Petri Net Relaxation for Infeasibility Explanation and Sequential Task Planning

P etri Net Relaxation for Infeasibilit y Explanation and Sequen tial T ask Planning Nhat Le 1 [0009 − 0007 − 1013 − 3711] , John G. Rogers 2 [0000 − 0002 − 6074 − 0823] , Claire N. Bonial 2 [0000 − 0002 − 3154 − 2852] , and Neil T. Dan tam 1 [0000 − 0002 − 0907 − 2241] 1 Colorado Sc ho ol of Mines, Golden CO 80401, USA {nguyencongnhat_le,ndantam}@mines.edu 2 DEV COM Army Researc h Lab oratory , A delphi, MD, USA {john.g.rogers59.civ,claire.n.bonial.civ}@army.mil Abstract. Plans often c hange due to c hanges in the situation or our understanding of the situation. Sometimes, a feasible plan ma y not ev en exist, and iden tifying such infeasibilities is useful to determine when requiremen ts need adjustment. Common planning approac hes focus on efficien t one-shot planning in feasible cases rather than updating domains or detecting infeasibilit y . W e prop ose a Petri net reachabilit y relaxation to enable robust inv ariant syn thesis, efficient goal-unreac hability detection, and helpful infeasibility explanations. W e further leverage incremen tal constrain t solvers to support goal and constrain t up dates. Empirically , compared to baselines, our system produces a comparable num b er of in v ariants, detects up to 2 × more infeasibilities, performs comp etitiv ely in one-shot planning, and outp erforms in sequen tial plan up dates in the tested domains. Keyw ords: T ask Planning · Algorithmic Completeness and Complexity 1 In tro duction The classic line “Kein Op erationsplan reich t mit einiger Sicherheit üb er das erste Zusammen treffen mit der feindlic hen Hauptmach t hinaus,” roughly translates to “no plan survives contact with the enemy” [ 58 ]. Across applications, plans must often c hange, b ecause the situation has changed, or b ecause a user did not initially understand or correctly sp ecify the situation. F urther, it is useful to identify cases when no plan exists, and why . Planning research has largely considered a given sp ecification or model, where our only w ant is to find a plan, ideally fast [ 2 , 17 , 78 ]. Dynamic navigation [ 50 , 75 , 84 ] and motion [ 18 , 57 ] planning are w ell-studied, yet adapting to analogous changes in task planning is less explored [ 71 ]. W e address this challenge of sequentially planning or explaining infeasibility and up dating task planning problems. W e r elax Petri net r e achability to identify invariants and explain infe asi- bilities in c ombinatorial and numeric task planning, c ouple d with incr emental c onstr aint solving for se quential planning to show impr ove d running times, im- pr ove d infe asibility dete ction, and new infe asibility explanation . T ransforming 2 Le, Rogers, Bonial, and Dantam planning problems to P etri nets (PNs) reveals useful problem structure. Relaxing PN reachabilit y to a linear program (LP) effectively iden tifies in v ariants and infeasible problems, and conflicting LP contrain ts offer useful explanations for planning infeasibilities. Incremen tal constraint solving efficiently incorporates certain forms of problem up dates. Compared to ev aluated baseline planners, we sho w comp etitiv e running times for some single-shot numeric domains, impro v ed running times for sequential planning on tested domains, improv ed infeasibilit y detection on tested domains, and new capabilities to explain infeasibility . 2 Related W ork T ask planning is well-established, largely evolving from the pioneering work on STRIPS [ 30 ]. Efficien t task planning approaches include heuristic search [ 37 , 41 , 69 ] and constraint-based metho ds [ 13 , 42 , 49 , 65 , 67 ]. Planning domains are often sp ecified using the Planning Domain Definition Language (PDDL) [ 25 , 36 , 56 ]. Prior approaches hav e also posed planning problems as integer programs (IPs) [ 72 , 80 ]. [ 81 , 82 ] prop ose an IP formulation for with an effective LP relaxation for plan-length estimation. W e develop a different LP formulation for in v ariant and infeasibilit y detection. Iden tifying infeasible planning problems has receiv ed less attention than planning for feasible cases. Some approaches fo cus on h uman-understandable explanations [ 73 , 79 ]. [ 28 ] pro duces unsolv ability certificates, fo cusing on mo del v erification. W e develop a p olynomial-time infeasibilit y test that is empirically robust and further iden tifies conflicting constraints. Prior work has addressed planning in dynamic and incomplete spaces. Building on heuristic searc h [ 35 ], incremental approaches [ 50 , 54 , 74 ] find optimal solutions in dynamic graphs under a fixed goal assumption, with extensions for changing targets [ 51 , 76 ]. W e tak e a constrain t-based approac h to leverage efficien t solution tec hniques to dynamically change goals and constraints [ 6 , 23 , 34 ]. 