Undecidability of future timeline-based planning over dense temporal domains
Laura Bozzelli, Alberto Molinari, Angelo Montanari, Adriano Peron
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L. Bozzelli, A. Molinari, A. Montanari & A. PeronThis work is licensed under theCreative Commons Attribution License.
Undecidability of future timeline-based planningover dense temporal domains
Laura Bozzelli Adriano Peron
University of Napoli “Federico II”, Napoli, Italy [email protected] [email protected]
Alberto Molinari Angelo Montanari
University of Udine, Udine, Italy [email protected] [email protected]
Planning is one of the most studied problems in computer science. In this paper, we consider thetimeline-based approach, where the domain is modeled by a set of independent, but interacting,components, identified by a set of state variables, whose behavior over time (timelines) is governedby a set of temporal constraints (synchronization rules). Timeline-based planning in the dense-timesetting has been recently shown to be undecidable in the general case, and undecidability relies on thehigh expressiveness of the trigger synchronization rules. In this paper, we strengthen the previousnegative result by showing that undecidability already holds under the future semantics of the triggerrules which limits the comparison to temporal contexts in the future with respect to the trigger.
Timeline-based planning (TP for short) represents a promising approach for real-time temporal planningand reasoning about execution under uncertainty [14, 12, 13, 8, 9, 11]. Compared to classical action-basedtemporal planning [15, 24], TP adopts a more declarative paradigm which is focused on the constraints thatsequences of actions have to fulfill to reach a fixed goal. In TP, the planning domain is modeled as a set ofindependent, but interacting, components, each one identified by a state variable . The temporal behaviourof a single state variable (component) is described by a sequence of tokens ( timeline ) where each tokenspecifies a value of variable (state) and the period of time during which the variable assumes that value.The overall temporal behaviour (set of timelines) is constrained by a set of synchronization rules whichspecify quantitative temporal requirements between the time events (start-time and end-time) of distincttokens. Synchronization rules have a very simple format: either trigger rules expressing invariants andresponse properties (for each token with a fixed state, called trigger , there exist tokens satisfying somemutual temporal relations) or trigger-less rules expressing goals (there exist tokens satisfying some mutualtemporal relations). Note that the way in which timing requirements are specified in the synchronizationrules corresponds to the “freeze” mechanism in the well-known timed temporal logic TPTL [2] whichuses the freeze quantifier to bind a variable to a specific temporal context (a token in the TP setting).TP has been successfully exploited in a number of application domains, including space missions,constraint solving, and activity scheduling (see, e.g., [21, 19, 16, 7, 10, 4]). A systematic study ofexpressiveness and complexity issues for TP has been undertaken only very recently both in the discrete-time and dense-time settings [17, 18, 6, 5]. In the discrete-time context, the TP problem is
EXPSPACE -complete, and is expressive enough to capture action-based temporal planning (see [17, 18]).On the other hand, despite the simple format of synchronization rules, the shift to a dense-time domaindramatically increases expressiveness, depicting a scenario which resembles that of the well-known timed a r X i v : . [ c s . F L ] A p r Complexity of timeline-based planning over dense domains linear temporal logics
MTL and
TPTL (under a pointwise semantics) which are undecidable in the generalsetting [2, 22]. In fact the TP problem is undecidable in the general case [6], and undecidability relieson the high expressiveness of the trigger rules (by restricting the formalism to only trigger-less rulesthe problem is just NP -complete [6]). Decidability can be recovered by suitable (syntactic/semantic)restrictions on the trigger rules. In particular, in [5], two restrictions are considered: (i) the first one limitsthe comparison to tokens whose start times follow the trigger start time ( future semantics of trigger rules ),and (ii) the second one is syntactical and imposes that a non-trigger token can be referenced at most oncein the timed constraints of a trigger rule ( simple trigger rules ). Note that the second restriction avoidscomparisons of multiple token time-events with a non-trigger reference time-event. Under the previoustwo restrictions, the TP problem is decidable with a non-primitive recursive complexity [5] and can besolved by a reduction to model checking of Timed Automata( TA ) [3] against MTL over finite timed words,the latter being a known decidable problem [23]. As in the case of
MTL [1], better complexity results,i.e.
