . , p_s} ⊆P.

• Each sequence of bits (b_(r,v2), . . . , b_(vi,vi+1)), for each i ∈ {1,2, . . . , k_s−1} (where v₁ , r and v_ks , s), can be seen as a partial scenario s_i over P; i.e., define s_i(v_j),b_(vj,vj+1), for everyj ∈ {1, . . . , i}.

• πT iscoherent (c-ps-tree)withΓif, for every leafsofπT, {Op_r,Op_v2, . . . ,Op_s}=OV_s⁺0 holds for every complete scenario s⁰ ∈ΣP such thatSub(s⁰, sk_s−1).

It is not difficult to see that a π-dynamicπ-ES induces one and only one c-ps-treeπT. So, the existence of a suitable c-ps- tree is a necessary condition for aπ-ES to beπ-dynamic. One may ask whether aπ-dynamicπ-ES can be reconstructed from its c-ps-tree; the following subsection answers affirmatively.

B. Verifying a c-ps-tree: on π-DC and HyTN-Consistency.

This subsection builds on the notion of c-ps-tree to work out the details of the relationship betweenπ-DC and HyTN- Consistency. Once this picture is in place, it will be easy to reduce to HyTN-Consistency the problem of deciding whether a given CSTN admits a validπ-dynamicπ-ES with a given c- ps-tree. This easy result already provides a first combinatorial

algorithm forπ-DC, though of doubly exponential complexity in|P|; a bound to be improved in later subsections, but that can help sizing the sheer dimensionality and depth of the problem.

Firstly, the notion of Expansionof CSTNs is recalled [5].

Definition 19(ExpansionhV_Γ^Ex,Λ^Ex_Γi). Consider a CSTNΓ = (V, A, L,O,OV, P). Consider the family of all (distinct) STNs (V_s, A_s), one for each scenarios∈Σ_P, defined as follows:

Vs,{vs|v∈V_s⁺}and As,{(us, vs, w)|(u, v, w)∈A⁺_s}.

TheexpansionhV_Γ^Ex,Λ^Ex_Γiof the CSTNΓis defined as follows:

hV_Γ^Ex,Λ^Ex_Γi, [

s∈ΣP

Vs, [

s∈ΣP

As

.

Notice,(V_Γ^Ex,Λ^Ex_Γ )is an STN with at most|V_Γ^Ex| ≤ |ΣP|·|V| nodes and at most|Λ^Ex_Γ| ≤ |ΣP| · |A| standard arcs.

We now show that the expansion of a CSTN can be enriched with some standard arcs and some hyperarcs in order to model theπ-DC property, by means of an HyTN denotedH^π₀^T(Γ).

Definition 20(HyTNH^π₀^T(Γ)). LetΓ = (V, A, L,O,OV, P) be a given CSTN. Let π_T be a given c-ps-tree overP.

Then, the HyTNH^π₀^T(Γ)is defined as follows:

• For every scenarios s₁, s₂ ∈Σ_P and u∈V_s⁺

1,s₂\ OV, define a hyperarcα=α₀(s₁;s₂;u)as follows (with the intention to modelH0(s1;s2;u), see Definition 13):

α=α0(s1;s2;u),htα, Hα, wαi, where:

– tα,us₁ is the tail of the hyperarc α;

– Hα,{us₂} ∪∆(s1;s2)is the set of the heads;

– wα(us₂),0;∀(v∈∆(s1;s2))wα(v),0.

Now, consider the expansion of the CSTNΓhV_Γ^Ex,Λ^Ex_Γi= S

s∈ΣPVs,S

s∈ΣPAs

(as in Definition 19). Then:

• For each internal nodexofπT,A⁰_xis a set of (additional) standard arcs defined as follows. Let πx = (r, . . . , x⁰) be the sequence of all and only the nodes along the path going from the root r to the parent x⁰ ofx in πT

(where we can assume r⁰ = r). Let Px ⊆ P be the corresponding literals,p_x excluded, i.e.,P_x,{pz∈P| z appears inπ_x and p_z is the label ofz inπ_T} \ {px}.

Lets_x be the partial scenario defined as follows:

sx:Px→ {0,1}:

λ , ifx=r;

pz7→b(z,z⁰), ifx6=r.

where z⁰ is the unique child of z in πT lying on πx. Let x0 (x1) be the unique child of x in πT such that bx,x₀= 0 (bx,x₁ = 1). For every completes⁰_x∈ΣP such thatSub(s⁰_x, sx), we define:

B_s⁰0 x,

(Op_x0)s⁰_x,(Op_x)s⁰_x,0 , ifs⁰_x(x) = 0;

(Opx1)s⁰_x,(Opx)s⁰_x,0 , ifs⁰_x(x) = 1.