3 Problem Definition W e address se quential task planning whic h inv olv es finding sequences of actions from a start to a goal, or determining no such sequence of actions exists, for a sequence of related planning problems. Often, planning is viewed as a one-shot pro cess, yet we may need to iteratively revise planning problems—e.g., changing goals or constraints—due to c hanges in the en vironment or to effectiv ely capture user in tent. W e define one-shot and then sequential planning. A planning problem P = ( X , A , f , I , G , O ) consists of state space X , actions A , transition function f : X × A → X , initial condition I ∈ X , goals G ⊂ X , and an ob jective O [ 52 ]. A feasible plan is a sequence of actions from the initial state I to a goal state in G follo wing applications of transition function f , and an optimal plan is a feasible plan that also maximizes (minimizes) ob jective O . W e consider sequential planning as a process of solving a sequence of related planning problems. W e begin with a given initial planning problem P 0 . Then, we 4. BA CKGR OUND 3 are given up date sequence U 1 , . . . , U n . The planning problem at the i th round is the recursiv e comp osition of the i th up date U i and the previous problem P i − 1 , P i = ( P 0 , if i = 0 compose ( P i − 1 , U i ) , if i ≥ 1 . (1) See Sec. 5.8 for the sp ecific up dates and comp ositions in our approach. A t each round i , we m ust solve P i either by (1) finding a feasible (optimal) plan or (2) determining that no plan exists and explaining, ξ i ∈ Ξ , the infeasibility , plan  P i  ∈ A ∗ ∪ ( { UNSA T } × Ξ ) . (2) See Sec. 5.5 for the infeasibilit y explanations we pro duce. Sequen tial planning terminates when plan ( P n ) pro duces a desired result. 4 Bac kground Planning Domain Definition L anguage The Planning Domain Definition Lan- guage (PDDL) is a de-facto standard notion for describing planning problems [ 56 ]. PDDL descriptions use first-order logic to sp ecify the planning problem, though w e consider the grounded or prop ositionalized form ( V , I , A, G ) , where V is a set of state v ariables that may b e Bo olean- or real-v alued, I is an initial state, A is a set of actions, and G is a set of goal states. State s is a binding to each state v ariable. Actions a ∈ A include preconditions pr e and effects eff . Constr aint Solvers W e consider tw o types of constraint formulations and solvers: Satisfiabilit y Mo dulo Theories (SMT) and Mixed Integer Programming (MIP). SMT extends Bo olean satisfiability with rules (theories) for domains such as en umerated types and linear arithmetic [ 7 , 24 ]. Compared to traditional SA T solv ers, SMT solvers provide a higher level interface with useful features for expressing constrain ts [ 16 , 22 , 83 ]. MIP extends numerical programming (con- strained optimization) to include not only real-v alued but also integer or Boolean decision v ariables. Though seeming to come from opp osite directions, SMT and MIP offer some conv ergen t capabilities; how ev er, algorithmic differences in solv ers sometimes yield substantially differen t p erformance. SMT solvers [ 6 , 23 ] are often based on a backtrac king searc h core [ 55 , 59 ], while MIP solv ers [ 34 , 43 ] are of- ten organized around a branch-and-cut framework [ 62 ]. Performance differences suggest adv an tages to supp orting multiple solver types [ 20 , 68 ]. A useful feature of many SMT solv ers is incr emental constraint solving, enabling certain additions and deletions of constraints at run-time to pro duce alternativ e solutions. There are t wo mechanisms for incremental solving: an assertion stack [ 32 ] and p er-c heck assumptions [ 26 ]. An incremental SMT solver main tains constraints using a stack of scop es, where each scop e is a con tainer for a set of constraints. New constraints are added to the scop e on top of the stack, and solvers offer an interface to push and p op scop es (removing constrain ts in an y p opped scop e). Per-c heck assumptions are constraints that hold only for a 4 Le, Rogers, Bonial, and Dantam sp ecific satisfiabilit y chec k. Solvers may retain or discard learned lemmas (e.g., additional conflict clauses) differently after a chec k with assumptions or popping a scop e, resulting in p erformance differences b et ween these t wo mechanisms. Constr aint-b ase d Planning Constrain t-based planners enco de the planning domain as a logical or numeric formula and use a constraint solver, often a Bo olean satisfiabilit y solver, to find a satisfying assignmen t corresp onding to a plan [ 47 , 52 ]. Commonly , the decision v ariables represent the state and actions for a fixed n umber of steps h . The formula asserts that the start state holds at step 0 , goal condition holds at step h , and v alid transitions o ccur b et ween eac h step k and k + 1 . Planners increase step count h (creating additional decision v ariables and form ula clauses) until finding a satisfying assignment enco ding a plan. A crucial part of efficient constraint-based planners (e.g., [ 49 , 65 ]) is iden tifying invariants [ 1 ], which are facts or constraints that m ust hold for all reachable states at one or more steps. One common in v ariant form is mutual exclusion (mutex)— t wo Bo olean state v ariables that cannot sim ultaneously b e true. Inv ariants narrow the search space b y eliminating decision v ariables or enabling solv ers to more rapidly iden tify bindings, resulting in empirical runtime improv ement [ 33 ]. Petri Nets Petri nets (PNs ) are a representation for transition systems that offer useful features for concurrency and shared resources [ 61 ]. A PN is a weigh ted, directed bipartite graph comprised of plac es and tr ansitions . Classically , a PN place contains a non-negative integer-v alued num b er of tokens , though we gen- eralize to Bo olean, integer, and real places. A transition may fir e to alter the tok en count of adjacen t places by edge w eigh ts. A transition may only fire if the pr e c ondition plac es hav e enough tokens, known as the transition cost. A marking is the num ber of tokens in each place, i.e., a PN state. Finally , we also considers a set of goal markings for PNs, defined b y a range of tokens on each place. PNs offer a close relationship to task planning domains b ey ond typical PN applications to concurrent systems [ 39 ] and in particular capture the implicit concurrency that prior work has found to b e critical for effectiv e p erformance [ 49 , 65 ]. F urther, the PNs inform the linear relaxation of Sec. 5.3 . 