EXPSPACE -completeness (resp.,
PSPACE -completeness) can be obtained by restricting also thetype of intervals used in the simple trigger rules in order to compare tokens: non-singular intervals (resp.,intervals unbounded or starting from 0).In this paper, we show that both the considered restrictions on the trigger rules are necessary torecovery decidability. The undecidability of the TP problem with simple trigger rules has been alreadyestablished in [6]. Here, we prove undecidability of the TP problem with arbitrary trigger rules under thefuture semantics.
Let N be the set of natural numbers, R + be the set of non-negative real numbers, and Intv be the set ofintervals in R + whose endpoints are in N ∪ { ∞ } . Moreover, let us denote by Intv ( , ∞ ) the set of intervals I ∈ Intv such that either I is unbounded, or I is left-closed with left endpoint 0. Such intervals I can bereplaced by expressions of the form ∼ n for some n ∈ N and ∼∈ { <, ≤ , >, ≥} . Let w be a finite wordover some alphabet. By | w | we denote the length of w . For all 0 ≤ i < | w | , w ( i ) is the i -th letter of w . In this section, we recall the TP framework as presented in [13, 17]. In TP, domain knowledge isencoded by a set of state variables, whose behaviour over time is described by transition functions andsynchronization rules.
Definition 1. A state variable x is a triple x = ( V x , T x , D x ) , where V x is the finite domain of the variable x , T x : V x → V x is the value transition function , which maps each v ∈ V x to the (possibly empty) set ofsuccessor values, and D x : V x → Intv is the constraint function that maps each v ∈ V x to an interval.A token for a variable x is a pair ( v , d ) consisting of a value v ∈ V x and a duration d ∈ R + such that d ∈ D x ( v ) . Intuitively, a token for x represents an interval of time where the state variable x takes value v .The behavior of the state variable x is specified by means of timelines which are non-empty sequences oftokens π = ( v , d ) . . . ( v n , d n ) consistent with the value transition function T x , that is, such that v i + ∈ T x ( v i ) for all 0 ≤ i < n . The start time s ( π , i ) and the end time e ( π , i ) of the i -th token (0 ≤ i ≤ n ) of the timeline π are defined as follows: e ( π , i ) = i ∑ h = d h and s ( π , i ) = i =
0, and s ( π , i ) = i − ∑ h = d h otherwise. SeeFigure 1 for an example. . Bozzelli, A. Molinari, A. Montanari & A. Peron x t = t = t = t = . x = a x = b x = c x = b Figure 1: An example of timeline ( a , )( b , )( c , . ) · · · for the state variable x = ( V x , T x , D x ) , where V x = { a , b , c , . . . } , b ∈ T x ( a ) , c ∈ T x ( b ) , b ∈ T x ( c ) . . . and D x ( a ) = [ , ] , D x ( b ) = [ , ] , D x ( c ) = [ , ∞ [ . . .Given a finite set SV of state variables, a multi-timeline of SV is a mapping Π assigning to each statevariable x ∈ SV a timeline for x . Multi-timelines of SV can be constrained by a set of synchronizationrules , which relate tokens, possibly belonging to different timelines, through temporal constraints onthe start/end-times of tokens (time-point constraints) and on the difference between start/end-times oftokens (interval constraints). The synchronization rules exploit an alphabet Σ of token names to refer tothe tokens along a multi-timeline, and are based on the notions of atom and existential statement . Definition 2. An atom is either a clause of the form o ≤ e , e I o ( interval atom ), or of the forms o ≤ e I n or n ≤ e I o ( time-point atom ), where o , o ∈ Σ , I ∈ Intv , n ∈ N , and e , e ∈ { s , e } .An atom ρ is evaluated with respect to a Σ -assignment λ Π for a given multi-timeline Π which is amapping assigning to each token name o ∈ Σ a pair λ Π ( o ) = ( π , i ) such that π is a timeline of Π and0 ≤ i < | π | is a position along π (intuitively, ( π , i ) represents