Also, for every complete s⁰_x, s⁰⁰_x ∈ Σ_P such that Sub(s⁰_x, s_x)and Sub(s⁰⁰_x, s_x), where s⁰_x6=s⁰⁰_x, we define:

C_s⁰0

x,s⁰⁰_x , (O_px)_s⁰

x,(O_px)_s⁰⁰

x,0 .

Finally,

A⁰_x, [

s⁰_x∈Σ_P:Sub(s⁰_x,s_x)

B_s⁰0

x∪ [

s⁰_x, s⁰⁰_x∈ΣP :s⁰_x6=s⁰⁰_x, Sub(s⁰_x, sx),Sub(s⁰⁰_x, sx)

C_s⁰0 x,s⁰⁰_x.

• Then, H^π₀(Γ) is defined as H^π₀(Γ) , hV_Γ^Ex,A_H^π

0(Γ)i, where,

A_H^π

0(Γ),Λ^Ex_Γ ∪ [

s₁,s₂∈Σ_P u∈V_s⁺_1,s2

αε(s1;s2;u)∪ [ x: internal node ofπT

A⁰_x.

Notice that the following holds: eachα_ε(s₁;s₂;u)has size

|αε(s₁;s₂;u)|= ∆(s₁;s₂) + 1≤ |P|+ 1.

The following theorem establishes the connection between the π-DC of CSTNs and the consistency of HyTNs.

Theorem 4. Given any CSTN Γ = hV, A, L,O,OV, Pi, it holds that the CSTN Γ is π-DC if and only if there exists a c-ps-tree π_T such that the HyTN H^π₀^T(Γ) is consistent.

Moreover, H^π₀^T(Γ) has at most|V_HπT

0 (Γ)| ≤ |Σ_P| |V|nodes,

|A_HπT

0 (Γ)| =O(|ΣP| |A|+|ΣP|²|V|) hyperarcs, and it has size at mostm_A

HπT

0 (Γ) =O(|ΣP| |A|+|ΣP|²|V| |P|).

Proof. (1) Firstly, we prove that the CSTNΓ isπ-DC if and only if there exists a c-ps-treeπ_T such that the HyTNH^π₀^T(Γ) is consistent.

(⇒) Letσ∈ S_Γbe a given viable andπ-dynamic execution strategy for the CSTN Γ. Since σisπ-dynamic, then for any two s1, s2 ∈ ΣP and any v ∈ V_s⁺_1,s2 the following holds on the execution time τ,[σ(s1)]^t_v and positionψ,[σ(s1)]^π_v:

Con(π-Hst(τ, ψ, s1, σ), s2)⇒[σ(s2)]^t_v =τ,[σ(s2)]^π_v =ψ.

It is easy to see that this induces one and only one c-ps-treeπT: indeed, due to Remark 1, there must be exactly oneOp⁰ ∈ OV, for some p⁰ ∈ P, which is executed at first (w.r.t. to both execution time and position) under all possible scenarios; then, depending on the boolean result of p⁰, a second observation p⁰⁰ can be differentiated, and it can occur at the same or at a subsequent time instant, but still at a subsequent position;

again, by Remark 1, there is exactly one Op⁰⁰ ∈ OV which comes first under all possible scenarios that agree onp⁰; and so on and so forth, thus forming a tree structure over P, rooted at p⁰, which is captured exactly by our notion of c-ps-tree.

Then, let φσ :V_Γ^Ex →R be the schedule ofH₀^πT(Γ)defined as:φσ(vs),[σ(s)]^t_v for every vs∈V_Γ^Ex, wheres∈ΣP and v ∈ V_s⁺. It is not difficult to check from the definitions, at this point, that all of the standard arc and hyperarc constraints of H^π₀^T(Γ)are satisfied byφσ, that is to say thatφσ must be feasible for H₀^πT(Γ). Hence,H^π₀^T(Γ) is consistent.

(⇐) Assume that there exists a c-ps-tree π_T such that the HyTN H^π₀^T(Γ) is consistent, and let φ : V_Γ^Ex → R be a feasible schedule forH^π₀^T(Γ). Then, letσ_φ,πT(s)∈ SΓbe the execution strategy defined as follows:

• [σ_φ,πT(s)]^t_v,φ(v_s),∀v_s∈V_Γ^Ex,s∈Σ_P,v∈V_s⁺;

• Let s⁰∈Σ_P be any complete scenario. Then,s⁰ induces exactly one path in π_T, in a natural way, i.e., by going

from the root r down to some leaf s. Notice that the sequence of labels (pr, pv₂, . . . , ps) can be seen as a bijection, i.e., πs :OV_s⁺0 {1, . . . ,|OV_s⁺0|}. Then, for anys⁰ ∈ΣP andv∈ OV_s⁺0, define[σφ,π_T(s⁰)]^π_v ,πs(v).