5 Metho d 5.1 P etri Nets for Planning Domains W e construct a PN from a grounded (prop ositionalized) PDDL domain (see Sec. 4 ). PN places corresp ond to PDDL state v ariables V , PN transitions corre- sp ond to PDDL actions A , and arcs enco de preconditions and effects. A ctions inv olving Bo olean state v ariables presen t three cases (see Figure 1 ). When the v ariable occurs only in the precondition, we create incoming and outgoing arcs weigh ted 1 for p ositiv e or − 1 negative preconditions (see Figure 1a ). When the v ariable o ccurs only in the effect, we create an arc from the transition to the place weigh ted 1 for p ositiv e or − 1 negative effects (see Figure 1b ). When the v ariable o ccurs in b oth the precondition and effect, we create an arc from 5. METHOD 5 p τ w w (a) p τ w (b) p τ w (c) Fig. 1: Petri net structure for PDDL preconditions and effects. (a) v ariable in precondition (not in effect). (b) v ariable in effect (not in precondition). (c) v ariable in precondition and effect. the place to the transition weigh ted 1 to flip the v ariable from true to false or − 1 to flip the v ariable from false to true (see Figure 1c ). A ctions inv olving numeric state v ariables require similar arc construction and further inference of b ounds (b o x constrain ts) that are implicitly sp ecified in PDDL. When the numeric v ariable v app ears in the effect with a constant increase or decrease, we create an arc from the transition to the place weigh ted b y the amount of increase or decrease ( Figure 1b ). Next, we infer the b ounds. Numeric v ariables that nev er app ear in decrease (increase) effects hav e a low er b ound (upp er b ound) equal to their starting v alue. When every action i that decreases v b y x i has precondition v ≥ y i , w e infer a low er bound of v as the minim um of y i − x i . When every action i that increases v b y x i has precondition v ≤ y i , w e infer an upp er b ound of v as the maximum of y i + x i . Initial state I corresp onds to the PN’s initial marking, where token counts at places equal the v alues of corresp onding state v ariables in I . Goal G corresp onds to a goal marking. A plan is a sequence of transitions that propagates the PN from the initial marking to a goal marking. This transformation defines the PN structure used throughout the pap er and underlies empirical results in Sec. 6 . 5.2 P etri Net Constrain ts Next, we construct sym b olic constraints for the PN dynamics. While these constrain ts are similar to existing constrain t-based planning metho ds (see Sec. 4 ), the PN form ulation supp orts the relaxation in Sec. 5.3 , offering further information ab out reac hability and inv ariants. F ollowing typical constraint-based metho ds, the decision v ariables represent place token coun ts (planning state v ariables) and transitions that fire (planning actions) o ver steps 0 to h , and the constraint form ula enco des the PN’s transition function along with start and goal conditions. Eac h numeric or integer place yields a linear constraint for token flow, p ⟨ k +1 ⟩ = p ⟨ k ⟩ + X i ∈ in w i τ i ⟨ k ⟩ − X j ∈ out w j τ j ⟨ k ⟩ , (3) where p ⟨ k ⟩ , p ⟨ k +1 ⟩ are the token counts of the place at steps k , k + 1 , τ i are the incoming transitions with arc weigh ts w i , and τ j are the outgoing transitions with arc weigh ts w j . Com bining the constraints ( 3 ) for eac h place pro duces a system of linear equations called the marking e quation [ 61 ], p ⟨ k +1 ⟩ = p ⟨ k ⟩ + C τ ⟨ k ⟩ , (4) 6 Le, Rogers, Bonial, and Dantam where p ⟨ k ⟩ , p ⟨ k +1 ⟩ are v ectors of tok en counts (markings) at steps k , k + 1 , τ ⟨ k ⟩ is a vector indicating transitions that fire, and C is the incidenc e matrix in which eac h entry C ij is the c hange in token count at place p i when transition τ j fires. Constrain ts for Bo olean places take a different form b ecause we consider as v alid effects rebinding a true Bo olean place to true, or false to false. F or arc w eights of 1, constraints for a Bo olean place take the following form, ∀ i, τ in ,i ⟨ k ⟩ = ⇒ p ⟨ k +1 ⟩ ∀ j, τ out ,j ⟨ k ⟩ = ⇒ p ⟨ k ⟩ ∧ ¬ p ⟨ k +1 ⟩ ∀ ℓ, τ inout ,ℓ ⟨ k ⟩ = ⇒ p ⟨ k ⟩ ∧ p ⟨ k +1 ⟩ , (5) where τ in , τ out , τ inout are the adjacent transitions with incoming, outgoing, and b oth incoming and outgoing arcs to p . Negative arc weigh t reverse the p olarities of p in the implication’s conclusion. W e exclude conflicting transitions from simultaneously firing. Conflicts may o ccur if one transition mo difies a place in a wa y that could negate a precondition of another transition. This m utex constraint is, τ group , 1 + . . . + τ group ,m ≤ 1 . (6) W e note that ( 6 ) is different from typical mutex constraints of pairwise disjunc- tions ¬ τ i ∨ ¬ τ j , e.g., used by [ 48 , 67 ]. Pairwise disjunctions pro duce quadratically- sized constraints, whereas the summation ( 6 ) is linear in the num b er of v ariables. While classic SA T solv ers often required the quadratically-sized disjunctions, some mo dern SMT solvers supp ort pseudo-Bo olean constraints [ 23 ] to directly , compactly , and efficiently represen t ( 6 ). W e also add b ounds (b o x constraints) when kno wn for in teger and real places, min( p ) ≤ p ≤ max( p ) . (7) Finally , we assert the start state at step 0 and the goal at final step h . The constraints describ ed in this section are adequate to correctly sp ecify the transitions of a PN. How ev er, the p erformance of current solvers [ 23 , 34 ] on these constrain ts scales p oorly and is not practical b ey ond small problems. W e address this p erformance concern and further offer new capabilities for reac hability iden tification through the linear relaxation in the next section. 