the token of Π referenced by the name o ). An interval atom o ≤ e , e I o is satisfied by λ Π if e ( λ Π ( o )) − e ( λ Π ( o )) ∈ I . A point atom o ≤ eI n (resp., n ≤ eI o ) is satisfied by λ Π if n − e ( λ Π ( o )) ∈ I (resp., e ( λ Π ( o )) − n ∈ I ). Definition 3. An existential statement E for a finite set SV of state variables is a statement of the form: E : = ∃ o [ x = v ] · · · ∃ o n [ x n = v n ] . C where C is a conjunction of atoms, o i ∈ Σ , x i ∈ SV , and v i ∈ V x i for each i = , . . . , n . The elements o i [ x i = v i ] are called quantifiers . A token name used in C , but not occurring in any quantifier, is said tobe free . Given a Σ -assignment λ Π for a multi-timeline Π of SV , we say that λ Π is consistent with theexistential statement E if for each quantified token name o i , λ Π ( o i ) = ( π , h ) where π = Π ( x i ) and the h -th token of π has value v i . A multi-timeline Π of SV satisfies E if there exists a Σ -assignment λ Π for Π consistent with E such that each atom in C is satisfied by λ Π . Definition 4. A synchronization rule R for a finite set SV of state variables is a rule of one of the forms o [ x = v ] → E ∨ E ∨ . . . ∨ E k , (cid:62) → E ∨ E ∨ . . . ∨ E k , where o ∈ Σ , x ∈ SV , v ∈ V x , and E , . . . , E k are existential statements . In rules of the first form ( triggerrules ), the quantifier o [ x = v ] is called trigger , and we require that only o may appear free in E i (for i = , . . . , n ). In rules of the second form ( trigger-less rules ), we require that no token name appears free.Intuitively, a trigger o [ x = v ] acts as a universal quantifier, which states that for all the tokens of thetimeline for the state variable x , where the variable x takes the value v , at least one of the existentialstatements E i must be true. Trigger-less rules simply assert the satisfaction of some existential statement.The semantics of synchronization rules is formally defined as follows. Definition 5.
Let Π be a multi-timeline of a set SV of state variables. Given a trigger-less rule R of SV , Π satisfies R if Π satisfies some existential statement of R . Given a trigger rule R of SV with trigger Complexity of timeline-based planning over dense domains o [ x = v ] , Π satisfies R if for every position i of the timeline Π ( x ) for x such that Π ( x ) = ( v , d ) ,there is an existential statement E of R and a Σ -assignment λ Π for Π which is consistent with E such that λ Π ( o ) = ( Π ( x ) , i ) and λ Π satisfies all the atoms of E .In the paper, we focus on a stronger notion of satisfaction of trigger rules, called satisfaction underthe future semantics . It requires that all the non-trigger selected tokens do not start strictly before thestart-time of the trigger token. Definition 6.
A multi-timeline Π of SV satisfies under the future semantics a trigger rule R = o [ x = v ] → E ∨ E ∨ . . . ∨ E k if Π satisfies the trigger rule obtained from R by replacing each existentialstatement E i = ∃ o [ x = v ] · · · ∃ o n [ x n = v n ] . C with ∃ o [ x = v ] · · · ∃ o n [ x n = v n ] . C ∧ (cid:86) ni = o ≤ s , s [ , + ∞ [ o i .A TP domain P = ( SV , R ) is specified by a finite set SV of state variables and a finite set R ofsynchronization rules modeling their admissible behaviors. Trigger-less rules can be used to expressinitial conditions and the goals of the problem, while trigger rules are useful to specify invariants andresponse requirements. A plan of P is a multi-timeline of SV satisfying all the rules in R . A future plan ofP is defined in a similar way, but we require that the fulfillment of the trigger rules is under the futuresemantics. We are interested in the Future TP problem consisting in checking for a given TP domain P = ( SV , R ) , the existence of a future plan for P . In this section, we establish the following result.
Theorem 7.
