Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Proposition Finite branching is undecidable for post-effective WSTS. 5 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Proposition Finite branching is undecidable for post-effective WSTS. Proof Let S i = ( N , − → S i , ≤ ) be the WSTS such that: x − → S i x + 1 if TM i does not halt within ≤ x steps, x − → S i 0 , 1 , 2 , . . . otherwise. 5 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Proposition Finite branching is undecidable for post-effective WSTS. Proof Let S i = ( N , − → S i , ≤ ) be the WSTS such that: x − → S i x + 1 if TM i does not halt within ≤ x steps, x − → S i 0 , 1 , 2 , . . . otherwise. S i is post-effective (the cardinal of Post S i ( x ) is computable). S i has strong and strict monotony since x − → S i x + 1 for every x ∈ N . 5 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Proposition Finite branching is undecidable for post-effective WSTS. Proof Let S i = ( N , − → S i , ≤ ) be the WSTS such that: x − → S i x + 1 if TM i does not halt within ≤ x steps, x − → S i 0 , 1 , 2 , . . . otherwise. S i is post-effective (the cardinal of Post S i ( x ) is computable). S i has strong and strict monotony since x − → S i x + 1 for every x ∈ N . TM i halts iff there exist x ∈ N and an execution 0 ∗ − → S i x such that Post S i ( x ) is infinite. The halting problem thus Turing-reduces to the infinite branching problem. 5 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., 6 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination, 6 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination, Coverability, 6 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Objective We want to study the usual reachability problems for these infinitely branching systems, e.g., Termination, Coverability, Boundedness. 6 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . ∃ x 0 − → x 1 − → x 2 − → . . . ? Question: 7 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . ∃ x 0 − → x 1 − → x 2 − → . . . ? Question: Theorem (Finkel ICALP’87) Termination is decidable for finitely branching WSTS with transitive monotony. 7 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . ∃ x 0 − → x 1 − → x 2 − → . . . ? Question: Theorem (deduced from Dufourd, Jančar & Schnoebelen ICALP’99) Termination is undecidable for infinitely branching WSTS. 7 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Strong termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . Question: ∃ k bounding length of executions from x 0 ? 8 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Strong termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . Question: ∃ k bounding length of executions from x 0 ? Remark Strong termination and termination are the same in finitely branching WSTS. 8 / 24
Introduction Overview WSTS completion WSTS Applications Reachability problems Conclusion Strong termination Input : ( X , − → , ≤ ) a WSTS, x 0 ∈ X . Question: ∃ k bounding length of executions from x 0 ? Theorem Strong termination is decidable for infinitely branching WSTS under some assumptions. 8 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Issues with finite branching techniques Some techniques for WSTS based on finite reachability trees; impossible for infinite branching. Some rely on upward closed sets; what about downward closed, in particular with infinite branching? 9 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Issues with finite branching techniques Some techniques for WSTS based on finite reachability trees; impossible for infinite branching. Some rely on upward closed sets; what about downward closed, in particular with infinite branching? A tool Develop from the WSTS completion introduced by Finkel & Goubault-Larrecq in STACS’09 and ICALP’09. 9 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Ideals I ⊆ X is an ideal if downward closed: I = ↓ I , 10 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Ideals I ⊆ X is an ideal if downward closed: I = ↓ I , directed: a , b ∈ I = ⇒ ∃ c ∈ I s.t. a ≤ c and b ≤ c . 10 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Ideals I ⊆ X is an ideal if downward closed: I = ↓ I , directed: a , b ∈ I = ⇒ ∃ c ∈ I s.t. a ≤ c and b ≤ c . 10 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Ideals I ⊆ X is an ideal if downward closed: I = ↓ I , directed: a , b ∈ I = ⇒ ∃ c ∈ I s.t. a ≤ c and b ≤ c . 10 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem (Finkel & Goubault-Larrecq ICALP’09; Goubault-Larrecq ’14) � D downward closed = ⇒ D = Ideals finite 11 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem (Finkel & Goubault-Larrecq ICALP’09; Goubault-Larrecq ’14) � D downward closed = ⇒ D = Ideals finite Corollary Every downward closed set decomposes canonically as the union of its maximal ideals. 