5.3 Relaxed Petri Net Reac hability W e describ e a linear relaxation of PN reac hability that offers an effectiv e analysis to ol for planning problems. PN reachabilit y is well-studied and computationally hard [ 11 , 15 ]—an unsurprising observ ation at this p oin t given the reduction of task planning (also computationally hard [ 14 ]) to PN reachabilit y . W e extend linear relaxations of PN reachabilit y to address Bo olean state v ariables and show the effectiv e identification of planning inv ariants and infeasibilities. The idea of the relaxation is to sum flows ov er all steps, considering only initial marking, final marking, and ho w many times each transition fired. Intuitiv ely , 5. METHOD 7 this relaxation captures a conserv ation requirement: tokens are not created ex nihilo , but rather mo ved b et ween places as transitions fire. The relaxation do es not, how ev er, fully capture order requirements and th us may optimistically (and incorrectly) determine a certain state is reachable when sums of flows could indicate so, but when no v alid ordering of the indicated transitions exists, i.e., the relaxed solution requires some places to b e negative or out-of-b ounds at in termediate steps. W e now derive the relaxed constraint equations. Consider the marking equation ( 4 ) applied from steps 0 to h , p ⟨ h ⟩ = p ⟨ 0 ⟩ + C τ ⟨ 0 ⟩ + . . . + C τ ⟨ h − 1 ⟩ = p ⟨ 0 ⟩ + C h − 1 X k =0 τ ⟨ k ⟩ (8) A c hange of v ariables ˜ τ = P h − 1 k =0 τ ⟨ k ⟩ yields the system of equations, p ⟨ h ⟩ = p ⟨ 0 ⟩ + C ˜ τ . (9) Our in terpretation of Bo olean places requires an additional consideration. Un- der ( 5 ) , transitions ma y fire that bind a Bo olean place to true (false), even if that place is already true (false). If we would accumulate those firings as in equation ( 9 ) and some other transition uses that place in a negativ e (p ositiv e) precondition, then the relaxation could miss a reachable state. Effectively , rebindings turn ( 9 ) in to an inequality , whic h we address with non-negativ e slack v ariables. F or each place p i , w e create a non-negative slack v ariable s + i that incremen ts the relaxed coun t and a non-negative sl a ck v ariable s − i that decrements the relaxed count. If p i is Boolean and no transition can rebind p i to false, s + i = 0 . If p i is Boolean and no transition can rebind p i to true, s − i = 0 . If p i is real or in teger, s + i = s − i = 0 . T ogether, the slack v ariables form vectors s + ∈ R m + and s − ∈ R m + , pro ducing, p ⟨ h ⟩ = p ⟨ 0 ⟩ + C ˜ τ + s + − s − . (10) Relaxing in tegers and Bo oleans to reals in ( 10 ) pro duces a linear system. Prop osition 1. A marking p ⟨ h ⟩ r esulting in infe asible c onstr aints for the r elaxe d system dynamics ( 10 ) implies that p ⟨ h ⟩ is unr e achable at any step under the original system dynamics ( 4 ) and ( 5 ) . Pr o of. W e prov e b y contradiction. Assume p ⟨ h ⟩ mak es relaxation ( 10 ) infeasible but is reachable under the original dynamics ( 4 ) and ( 5 ) . Reachabilit y under ( 4 ) and ( 5 ) implies that there exists a sequence of transition firings to progressively apply ( 4 ) and ( 5 ) to pro duce p ⟨ h ⟩ . Summing those transition firings and selecting v alues for s + , s − satisfies ( 10 ). So ( 10 ) m ust b e feasible. Contradiction. Prop osition 2. The time c omplexity to che ck fe asibility of ( 10 ) is p olynomial in the numb er of plac es and tr ansitions in the PN. Pr o of. Relaxation ( 10 ) consists of linear equations, non-negativit y constrain ts on ˜ τ , s + , s − , and bindings or b o x constrain ts on p ⟨ h ⟩ , forming a linear program with decision v ariables ( ˜ τ , s + , s − , p ⟨ h ⟩ ) and constraints ( C ) polynomial in the n umber of places and transitions. Linear programming is p olynomial time [ 45 ]. W e next apply ( 10 ) to inv ariants in Sec. 5.4 and infeasibilities in Sec. 5.5 . 8 Le, Rogers, Bonial, and Dantam 5.4 In v arian ts from Linear Relaxations W e apply the PN reachabilit y relaxation ( 10 ) to find in v ariants for planning domains. In v ariants are imp ortan t for efficiency of constraint-based planning [ 49 , 65 ]. W e fo cus particularly on mutex inv ariants. T wo places are mutex when they cannot both b e simultaneously true. F or places p u and p v and ( 10 ), m utex exists when,  p u ⟨ h ⟩ ∧ p v ⟨ h ⟩  | =  p ⟨ h ⟩  = p ⟨ 0 ⟩ + C ˜ τ + s + − s −  . (11) Chec king ( 11 ) simplifies to a linear program containing only tw o rows of ( 10 ),  p u ⟨ h ⟩ p v ⟨ h ⟩  =  p u ⟨ 0 ⟩ p v ⟨ 0 ⟩  +  C u, : C v , :  ˜ τ +  s + u s + v  −  s − u s − v  . (12) By Prop osition 1 , if ( 12 ) is infeasible, there is no reachable state where b oth p u and p v are true, so p u and p v m ust b e mutex. W e note that ( 12 ) offers tw o p ossible sources of parallelism. First, individual in v ariant chec ks ma y b enefit from SIMD parallelism on CPUs or GPUs [ 4 ]. Second, chec king inv arian ts across differen t pairs of places is embarrassingly parallel, e.g. different chec ks may run in different threads using all av ailable cores. Next, we use the collected pairwise mutexes ( ¬ p u ∨ ¬ p v ) to construct more efficien tly-solv able mutex groups. Pairwise m utexes can yield a quadratic n umber of constraints that are exp ensiv e to solve [ 66 ]. Instead, we reduce the num b er of constrain ts by constructing mutex groups of the form, X i ∈ group p i ≤ 1 , (13) whic h are directly handled b y SMT solv ers with pseudo-Bo olean supp ort [ 23 ] and MIP solvers. Constructing maximal mutex groups is PSP ACE-Complete [ 31 ], so w e take a greedy approach that progressiv ely grows a mutex group when we can add another v ariable that is mutually exclusive with all current group members. F rom a mutex group, we can in some cases iden tify a more precise and efficien tly solv able one-hot inv arian t when some place in the group is initially true and every transition that disables a place in the group enables another place in the group. The resulting one-hot in v ariant is, X i ∈ group p i = 1 . (14) Exp erimen ts in Sec. 6.1 show that relaxation ( 10 ) generates similar inv ariant coun ts as baselines for tested domains while further supp orting n umeric domains. 