Future TP with one state variable is undecidable even if the intervals are in Intv ( , ∞ ) . Theorem 7 is proved by a polynomial-time reduction from the halting problem for Minsky -countermachines [20]. Such a machine is a tuple M = ( Q , q init , q halt , ∆ ) , where Q is a finite set of (control)locations, q init ∈ Q is the initial location, q halt ∈ Q is the halting location, and ∆ ⊆ Q × L × Q is a transitionrelation over the instruction set L = { inc , dec , zero } × { , } .We adopt the following notational conventions. For an instruction op = ( , c ) ∈ L , let c ( op ) : = c ∈{ , } be the counter associated with op . For a transition δ ∈ ∆ of the form δ = ( q , op , q (cid:48) ) , we define from ( δ ) : = q , op ( δ ) : = op , c ( δ ) : = c ( op ) , and to ( δ ) : = q (cid:48) . Without loss of generality, we make theseassumptions: • for each transition δ ∈ ∆ , from ( δ ) (cid:54) = q halt and to ( δ ) (cid:54) = q init , and • there is exactly one transition in ∆ , denoted δ init , having as source the initial location q init .An M -configuration is a pair ( q , ν ) consisting of a location q ∈ Q and a counter valuation ν : { , } → N . M induces a transition relation, denoted by −→ , over pairs of M -configurations defined as follows. Forconfigurations ( q , ν ) and ( q (cid:48) , ν (cid:48) ) , ( q , ν ) −→ ( q (cid:48) , ν (cid:48) ) if for some instruction op ∈ L , ( q , op , q (cid:48) ) ∈ ∆ andthe following holds, where c ∈ { , } is the counter associated with the instruction op : (i) ν (cid:48) ( c (cid:48) ) = ν ( c (cid:48) ) if c (cid:48) (cid:54) = c ; (ii) ν (cid:48) ( c ) = ν ( c ) + op = ( inc , c ) ; (iii) ν ( c ) > ν (cid:48) ( c ) = ν ( c ) − op = ( dec , c ) ; and(iv) ν (cid:48) ( c ) = ν ( c ) = op = ( zero , c ) .A computation of M is a non-empty finite sequence C , . . . , C k of configurations such that C i −→ C i + for all 1 ≤ i < k . M halts if there is a computation starting at the initial configuration ( q init , ν init ) , where ν init ( ) = ν init ( ) =
0, and leading to some halting configuration ( q halt , ν ) . The halting problem is todecide whether a given machine M halts, and it is was proved to be undecidable [20]. We prove thefollowing result, from which Theorem 7 directly follows. Proposition 8.
One can construct (in polynomial time) a TP instance (domain) P = ( { x M } , R M ) wherethe intervals in P are in Intv ( , ∞ ) such that M halts iff there exists a future plan for P. . Bozzelli, A. Molinari, A. Montanari & A. Peron Proof.
First, we define a suitable encoding of a computation of M as the untimed part of a timeline (i.e.,neglecting tokens’ durations and accounting only for their values) for x M . For this, we exploit the finiteset of symbols V : = V main ∪ V sec corresponding to the finite domain of the state variable x M . The set of main values V main is the set of M -transitions, i.e. V main = ∆ . The set of secondary values V sec is definedas V sec : = ∆ × { , } × { , beg , end } , where beg , and end are three special symbols used as markers.Intuitively, in the encoding of an M -computation a main value keeps track of the transition used in thecurrent step of the computation, while the set V sec is used for encoding counter values.For c ∈ { , } , a c-code for the main value δ ∈ ∆ is a finite word w c over V sec of the form ( δ , c , beg ) · ( δ , c , ) h · ( δ , c , end ) for some h ≥ h = op ( δ ) = ( zero , c ) . The c -code w c encodes the valuefor counter c given by h (or equivalently | w c | − ( δ , c , ) encode units in the value of counter c , while the symbol ( δ , c , beg ) (resp., ( δ , c , end ) ) is only used as left(resp., right) marker in the encoding.A configuration-code w for a main value δ ∈ ∆ is a finite word over V of the form w = δ · w · w suchthat for each counter c ∈ { , } , w c is a c -code for the main value δ . The configuration-code w encodesthe M -configuration ( from ( δ ) , ν ) , where ν ( c ) = | w c | − c ∈ { , } . Note that if op ( δ ) = ( zero , c ) ,then ν ( c ) = computation -code is a non-empty sequence of configuration-codes π = w δ · · · w δ k , where for all 1 ≤ i ≤ k , w δ i is a configuration-code with main value δ i , and whenever i < k , it holds that to ( δ i ) = from ( δ i + ) .Note that by our assumptions to ( δ i ) (cid:54) = q halt for all 1 ≤ i < k , and δ j (cid:54) = δ init for all 1 < j ≤ k . Thecomputation-code π is initial if the first configuration-code w δ has the main value δ init and