It is not difficult to check from the definitions, at this point, that sinceφis feasible forH^π₀^T(Γ), thenσφ,π_T must be viable andπ-dynamic for the CSTNΓ. Hence, the CSTNΓisπ-DC.

(2) The size bounds forH₀^πT(Γ)follow from Definition 20.

In Fig. 7, Algorithm 1 presents the pseudocode for con- structing the HyTN H₀^πT(Γ), as prescribed by Definition 20.

Algorithm 1: construct_H(Γ, πT)

Input: a CSTNΓ,hV, A, L,O,OV, Pi, a c-ps-treeπT

coherent withΓ.

1 foreach(s∈ΣP) do

2 Vs← {vs|v∈V_s⁺};

3 As← {as|a∈A⁺s};

4 VΓ^Ex← ∪s∈Σ_PVs;

5 Λ^ExΓ ← ∪s∈Σ_PAs;

6 foreach(s1, s2∈ΣP,s16=s2)do

7 foreach(u∈Vs⁺₁,s₂\ OV)do

8 tα←us₁;

9 Hα← {us₂} ∪(∆(s1;s2));

10 wα(us₂)←0;

11 foreachv∈∆(s1;s2)do

12 wα(vs₁)←0;

13 α0(s1;s2;u)← htα, Hα, wαi;

14 foreach(x: internal node ofπT)do

15 A⁰_x←as defined in Definition 20;

16 A_HπT

0 (Γ)←Λ^ExΓ ∪ [

s₁,s₂∈Σ_P u∈V_s⁺

1,s2

α0(s1;s2;u)∪ [ x:internal node ofπT

A⁰x;

17 H^π₀^T(Γ)← hV_Γ^Ex,A_HπT 0 (Γ)i;

18 returnH^π₀^T(Γ);

IfΓisπ-DC, there is an integralπ-dynamicπ-ES, as below.

Proposition 5. AssumeΓ =hV, A, L,O,OV, Pito beπ-DC.

Then, there is someπ-ESσ∈ S_Γ which is viable,π-dynamic, and integral, namely, for every s∈ Σ_P and every v ∈ V_s⁺, the following integrality property holds:

[σ(s)]^t_v∈

0,1,2, . . . ,MΓ ⊆N, whereMΓ, |ΣP||V|+|ΣP||A|+|ΣP|²|V|

W.

Proof. By Theorem 4, sinceΓisπ-DC, there exists some c-ps- tree πT such that the HyTNH^π₀^T(Γ)is consistent; moreover, by Theorem 4 again, H^π₀^T(Γ) has |V_HπT

0 (Γ)| ≤ |ΣP| |V| nodes and|A_HπT

0 (Γ)| ≤ |ΣP| |A|+|ΣP|²|V|hyperarcs. Since H^π₀^T(Γ) is consistent, it follows from Theorem 4 [also see Lemma 1 and Theorem 8 in [4]] that H₀^πT(Γ) admits an integral and feasible schedule φsuch that:

φ:V_HπT

0 (Γ)→

0,1,2, . . . ,M_Γ ,

whereMΓ≤(|V_HπT

0 (Γ)|+|A_HπT

0 (Γ)|)W. Therefore, it holds that MΓ≤(|ΣP| |V|+|ΣP| |A|+|ΣP|²|V|)W.

Given a CSTN Γ and some c-ps-treeπ_T, it is thus easy to check whether there exists some π-ES forΓ whose ordering relations are exactly the same as those prescribed by π_T. Indeed, it is sufficient to constructH^π₀^T(Γ)with Algorithm 1, then checking the consistency of H₀^πT(Γ)with the algorithm mentioned in Theorem 1. This results into Algorithm 2. The corresponding time complexity is also that of Theorem 1.

Algorithm 2: check_π-DC_on_c-ps-tree(Γ, πT) Input: a CSTNΓ,hV, A, L,O,OV, Pi, a c-ps-treeπT

coherent withΓ.

1 H^π₀^T(Γ)←construct_H(Γ, πT);//ref. Algorithm 1 2 φ←check_HyTN-Consistency(H^π₀^T(Γ));//ref. Thm 1 3 if(φis a feasible schedule ofH^π₀^T(Γ))then

4 returnhYES, φ, πTi;

5 returnNO;

Fig. 7: Checking π-DC given a c-ps-treeπ_T, by reduction to HyTN-Consistency.

Notice that, in principle, one could generate all of the possi- ble c-ps-treesπT givenP, one by one, meanwhile checking for the consistency state of H^π₀^T(Γ)with Algorithm 2. However, it is not difficult to see that, in general, the total number f_|P|

of possible c-ps-trees over P is not singly-exponential in|P|.