5.5 Infeasibilit y Explanations from Linear Relaxations W e apply the PN reachabilit y relaxation ( 10 ) to identify and explain planning infeasibilities. Planning is infeasible when there is no plan from the start to the 5. METHOD 9 goal. W e consider a planning infeasibility explanation to b e a minimal set of clauses causing the infeasibilit y , i.e., leading to unsatisfiable constraints. W e chec k planning feasibility using relaxation ( 10 ) under the goal G , G | =  p ⟨ h ⟩  = p ⟨ 0 ⟩ + C ˜ τ + s + − s −  . (15) By Prop osition 1 , infeasibility in ( 15 ) means the planning problem is infeasible. Infeasibilit y explanations relate to unsatisfiable cores [ 53 ] and Irreducible Inconsisten t Subsystems (I ISs) [ 19 ], which b oth identify minimal, conflicting sets of constrain ts. Identifying all such inconsistencies is computationally hard [ 3 ]. W e fo cus on finding inconsistent sets of goal conditions. Goal conditions offer p oten tially useful feedbac k since they directly relate to an originally sp ecified planning problem, while other constrain ts arise from an inv olved sequence of transformations. Sp ecifically , an explanation E ⊆ G is a subset of goal conditions causing ( 10 ) to b e infeasible, but where ( 10 ) is feasible for any strict subset of E . W e wan t to find all such subsets E . The difficulty of finding all inconsistent goal sets dep ends on the num b er of goal conditions. F or few goal conditions, we may enumerate com binations in increasing cardinality , terminating a branch on infeasibilit y , and collecting all infeasible combinations. F or many goal conditions, equation ( 10 ) extends to an in teger program [ 3 ] that minimizes the num b er of goal conditions to disable, min ℓ X i =1 y i s.t. p ⟨ h ⟩ = p ⟨ 0 ⟩ + C ˜ τ + s + − s − ¬ y i = ⇒ G i , ∀ i ∈ 1 , . . . , ℓ , (16) where Bo olean decision v ariables y i disable corresp onding goal conditions G i . Finding alternative solutions to ( 16 ) b y asserting previously disabled conditions are not join tly disabled yields sets of inconsistent goal conditions. Exp erimen ts in Sec. 6.2 show that relaxation ( 10 ) outp erforms all baselines at infeasibilit y detection for tested domains and further explains infeasibilities. 5.6 F orw ard and Backw ard Reachable Sets Next, w e compute p er-step reachable sets forward from the start and backw ard from the goal. These sets are optimistic approximations: states outside the set are unreac hable, while states in the set are possibly (but not necessarily) reac hable. W e represent a reac hable set using bindings for a subset of state v ariables, so the set con tains states not conflicting with the bindings. Reachable sets help identify constan ts, eliminate decision v ariables, and b ound the minimum step horizon h . W e describ e reachable set propagation for forw ard reachabilit y . Backw ard reac hability is similar but swaps use of preconditions with effects and start with goal states. W e propagate known bindings from one step to the next according to which transitions can or cannot fire. Conceptually , this approach is similar to 10 Le, Rogers, Bonial, and Dantam planning graphs and other unfolding techniques [ 12 , 39 ]. How ev er, we propagate sets via p artial evaluation support complex expressions and numeric state. Set propagation uses functions partial-eval and pa rtial-sat . F unction pa rtial-eval tak es an SMT-LIB [ 7 ] expression and v ariable bindings to pro duce an equiv alen t (partially ev aluated) expression b y substituting bindings and algebraically simpli- fying. F unction partial-sat is an incomplete pro cedure for Bo olean and arithmetic satisfiabilit y that uses unit propagation and pure literal elimination [ 55 , 59 ] for satisfiabilit y tests that are p olynomial-time [ 46 ] but may return UNKNOWN . The first step of reachable set propagation identifies which transitions cannot fire. F or transition τ j , if its precondition e under curren tly known bindings B ⟨ k ⟩ is UNSA T , the transition cannot fire, and we add ¬ τ j ⟨ k ⟩ to B ⟨ k ⟩ ,  UNSA T = pa rtial-sat  pa rtial-eval ( e, B ⟨ k ⟩ )  = ⇒ ¬ τ j ⟨ k ⟩ . (17) The second step of propagation tests whether b ound state v ariables can change at the next step. F or b ound place p i , if its up date e from ( 3 ) or ( 5 ) and distinct next v alue under bindings B ⟨ k ⟩ are UNSA T , we propagate the binding,  UNSA T = pa rtial-sat  pa rtial-eval  e ∧  p i ⟨ k ⟩  = p i ⟨ k +1 ⟩  , B ⟨ k ⟩  = ⇒  p i ⟨ k ⟩ = p i ⟨ k +1 ⟩  . (18) The result of propagating reachable sets to a fixp oin t is a mapping of per- step v ariable bindings forward from the start and backw ard from the goal. Any bindings present in the forward reac hability fixp oin t represent constan ts. W e use these bindings to eliminate the decision v ariable for any known binding in the constraint form ulation. A low er bound h on steps in feasible plans is the maximum of the forw ard reac hable step consistent with the start and the bac kward reachable step consistent with the goal. 