encodes theinitial configuration, and it is halting if for the last configuration-code w δ k in π , it holds that to ( δ k ) = q halt .For all 1 ≤ i ≤ k , let ( q i , ν i ) be the M -configuration encoded by the configuration-code w δ i and c i = c ( δ i ) .The computation-code π is well-formed if, additionally, for all 1 ≤ j < k , the following holds: • ν j + ( c ) = ν j ( c ) if either c (cid:54) = c j or op ( δ j ) = ( zero , c j ) ( equality requirement ); • ν j + ( c j ) = ν j ( c j ) + op ( δ j ) = ( inc , c j ) ( increment requirement ); • ν j + ( c j ) = ν j ( c j ) − op ( δ j ) = ( dec , c j ) ( decrement requirement ).Clearly, M halts iff there exists an initial and halting well-formed computation-code. Definition of x M and R M . We now define a state variable x M and a set R M of synchronization rules for x M with intervals in Intv ( , ∞ ) such that the untimed part of every future plan of P = ( { x M } , R M ) is an initialand halting well-formed computation-code. Thus, M halts iff there is a future plan of P .Formally, variable x M is given by x M = ( V = V main ∪ V sec , T , D ) , where for each v ∈ V , D ( v ) =] , ∞ [ .Thus, we require that the duration of a token is always greater than zero ( strict time monotonicity ). Thevalue transition function T of x M ensures the following property. Claim . The untimed parts of the timelines for x M whose first token has value δ init correspond to theprefixes of initial computation-codes. Moreover, δ init / ∈ T ( v ) for all v ∈ V .By construction, it is a trivial task to define T so that the previous requirement is fulfilled.Let V halt = { δ ∈ ∆ | to ( δ ) = q halt } . By Claim 9 and the assumption that from ( δ ) (cid:54) = q halt for eachtransition δ ∈ ∆ , in order to enforce the initialization and halting requirements, it suffices to ensure that atimeline has a token with value δ init and a token with value in V halt . This is captured by the trigger-lessrules (cid:62) → ∃ o [ x M = δ init ] . (cid:62) and (cid:62) → (cid:87) v ∈ V halt ∃ o [ x M = v ] . (cid:62) .Finally, the crucial well-formedness requirement is captured by the trigger rules in R M which expresspunctual time constraints . We refer the reader to Figure 2, that gives an intuition on the properties Such punctual contrains are expressed by pairs of conjoined atoms whose intervals are in
Intv ( , ∞ ) . Complexity of timeline-based planning over dense domains = 1 = 1= 1 δ δ (cid:48) ( δ , , e n d ) ( δ , , ) ( δ , , b e g ) ( δ , , e n d ) ( δ , , ) ( δ , , b e g ) ( δ (cid:48) , , e n d ) ( δ (cid:48) , , ) ( δ (cid:48) , , b e g ) ( δ (cid:48) , , e n d ) ( δ (cid:48) , , ) ( δ (cid:48) , , b e g ) ( δ (cid:48) , , ) = 1 w w (cid:48) Figure 2: The figure shows two adjacent configuration-codes, w (highlighted in cyan) and w (cid:48) (in green),the former for δ = ( q , ( inc , ) , q (cid:48) ) ∈ ∆ and the latter for δ (cid:48) = ( q (cid:48) , . . . ) ∈ ∆ ; w encodes the M -configuration ( q , ν ) where ν ( ) = ν ( ) =
1, and w (cid:48) the M -configuration ( q (cid:48) , ν (cid:48) ) where ν (cid:48) ( ) = ν (cid:48) ( ) = op ( δ ) = ( inc , ) , the value of counter 2 does not change in this computation step, and thus the valuesfor counter 2 encoded by w and w (cid:48) must be equal. To this aim the “equality requirement” (represented byblue lines with arrows) sets a one-to-one correspondence between pairs of tokens associated with counter2 in w and w (cid:48) (more precisely, a token tk with value ( δ , , ) in w is followed by a token tk (cid:48) with value ( δ (cid:48) , , ) in w (cid:48) such that s ( tk (cid:48) ) − s ( tk ) = e ( tk (cid:48) ) − e ( tk ) = tk (cid:48) with value ( δ (cid:48) , , ) is in w (cid:48) in the placewhere the token tk with value ( δ , , beg ) was in w (i.e., s ( tk (cid:48) ) − s ( tk ) = e ( tk (cid:48) ) − e ( tk ) = tk (cid:48)(cid:48) with value ( δ (cid:48) , , beg ) is “anticipated”, in such a way that e ( tk (cid:48)(cid:48) ) − s ( tk ) = δ (cid:48) in w (cid:48) has a shorter duration than that with value δ in w , leaving space for tk (cid:48)(cid:48) , so as to represent the unit added by δ to counter 1. Clearly density of the timedomain plays a fundamental role here.enforced by the rules we are about to define. In particular, we essentially take advantage of the densetemporal domain to allow for the encoding of arbitrarily large values of counters in one time units. Trigger rules for 1-Time distance between consecutive main values.