11 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Completion → S , ≤ ) is � S = ( � The completion of S = ( X , − X , − → � S , ⊆ ) such that 12 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Completion → S , ≤ ) is � S = ( � The completion of S = ( X , − X , − → � S , ⊆ ) such that � X = Ideals ( X ) , 12 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Completion → S , ≤ ) is � S = ( � The completion of S = ( X , − X , − → � S , ⊆ ) such that � X = Ideals ( X ) , I − → � S J if ↓ Post ( I ) = . . . ∪ J ∪ . . . � �� � canonical decomposition 12 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem → S , ≤ ) be a WSTS, then � S = ( � Let S = ( X , − X , − → � S , ⊆ ) is such that: � S is finitely branching, 13 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem → S , ≤ ) be a WSTS, then � S = ( � Let S = ( X , − X , − → � S , ⊆ ) is such that: � S is finitely branching, � S has (strong) monotony, 13 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem → S , ≤ ) be a WSTS, then � S = ( � Let S = ( X , − X , − → � S , ⊆ ) is such that: � S is finitely branching, � S has (strong) monotony, � S is not always a WSTS (Jančar IPL’99) . 13 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem → S , ≤ ) be a WSTS, then � S = ( � Let S = ( X , − X , − → � S , ⊆ ) is such that: � S is finitely branching, � S has (strong) monotony, � S is not always a WSTS (Jančar IPL’99) . Jančar IPL’99 A wqo ≤ is a ω 2 -wqo iff ≤ # is a wqo, where ≤ # is the Hoare ordering defined by A ≤ # B iff ↑ B ⊆↑ A . 13 / 24
Introduction WSTS completion Ideals Applications Completion Conclusion Theorem → S , ≤ ) be a WSTS, then � S = ( � Let S = ( X , − X , − → � S , ⊆ ) is such that: � S is finitely branching, � S has (strong) monotony, � S is not always a WSTS (Jančar IPL’99) . Jančar IPL’99 A wqo ≤ is a ω 2 -wqo iff ≤ # is a wqo, where ≤ # is the Hoare ordering defined by A ≤ # B iff ↑ B ⊆↑ A . Theorem Let S be a WSTS, then � S is a WSTS iff S is a ω 2 -WSTS. 13 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal J ⊇ ↓ y 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , then for every y ∈ J 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , then for every y ∈ J there exists x ∈ I 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , then for every y ∈ J there exists x ∈ I such that → S y ′ ≥ y . ∗ − x 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS with transitive monotony, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , then for every y ∈ J there exists x ∈ I such that → S y ′ ≥ y . ≥ k − − x 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relating executions of S and � S Let S = ( X , − → S , ≤ ) be a WSTS with strong monotony, then k if x − → S y , then for every ideal I ⊇ ↓ x there exists an ideal k J ⊇ ↓ y such that I − → � S J , k if I − → � S J , then for every y ∈ J there exists x ∈ I such that → S y ′ ≥ y . k − x 14 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Relations between S and � S A generality The completion � S = ( � X , − → � S , ⊆ ) computes exactly the downward closure of the reachability set of its original system S = ( X , − → S , ≤ ) . An equality We have: Post ∗ S ( ↓ x )= ↓ Post ∗ S ( x ) . � In fact, it is more exactly: Theorem If Post ∗ S ( ↓ x ) = { J 1 , . . . , J n } then ↓ Post ∗ S ( x ) = J 1 ∪ . . . ∪ J n . � 15 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that � S is a post-effective WSTS. 16 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that � S is a post-effective WSTS. Post-effectiveness Possible to compute cardinality of ) = Post( , , , . . . 16 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that � S is a post-effective WSTS. Proof Executions bounded in S iff bounded in � S . 16 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Theorem Strong termination is decidable for infinitely branching WSTS with transitive monotony and such that � S is a post-effective WSTS. Proof Executions bounded in S iff bounded in � S . � S finitely branching, can decide termination in � S by Finkel ICALP’87, Finkel & Schnoebelen TCS’01. 16 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . Input : → x ′ ≥ x ? ∗ − Question: x 0 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x 0 ∈ ↑ Pre ∗ ( ↑ x ) ? Question: Backward method (Abdulla, Cerans, Jonsson & Tsay IC’00) Compute ↑ Pre ∗ ( ↑ x ) iteratively assuming ↑ Pre ( ↑ x ) computable. 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x 0 ∈ ↑ Pre ∗ ( ↑ x ) ? Question: Backward method (Abdulla, Cerans, Jonsson & Tsay IC’00) Compute ↑ Pre ∗ ( ↑ x ) iteratively assuming ↑ Pre ( ↑ x ) computable. 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed, x 0 ∈ D and ↓ Post S ( D ) ⊆ D Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D = I 1 ∪ . . . ∪ I k Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed, x 0 ∈ D Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed, ↓ x 0 ⊆ I 1 ∪ . . . ∪ I k Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed, ∃ j s.t. ↓ x 0 ⊆ I j Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Coverability Input : ( X , − → , ≤ ) a WSTS, x 0 , x ∈ X . x ∈ ↓ Post ∗ ( x 0 ) ? Question: Forward method Coverability: ∗ Enumerate executions ↓ x 0 − → � S I , Accept if x ∈ I . Non coverability: Enumerate D ⊆ X downward closed, x 0 ∈ D and ↓ Post S ( D ) ⊆ D Reject if x �∈ D . 