5.7 Solutions via SMT and MILP Solvers W e solve constraints to find plans using SMT [ 23 ] or MILP [ 34 ] solvers. Differen t solv er algorithms or implementations perform better for different problems, suggesting adv antages for a p ortfolio of solv ers [ 20 , 68 ]. SMT solv ers directly supp ort the high-level constraint expressions from the previous subsections [ 7 ], and we leverage incremental solving to progressiv ely extend the step horizon. MILP solv ers op erate on separately-sp ecified line ar constrain ts, so we translate the SMT-LIB expressions to conjunctions of linear constrain ts. W e first describ e extending the step horizon and then translating SMT-LIB to MILP constraints. Incremen tally Solving Horizon Extension Generally , constrain t-based plan- ners chec k satisfiabilit y for a b ounded num ber of steps h and increase h un til finding a v alid plan. Early formulations c onsidered each b ound h separately [ 47 ]. Madagascar iden tified commonalities b et ween different b ounds h [ 67 ]. Prior use 5. METHOD 11 of SMT solvers for planning used the assertion stack [ 21 , 22 ]. Ho wev er, p opping an assertion scop e may cause solvers to discard prior work [ 10 ]. W e prop ose instead to use p er-c hec k assumptions for more effective incremental solving. Horizon extension inv olves (1) declaring new decision v ariables and constan ts 3 , (2) asserting the transition function at that step, and (3) c hecking satisfiabilit y of the goal at the next step. Extending the horizon to k , w e declare decisions v ariables for each place p i ⟨ k ⟩ and transition p i ⟨ k ⟩ at that step. If forward reachabilit y (see Sec. 5.6 ) indicated a known binding, then we declare instead constan ts (i.e., true , false , or a num b er). Next, we assert that the transition function (see Sec. 5.2 ) holds from steps k − 1 to k and that inv ariants (see Sec. 5.4 ) hold at step k . Finally , w e chec k satisfiabilit y with assumptions ( c heck-sat-assuming ) that the goals holds at step k and known bindings from backw ards reachabilit y (see Sec. 5.6 ) hold at steps prior to k . W e rep eatedly increase the horizon un til goal is satisfied. T ransforming SMT-LIB to MILP Constrain ts W e translate SMT-LIB to MILP constraints using a v ariation of the T seitin [ 77 ] or Plaisted-Greenbaum transformation [ 63 ] transformation to construct conjunctive normal form (CNF) expressions, and we further eliminates some decision v ariables and apply indicator constrain ts for disjunctions of linear relations [ 9 ]. The T seitin and Plaisted-Greenbaum transformations construct CNF by recur- siv ely introducing auxiliary v ariables for sub expressions. T seitin in tro duces auxil- iary v ariables γ that equal sub expressions op ( ℓ 1 , . . ., ℓ n ) o ver literals ℓ 1 , . . . , ℓ n . Plaisted and Greenbaum observed that satisfiability requires only one direction of the equalit y biconditional, which from negation normal form, b ecomes, op ( ℓ 1 , . . ., ℓ n ) ⇝ γ = ⇒ op ( ℓ 1 , . . ., ℓ n ) . (19) Equation ( 19 ) reduces directly to conjunctions of disjunctions. W e further eliminate auxiliary v ariables in some cases. Conjunctions inv olving auxiliary v ariables γ i require all to hold, so we eliminate the γ i ’s by rewriting their implications using γ ′ ,  ^ γ i  ∧  ^ ℓ j  ⇝  γ ′ = ⇒ ^ ℓ j  , ( γ i = ⇒ ϕ ⇝ γ ′ = ⇒ ϕ ) . (20) Disjunctions in v olving a single auxiliary v ariable γ require either γ or some literal ℓ j to hold, so w e eliminate γ by replacing implications of γ with γ ′ , γ ∨ _ ℓ j ⇝  γ = ⇒ ϕ ⇝ γ ′ = ⇒ ϕ ∨ _ ℓ j  . (21) Reducing the resulting CNF to linear constrain ts is well-established [ 46 ]. Supp ort for logical expressions o ver numeric relations ( a T x ≤ b ) requires additional consideration. Generally , such numeric relations p ose dis junctiv e pro- grams [ 5 ]. W e incorp orate n umeric relations into the CNF transformation by 3 The SMT-LIB standard uses the term “constant” [ 7 ] for what optimization literature calls a “decision v ariable.” 12 Le, Rogers, Bonial, and Dantam creating auxiliary v ariables that imply the relation, γ = ⇒ a T x ≤ b . (22) Some solvers supp ort implication ( 22 ) directly as an indicator constraint [ 34 ]. F or solv ers lacking indicator constraints [ 43 ], we apply the Big M metho d [ 8 ] coupled with in terv al arithmetic [ 44 ] to find tight M v alues, a T x + M γ ≤ b + M , M = s ( a T x − b ) , (23) where a T x is the upp er b ound of a T x computed via interv al arithmetic based on constan t co efficien ts a and b o x constraints for decision v ariables x , and s > 1 is a scaling factor to tolerate floating p oin t error (our implementation uses s = 2 ). 5.8 Up dates and Incremen tal Sequential Planning W e consider tw o forms of up dates for sequential planning problems: changing goals and adding constraints. Changing goals may b e useful when an initial problem is infeasible, yet a subset of conflicting goals (see Sec. 5.5 ) is still desirable. Adding constrain ts may b e useful when a candidate plan is undesirable due to an initially unsp ecified requirement. While more general up dates are p ossible—e.g., changes to state v ariables, preconditions, or effects—these restricted up date forms are efficien t to implement via incremental constraint solving. Changing Go als A change of goals op erates on up date U con taining a list of added clauses g + i and deleted clauses g − j , U = ( g + 1 , . . ., g + m ) , ( g − 1 , . . ., g − n ) . (24) Goal changes modify the PN’s goal marking, but the PN is otherwise unchanged. The relaxed marking equation ( Sec. 5.3 ) and inv ariants ( Sec. 5.4 ) are unmo di- fied. W e rechec k feasibility ( Sec. 