We define non-simple triggerrules requiring that the overall duration of the sequence of tokens corresponding to a configuration-codeamounts exactly to one time units. By Claim 9, strict time monotonicity, and the halting requirement, itsuffices to ensure that each token tk having a main value in V main \ V halt is eventually followed by a token tk (cid:48) such that tk (cid:48) has a main value and s ( tk (cid:48) ) − s ( tk ) = v ∈ V main \ V halt , we write the non-simpletrigger rule with intervals in Intv ( , ∞ ) : o [ x M = v ] → (cid:95) u ∈ V main ∃ o (cid:48) [ x M = u ] . o ≤ s , s [ , + ∞ [ o (cid:48) ∧ o ≤ s , s [ , ] o (cid:48) . Trigger rules for the equality requirement.
In order to ensure the equality requirement, we exploitthe fact that the end time of a token along a timeline corresponds to the start time of the next token (ifany). Let V = sec be the set of secondary states ( δ , c , t ) ∈ V sec such that to ( δ ) (cid:54) = q halt , and either c (cid:54) = c ( δ ) or op ( δ ) = ( zero , c ) . Moreover, for a counter c ∈ { , } and a tag t ∈ { beg , , end } , let V tc ⊆ V sec be the setof secondary states given by ∆ × { c } × { t } . We require the following:(*) each token tk with a ( V tc ∩ V = sec ) -value is eventually followed by a token tk (cid:48) with a V tc -value such . Bozzelli, A. Molinari, A. Montanari & A. Peron s ( tk (cid:48) ) − s ( tk ) = t (cid:54) = end , then e ( tk (cid:48) ) − e ( tk ) = Intv ( , ∞ ) : • for each v ∈ V tc ∩ V = sec and t (cid:54) = end , o [ x M = v ] → (cid:95) u ∈ V tc ∃ o (cid:48) [ x M = u ] . o ≤ s , s [ , + ∞ [ o (cid:48) ∧ o ≤ s , s [ , ] o (cid:48) ∧ o ≤ e , e [ , + ∞ [ o (cid:48) ∧ o ≤ e , e [ , ] o (cid:48) ; • for each v ∈ V endc ∩ V = sec , o [ x M = v ] → (cid:95) u ∈ V endc ∃ o (cid:48) [ x M = u ] . o ≤ s , s [ , + ∞ [ o (cid:48) ∧ o ≤ s , s [ , ] o (cid:48) . We now show that Condition (*) together with strict time monotonicity and 1-Time distance betweenconsecutive main values ensure the equality requirement. Let π be a timeline of x M satisfying all the rulesdefined so far, w δ and w δ (cid:48) two adjacent configuration-codes along π with w δ preceding w δ (cid:48) (note that to ( δ ) (cid:54) = q halt ), and c ∈ { , } a counter such that either c (cid:54) = c ( δ ) or op ( δ ) = ( zero , c ) . Let tk · · · tk (cid:96) + (resp., tk (cid:48) · · · tk (cid:48) (cid:96) (cid:48) + ) be the sequence of tokens associated with the c -code of w δ (resp., w δ (cid:48) ). We needto show that (cid:96) = (cid:96) (cid:48) . By construction tk and tk (cid:48) have value in V begc , tk (cid:96) + and tk (cid:48) (cid:96) (cid:48) + have value in V endc ,and for all 1 ≤ i ≤ (cid:96) (resp., 1 ≤ i (cid:48) ≤ (cid:96) (cid:48) ), tk i has value in V c (resp., tk (cid:48) i (cid:48) has value in V c ). Then strict timemonotonicity, 1-Time distance between consecutive main values, and Condition (*) guarantee the existenceof an injective mapping g : { tk , . . . , tk (cid:96) + } → { tk (cid:48) , . . . , tk (cid:48) (cid:96) (cid:48) + } such that g ( tk ) = tk (cid:48) , g ( tk (cid:96) + ) = tk (cid:48) (cid:96) (cid:48) + ,and for all 0 ≤ i ≤ (cid:96) , if g ( tk i ) = tk (cid:48) j (note that j < (cid:96) (cid:48) + g ( tk i + ) = tk (cid:48) j + (we recall that the endtime of a token is equal to the start time of the next token along a timeline, if any). These properties ensurethat g is surjective as well. Hence, g is a bijection and (cid:96) (cid:48) = (cid:96) . Trigger rules for the increment requirement.