Indeed, a moment’s reflection revelas that for every n >1 it holds thatf_n =n·f_n−1² , andf₁= 1. So, any algorithm based on the exhaustive exploration of the whole space comprising all of the possible c-ps-trees overPwould not have a (pseudo) singly-exponential time complexity in |P|. Nevertheless, we have identified another solution, that allows us to provide a sound-and-complete (pseudo) singly-exponential time π- DC-Checking procedure: it is a simple and self-contained reduction fromπ-DC-Checking to DC-Checking. This allows us to provide the first sound-and-complete (pseudo) singly- exponential time π-DC-Checking algorithm which employs our previous DC-Checking algorithm (i.e., that underlying Theorem 3) in a direct manner, as a black box, thus avoiding a more fundamental restructuring of it.

C. A Singly-Exponential Time π-DC-Checking Algorithm This section presents a sound-and-complete (pseudo) singly- exponential time algorithm for solving π-DC, also producing a viable and π-dynamic π-ES whenever the input CSTN is really π-DC. The main result of this section goes as follows.

Theorem 5. There exists an algorithm for checking π-DC on any input given CSTN Γ = (V, A, L,O,OV, P) with the following (pseudo) singly-exponential time complexity:

O

|ΣP|⁴|A|²|V|³+|ΣP|⁵|A||V|⁴|P|+|ΣP|⁶|V|⁵|P| W.

Moreover, whenΓisπ-DC, the algorithm also returns a viable and π-dynamicπ-ES for Γ. Here,W ,max_a∈A|w_a|.

The algorithm mentioned in Theorem 5 consits of a simple reduction fromπ-DC to (classical) DC in CSTNs.

Basically, the idea is to give a small margin γ so that the planner can actually do before, in the sense of the time value [σ(s)]v, what he did “before” in the orderingπ. Given any ES in the relaxed network, the planner would then turn it into a π-ES for the original network (which has some more stringent constraints), by rounding-down each time value[σ(s)]_v to the largest integer less than or equal to it, i.e.,

[σ(s)]_v . The problem is that one may (possibly) violate some constraints when there is a “leap” in the rounding (i.e., a difference of one unit, in the rounded value, w.r.t. what one would have wanted).

Anyhow, we have identified a technique that allows us to get around this subtle case, provided thatγis exponentially small.

Definition 21. Relaxed CSTNΓ⁰. LetΓ =hV, A, L,O,OV, Pi be any CSTN with integer constraints. Letγ∈(0,1)be a real.

Define Γ⁰_γ ,hV, A⁰_γ, L,O,OV, Pito be a CSTN that differs from Γ only in the numbers appearing in the constraints.

Specifically, each constraint hu−v ≤δ, `i ∈ A is replaced inΓ<sup>0</sup><sub>γ</sub> by a slightly relaxed constraint, hu−v≤δ<sup>0</sup><sub>γ</sub>, `i ∈A⁰_γ, where:

δ_γ⁰ ,δ+|V| ·γ.

The following two lemmata hold for any CSTNΓ.

Lemma 1. Let γbe any real in(0,|V|⁻¹).

IfΓ isπ-DC, then Γ⁰_γ is DC.

Proof. Since Γ is π-DC, by Proposition 5, there exists an integral, viable andπ-dynamic,π-ESσforΓ. Let us fix some real γ ∈(0,|V|⁻¹). Define the ES σ⁰_γ ∈ SΓ⁰_γ as follows, for everys∈ΣP andv∈V_s⁺:

[σ_γ⁰(s)]v,[σ(s)]^t_v+ [σ(s)]^π_v ·γ.

Since[σ(s)]^π_v ≤ |V|, then:

[σ(s)]^π_v·γ <|V| · |V|⁻¹= 1,

and so the total ordering of the values[σ_γ⁰(s)]v, for a givens∈ ΣP, coincides with[σ(s)]^π. Hence, the fact thatσ⁰_γis dynamic follows directly from the π-dynamicity of σ. Moreover, no LTC(u−v≤δ_γ⁰, `)of Γ⁰_γ is violated in any scenarios∈Σ_P since, if∆⁰_γ,u,v ,[σ_γ⁰(s)]u−[σ_γ⁰(s)]v then:

∆⁰_γ,u,v = [σ(s)]^t_u+ [σ(s)]^π_u·γ

− [σ(s)]^t_v+ [σ(s)]^π_v ·γ

≤[σ(s)]^t_u−[σ(s)]^t_v+|V| ·γ

≤δ+|V| ·γ=δ⁰.

So,σ_γ⁰ is viable. Sinceσ⁰_γis also dynamic, thenΓ⁰_γ is DC.

The next lemma shows that the converse direction holds as well, but for (exponentially) smaller values of γ.

Lemma 2. Let γbe any real in(0,|ΣP|⁻¹· |V|⁻²).

IfΓ⁰_γ is DC, thenΓ isπ-DC.

Proof. Letσ⁰_γ ∈ S_Γ⁰

γ be some viable and dynamic ES forΓ⁰_γ.