5.5 ). F orw ard reachabilit y is unmo dified, and w e recompute backw ard reachabilit y ( Sec. 5.6 ). Finally , the updated goal and bac kward reachabilit y bindings b ecome the p er-c heck assumptions ( Sec. 5.7 ). A dding Constr aints An addition of constraints op erates on up date U con taining a list of logical expressions or linear relations e i that m ust hold across all steps, U = ( e 1 , . . ., e n ) . (25) This up date creates additional constraints for the marking equation ( Sec. 5.2 ), e 1 ⟨ k +1 ⟩ ∧ . . . ∧ e n ⟨ k +1 ⟩ , and corresp ondingly for the linear relaxation ( Sec. 5.3 ), e 1 ⟨ h ⟩ ∧ . . . ∧ e n ⟨ h ⟩ . W e must recompute the inv ariants ( Sec. 5.4 ), goal reachabilit y ( Sec. 5.5 ), and reachable sets ( Sec. 5.6 ) under these new constraints, noting that these prepro cessing steps are all p olynomial time. Each additional constraint e i along with new inv ariants and known bindings from forward reachabilit y b ecome new assertions for the constrain t solv er, noting that these new assertions are t ypically efficiently handled [ 10 , 23 , 26 ]. Finally , an y new known bindings from bac kward reachabilit y b ecame new assertions for satisfiability chec ks. Exp erimen ts in Sec. 6.3 show that the incremental constraint approach out- p erforms all baselines for sequential planning in the tested domains. 6. EXPERIMENTS AND DISCUSSION 13 6 Exp erimen ts and Discussion W e ev aluate our approach for (1) inv ariant generation, (2) infeasibility detection, and (3) sequential planning, and we compare to the follo wing efficien t and state-of-the-art baselines: constraint-based Madagascar (Mp) [ 67 ], heuristic F ast Do wnw ard (FD) [ 37 ], numeric Metric-FF (FF) [ 40 ] and ENHSP [ 69 ]. Our results sho w that our approac h found similar num b ers of inv ariants as Mp and FD in classical domains and outp erformed baselines on infeasibilit y detection and sequen tial planning for the tested domains. Our implementation extends TMKit [ 21 ]. W e solve constrain ts for the re- laxation ( 10 ) and planning ( Sec. 5.7 ) using Z3 [ 23 ] and Gurobi [ 34 ]. The tested domains come from past International Planning Comp etitions (IPCs) [ 2 , 60 ], are auto-generated by [ 70 ], or randomly generated b y ourselves. W e ran our b enc hmarks using one core of an AMD Ryzen 9 7950X3D CPU under Debian 12. 6.1 In v arian t Generation 1X 3X 5X 7X 9X Problem Size 0 1 2 3 4 5 Number of Inva riants × 10 3 TMKit Mp FD Hanoi TSP Logistics Fig. 2: T wo-literal in v arian ts from our PN relaxation and baselines. All metho ds pro duce similar n umbers of inv ariants. W e ev aluate inv arian t generation from Sec. 5.4 for increasing problem size in three com bi- natorial problems. W e compare against Mp and FD b ecause the other baselines do not pro duce inv ariants. W e only ev aluate classical (not numeric) domains, b ecause Mp and FD only supp ort classical domains. Figure 2 plots in v ariant counts. Our PN relaxation pro duces similar n um b ers of inv ari- an ts as Mp and FD in the tested domains, and Prop osition 1 guarantees in v ariant v alidity . Mp yields slightly more inv ariants than ours and FD, while FD produces the same num b er of in v ariants as ours. Eac h ev aluated planner generates inv ari- an ts differently , and the additional inv ariants from Mp can b e explained b y its unique in- v arian t syn thesis. Our metho d chec ks inconsistency of relaxed PN flow to iden tify pairwise inv ariants. FD c hecks the monotonicity of eac h candidate across ac- tions and iteratively strengthens a non-monotonic inv ariant by adding axioms to main tain its monotonicity [ 38 ]. Mp chec ks a candidate inv arian t through op erator regressions while maintaining the consistency of the regressions with other candidate inv ariants [ 64 ], and a similar iterative approach [ 64 ] generates more classes of inv ariants than FD [ 38 ]. Our method is similar to FD in that we c heck candidate inv ariants indep enden tly . 6.2 Infeasibilit y Detection W e ev aluate infeasibilit y detection from Sec. 5.5 on classical problems from Unsolv abilit y IPC 2016 [ 60 ], and we auto-generate additional unsolv able numeric 14 Le, Rogers, Bonial, and Dantam 1 3 5 7 9 0 2 Time (s) Bottleneck (1) 1 3 5 7 9 0 500 Chessb oa rd (2) 1 3 5 7 9 0 250 P egsol Ro w5 (3) 1 3 5 7 9 # Infeasibilites 0 250 Time (s) Delivery (4) 1 3 5 7 9 # Infeasibilites 0 10 Counter (5) (1) (2) (3) (4) (5) All TMKit Gurobi 10 10 10 8 10 48 TMKit Z3 10 10 8 8 10 46 FF 10 4 4 2 0 20 FF BFS 10 4 4 2 0 20 FF BFS+H 10 10 9 3 0 32 ENHSP 10 5 5 9 5 34 FD 10 5 5 - - 20 Mp 0 0 0 - - 0 Detection Count T able TMKit Gurobi TMKit Z3 FF FF BFS FF BFS+H ENHSP SA T FD Mp Fig. 3: A v erage infeasibilit y detection times ov er 10 iterations and counts. Absen t data p oin ts exceeded a 10 -minute timeout or 8 GiB memory limit. Dashes in the table indicates unsupp orted numeric domains for Mp and FD. problems (Delivery and Coun ter). Each tested domain has 10 problems of increas- ing size. Planners hav e a 10 minute timeout 8 GiB memory limit. Our reachabilit y relaxation identifies up to twice as many infeas ibilities as the baselines in the tested domains, finding 48 infeasibilities out of 50 cases (see Figure 3 ). Our relaxation is particularly robust at infeasibility detection in n umeric domains. FF’s p erformance is comparable to ours in combinatorial problems but w orse on numeric problems. ENHSP exhibits an av erage p erformance on b oth trac ks, although it finds one more infeasibilit y than ours in the Delivery domain. Differen t procedures contribute to the v arying detection capabilities. Heuristic planners use a heuristic function (relaxation) to