Let V inc sec be the set of secondary states ( δ , c , t ) ∈ V sec such that to ( δ ) (cid:54) = q halt and op ( δ ) = ( inc , c ) . By reasoning like in the case of the rules ensuring the equalityrequirement, in order to express the increment requirement, it suffices to enforce the following conditionsfor each counter c ∈ { , } :(i) each token tk with a ( V begc ∩ V inc sec ) -value is eventually followed by a token tk (cid:48) with a V begc -value suchthat e ( tk (cid:48) ) − s ( tk ) = tk (cid:48) and the start time oftoken tk is exactly 1);(ii) for each t ∈ { beg , } , each token tk with a ( V tc ∩ V inc sec ) -value is eventually followed by a token tk (cid:48) with a V c -value such that s ( tk (cid:48) ) − s ( tk ) = e ( tk (cid:48) ) − e ( tk ) = ( V begc ∩ V inc sec ) -value is associated with atoken with V c -value anyway;(iii) each token tk with a ( V endc ∩ V inc sec ) -value is eventually followed by a token tk (cid:48) with a V endc -value suchthat s ( tk (cid:48) ) − s ( tk ) = w and w (cid:48) are two adjacent configuration-codes along a timeline of x M , with w preceding w (cid:48) ,(i) and (ii) force a token tk (cid:48) with a V c -value in w (cid:48) to “take the place” of the token tk with ( V begc ∩ V inc sec ) -valuein w (i.e., they have the same start and end times). Moreover a token with V begc -value must immediatelyprecede tk (cid:48) in w (cid:48) .These requirements can be expressed by non-simple trigger rules with intervals in Intv ( , ∞ ) similar tothe ones defined for the equality requirement. Complexity of timeline-based planning over dense domains
Trigger rules for the decrement requirement.
For capturing the decrement requirement, it suffices toenforce the following conditions for each counter c ∈ { , } , where V dec sec denotes the set of secondarystates ( δ , c , t ) ∈ V sec such that to ( δ ) (cid:54) = q halt and op ( δ ) = ( dec , c ) :(i) each token tk with a ( V begc ∩ V dec sec ) -value is eventually followed by a token tk (cid:48) with a V begc -value suchthat s ( tk (cid:48) ) − e ( tk ) = tk (cid:48) and the end time oftoken tk is exactly 1);(ii) each token tk with a ( V c ∩ V dec sec ) -value is eventually followed by a token tk (cid:48) with a V tc -value where t ∈ { beg , } such that s ( tk (cid:48) ) − s ( tk ) = e ( tk (cid:48) ) − e ( tk ) = tk with a ( V endc ∩ V dec sec ) -value is eventually followed by a token tk (cid:48) with a V endc -value suchthat s ( tk (cid:48) ) − s ( tk ) = Intv ( , ∞ ) as done before for expressing the equality requirement.By construction, the untimed part of a future plan of P = ( { x M } , R M ) is an initial and halting well-formed computation-code. Vice versa, by exploiting denseness of the temporal domain, the existenceof an initial and halting well-formed computation-code implies the existence of a future plan of P . Thisconcludes the proof of Proposition 8. References [1] R. Alur, T. Feder & T. A. Henzinger (1996):
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