17 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Prebasis computability Prebasis computability is sufficient , but not necessary , to ensure decidability of coverability. 18 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Prebasis computability Prebasis computability is sufficient , but not necessary , to ensure decidability of coverability. Coverability is decidable in F 1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x . 18 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Prebasis computability Prebasis computability is sufficient , but not necessary , to ensure decidability of coverability. Coverability is decidable in F 1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x . Prebasis is not computable for F 1 Let S i = ( N , − → S i , ≤ ) be the WSTS such that: x − → S i 0 if TM i does not halt on its encoding in ≤ x steps, x − → S i 1 otherwise. 18 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Prebasis computability Prebasis computability is sufficient , but not necessary , to ensure decidability of coverability. Coverability is decidable in F 1 The algorithm consists to enumerate strictly increasing reachable sequences until finding an y ≥ x . Prebasis is not computable for F 1 Let S i = ( N , − → S i , ≤ ) be the WSTS such that: x − → S i 0 if TM i does not halt on its encoding in ≤ x steps, x − → S i 1 otherwise. Then S i ∈ F 1 and S i is effective. 18 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Three Pre sets Pre S i ( 0 ) = { x ∈ N : TM i does not halt in ≤ x steps } , Pre S i ( 1 ) = { x ∈ N : TM i halts in ≤ x steps } , Pre S i ( x ) = ∅ for x ≥ 2. 19 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Three Pre sets Pre S i ( 0 ) = { x ∈ N : TM i does not halt in ≤ x steps } , Pre S i ( 1 ) = { x ∈ N : TM i halts in ≤ x steps } , Pre S i ( x ) = ∅ for x ≥ 2. Conclusion: prebasis is not computable for F 1 Therefore, ↑ Pre S i ( ↑ 1 ) = ↑ Pre S i ( 1 ) = Pre S i ( 1 ) . If an algorithm outputting a finite basis of ↑ Pre S i ( ↑ 1 ) existed, then it would be possible to decide whether Pre S i ( 1 ) = ∅ . 19 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Three Pre sets Pre S i ( 0 ) = { x ∈ N : TM i does not halt in ≤ x steps } , Pre S i ( 1 ) = { x ∈ N : TM i halts in ≤ x steps } , Pre S i ( x ) = ∅ for x ≥ 2. Conclusion: prebasis is not computable for F 1 Therefore, ↑ Pre S i ( ↑ 1 ) = ↑ Pre S i ( 1 ) = Pre S i ( 1 ) . If an algorithm outputting a finite basis of ↑ Pre S i ( ↑ 1 ) existed, then it would be possible to decide whether Pre S i ( 1 ) = ∅ . But Pre S i ( 1 ) = ∅ iff TM i does not halt. The halting problem thus Turing-reduces to the prebasis computation. 19 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. 20 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c 0 labelled x 0 . If Post S ( x 0 ) is infinite, then we return “unbounded”, otherwise we mark c 0 and for every x ∈ Post S ( x 0 ) we add a child labelled x to c 0 . 20 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c 0 labelled x 0 . If Post S ( x 0 ) is infinite, then we return “unbounded”, otherwise we mark c 0 and for every x ∈ Post S ( x 0 ) we add a child labelled x to c 0 . If c has an ancestor c ′ labelled x ′ such that x ′ < x , we return “unbounded”. Otherwise, if c has an ancestor c ′ labelled x ′ such that x ′ = x , we mark c . Otherwise, if Post S ( x ) is infinite, then we return “unbounded”. Otherwise we mark c and for every y ∈ Post S ( x ) we add a child labelled y to c . 20 / 24
Introduction WSTS completion Termination Applications Coverability Conclusion Boundedness for infinitely branching WSTS Boundedness is decidable for post-effective WSTS with strict monotony and a wpo. Proof We build a reachability tree T with root c 0 labelled x 0 . If Post S ( x 0 ) is infinite, then we return “unbounded”, otherwise we mark c 0 and for every x ∈ Post S ( x 0 ) we add a child labelled x to c 0 . If c has an ancestor c ′ labelled x ′ such that x ′ < x , we return “unbounded”. Otherwise, if c has an ancestor c ′ labelled x ′ such that x ′ = x , we mark c . Otherwise, if Post S ( x ) is infinite, then we return “unbounded”. Otherwise we mark c and for every y ∈ Post S ( x ) we add a child labelled y to c . T is finite and correct. 20 / 24
Introduction WSTS completion Applications Conclusion Further result for infinitely branching WSTS Strong maintainability is decidable for WSTS with strong monotony and such that � S is a post-effective WSTS. 21 / 24
Introduction WSTS completion Applications Conclusion Further work ∃ general class of infinitely branching WSTS with a Karp-Miller procedure? 22 / 24
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