detect dead-ends that cannot reac h the goal, though missed dead-ends may result in an exp ensiv e, exhaustive searc h. On the other hand, our LP relaxation enables polynomial time feasibilit y tests, and Figure 3 indicates that LP feasibility closely approximates planning feasibilit y . F rom Prop osition 1 , feasible domains never pro duce infeasible LPs. Bey ond infeasibility detection, our approach also explains infeasibility in direct connection to original planning domains. Our explanations are minimal sets of m utually unreachable goals (see Sec. 5.5 ), which conv ey helpful information to resolv e conflicts in subsequent plan up dates. Heuristic approac hes rep ort differen t infeasibility information in the form of unsolv abilit y certificates [ 28 ], i.e. dead-ends, whic h may offer some details on how to resolve infeasibilities. 6.3 Sequen tial Planning W e ev aluate sequen tial planning from Sec. 5.8 b y first considering time for traditional one-shot planning follow ed by cumulativ e time for sequen tial up dates 6. EXPERIMENTS AND DISCUSSION 15 1 3 5 7 9 0 25 Time (s) Blo ck Grouping 1 3 5 7 9 0 100 Counters 1 3 5 7 9 0 200 Delivery 1 3 5 7 # Solved Instance Count 0 5 Time (s) Rover 1 3 5 7 9 # Solved Instance Count 0 50 Sailing 1 3 5 7 # Solved Instance Count 0 250 Suga r TMKit Gurobi FF ENHSP SA T Fig. 4: A v erage one-shot planning times ov er 10 iterations. The figures sho w the n umbers of instances solved ov er time in numeric domains. Excluded data p oin ts exceeded the 10 -min ute timeout or 8 GiB memory limit. 0 10 20 30 Goal Up dates 0 10 CumulativeTime (s) Blo ck Grouping 0 10 20 30 Constraint Up dates 0 200 Blo ck Grouping 0 10 20 30 Goal Up dates 0 1000 Counters 0 10 20 30 Constraint Up dates 0 100 200 Counters Sequence 1 Sequence 2 Sequence 3 TMKit Gurobi FF ENHSP SA T Fig. 5: A v erage cumulativ e planning times ov er 10 iterations for sequential plan- ning. TMKit Gurobi outp erforms the baselines in most sequences. Excluded data p oin ts exceeded the 30 -minute timeout or 8 GiB memory limit. and planning. F or one-shot planning, we ev aluate six Simple Numeric Planning (SNP) domains from IPC 2023 [ 2 ]. Eac h domain consists of 10 difficult y-v arying problems. F or sequen tial planning, w e b enc hmark the planners on randomly generated up date sequences in the Blo c k Grouping and Counters domains. The sequences increase in the problem size. W e consider tw o forms of plan up dates: goal changes and constraint additions. Eac h sequence consists of 30 updates. W e sho w results for TMKit Gurobi, FF, and ENHSP SA T, which were most efficient v ariations of their resp ectiv e planners on these tests. Relativ e one-shot planner p erformance is domain dep enden t (see Figure 4 ). Our approac h outperforms FF in the Blo c k Grouping, Coun ters, Sailing, and Sugar domains and outp erforms ENHSP SA T in the Sugar domain. Inv arian t information and efficien t numeric constraint solving lead to our system’s one-shot 16 Le, Rogers, Bonial, and Dantam p erformance. Though one-shot p erformance was not the main focus of our work, w e do see efficient solutions for some tested problems. W e outp erforms the baselines in sequential planning in the tested domains (see Figure 5 ). TMKit Gurobi takes more time for some initial problems, but cum ulative times o ver sequential up dates are less than the baselines. In addition, while our constraint-based framework supports additional constrain ts, some baselines including Mp, FD, and FF could not handle these updates due to lac k of supp ort for PDDL global constraints. Our outp erformance for sequential planning comes from incremental solving and w arm starts. The baselines only support one-shot planning and thus replan p er up date. In contrast, we retain the constraints and reuse the previous search information to guide the next search. Gurobi as the underlying constraint solver applies the previous solution as a w arm start for the next search, resulting in efficien t up dates when changing goals or adding constraints [ 34 ]. 6.4 Limitations Our PN relaxation dep ends on chec king LP infeasibility , and floating p oin t error can lead to spurious infeasibilities [ 29 , 34 ]. Rational arithmetic (typical of SMT solv ers [ 23 ]) eliminates floating p oin t error at additional computational cost, though do es not address domains requiring irrational co efficien ts—e.g., π , trigonometric functions. The incremental constraint-based approach ma y present limitations in cer- tain c ases. Constrain t-based planning faces p erformance challenges under large n umbers of grounded state v ariables or long horizons [ 27 ], and which our current w ork do es not address. Con versely , on easy problems, constrain t solving ma y imp ose uncomp ensated ov erhead; Figure 5 shows FF outperforms our system on the smallest Counters problem. Incremental solving and warm starts help only after solving the first problem in a sequence, and heuristic approaches are often highly-effectiv e at one-shot planning. While our approach p erformed effectively in the exp erimen tal tests, there are cases it do es not address. 7 Conclusion W e presen ted an approach for infeasibility explanation and sequential planning that com bines a relaxation of Petri Net (PN) reachabilit y and incremental con- strain t solving. The PN reachabilit y relaxation is a linear program (LP); feasibility of this LP is necessary for planning feasibility , offering p olynomial time inv ariant and plan infeasibility tests. F urther, the LP offers infeasibility explanations in terms of conflicting constrain ts. 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