present and future of proving termination of rewriting
play

Present and Future of Proving Termination of Rewriting Albert Rubio - PowerPoint PPT Presentation

Nov 21, 2023 •457 likes •1.91k views

Present and Future of Proving Termination of Rewriting Albert Rubio Universitat Polit` ecnica de Catalunya Barcelona Albert Rubio - UPC RTA 2008 p.1/31 Overview of the talk Albert Rubio - UPC RTA 2008 p.1/31 Overview of the talk

  1. Termination Absence of infinite computations f : α × α ⇒ α g : ( α → α ) ⇒ α Let and in { x : α → α } ⊢ f ( g ( x ) , g ( x )) : α → @ ( x , g ( x )) : α � �� � f ( g ( λ y . f ( y , y ) ) , g ( λ y . f ( y , y ) )) : α → � �� � � �� � x x @ ( λ y . f ( y , y ) , g ( λ y . f ( y , y ))) : α Albert Rubio - UPC RTA 2008– p.6/31

  2. Termination Absence of infinite computations f : α × α ⇒ α g : ( α → α ) ⇒ α Let and in { x : α → α } ⊢ f ( g ( x ) , g ( x )) : α → @ ( x , g ( x )) : α � �� � f ( g ( λ y . f ( y , y ) ) , g ( λ y . f ( y , y ) )) : α → � �� � � �� � x x � �� � @ ( λ y . f ( y , y ) , g ( λ y . f ( y , y ) ) ) : α → β � �� � � �� � v λ y . u Albert Rubio - UPC RTA 2008– p.6/31

  3. Termination Absence of infinite computations f : α × α ⇒ α g : ( α → α ) ⇒ α Let and in { x : α → α } ⊢ f ( g ( x ) , g ( x )) : α → @ ( x , g ( x )) : α � �� � f ( g ( λ y . f ( y , y ) ) , g ( λ y . f ( y , y ) )) : α → � �� � � �� � x x � �� � @ ( λ y . f ( y , y ) , g ( λ y . f ( y , y ) ) ) : α → β � �� � � �� � v λ y . u f ( g ( λ y . f ( y , y )) , g ( λ y . f ( y , y ))) : α → . . . Albert Rubio - UPC RTA 2008– p.6/31

  4. Motivation Why studing termination (early days)? Albert Rubio - UPC RTA 2008– p.7/31

  5. Motivation Why studing termination (early days)? Knuth-Bendix completion and theorem proving Albert Rubio - UPC RTA 2008– p.7/31

  6. Motivation Why studing termination (early days)? Knuth-Bendix completion and theorem proving Termination of some particular TRS Albert Rubio - UPC RTA 2008– p.7/31

  7. Motivation Why studing termination (early days)? Knuth-Bendix completion and theorem proving Termination of some particular TRS rule-based algorithms Albert Rubio - UPC RTA 2008– p.7/31

  8. Motivation Why studing termination (early days)? Knuth-Bendix completion and theorem proving Termination of some particular TRS rule-based algorithms rule-based specification Albert Rubio - UPC RTA 2008– p.7/31

  9. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. Albert Rubio - UPC RTA 2008– p.8/31

  10. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). Albert Rubio - UPC RTA 2008– p.8/31

  11. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). 1978, Huet and Lankford. Undecidability of the termination of Term Rewriting Systems. Albert Rubio - UPC RTA 2008– p.8/31

  12. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). 1978, Huet and Lankford. Undecidability of the termination of Term Rewriting Systems. 1979, Dershowitz. Recursive Path Ordering (RPO). Albert Rubio - UPC RTA 2008– p.8/31

  13. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). 1978, Huet and Lankford. Undecidability of the termination of Term Rewriting Systems. 1979, Dershowitz. Recursive Path Ordering (RPO). 1979, Dershowitz Manna. Multiset ordering. Albert Rubio - UPC RTA 2008– p.8/31

  14. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). 1978, Huet and Lankford. Undecidability of the termination of Term Rewriting Systems. 1979, Dershowitz. Recursive Path Ordering (RPO). 1979, Dershowitz Manna. Multiset ordering. 1979, Lankford. One of the first references on polynomial interpretations. Albert Rubio - UPC RTA 2008– p.8/31

  15. A bit of history The beginings 1970, Manna and Ness. Reference for reduction orderings . One of the first papers on termination of rewriting. 1970, Knuth and Bendix. Knuth Bendix Ordering (KBO). 1978, Huet and Lankford. Undecidability of the termination of Term Rewriting Systems. 1979, Dershowitz. Recursive Path Ordering (RPO). 1979, Dershowitz Manna. Multiset ordering. 1979, Lankford. One of the first references on polynomial interpretations. 1980, Kamin and Levy. Lexicographic Path Ordering (LPO) and Semantic Path Ordering (SPO). Albert Rubio - UPC RTA 2008– p.8/31

  16. Reduction Orderings � �� � � �� � s ( s ( s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) )) → s ( s ( s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) )) ���� � �� � ���� � �� � n m n m s ( n ) + m → s ( n + m ) Albert Rubio - UPC RTA 2008– p.9/31

  17. Reduction Orderings � �� � � �� � s ( s ( s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) )) → s ( s ( s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) )) ���� � �� � ���� � �� � n m n m s ( n ) + m → s ( n + m ) s ( n ) + m ≻ s ( n + m ) Albert Rubio - UPC RTA 2008– p.9/31

  18. Reduction Orderings � �� � � �� � s ( s ( s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) )) → s ( s ( s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) )) ���� � �� � ���� � �� � n m n m s ( n ) + m → s ( n + m ) s ( n ) + m ≻ s ( n + m ) s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) ≻ s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) stable under ���� � �� � ���� � �� � substitution n m n m Albert Rubio - UPC RTA 2008– p.9/31

  19. Reduction Orderings � �� � � �� � s ( s ( s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) )) → s ( s ( s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) )) ���� � �� � ���� � �� � n m n m s ( n ) + m → s ( n + m ) s ( n ) + m ≻ s ( n + m ) s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) ≻ s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) stable under ���� � �� � ���� � �� � substitution n m n m � �� � � �� � s ( s ( s ( s ( 0 ) ) + s ( s ( s ( s ( 0 )))) )) ≻ s ( s ( s ( s ( 0 ) + s ( s ( s ( s ( 0 )))) ) )) monotonic ���� � �� � ���� � �� � n m n m Albert Rubio - UPC RTA 2008– p.9/31

  20. Reduction Orderings (cont.) ≻ is a reduction ordering if it’s well-founded � ∃ t 1 ≻ t 2 ≻ t 3 ≻ . . . stable under sustitution s ≻ t s σ ≻ t σ implies for every substitution σ monotonic s ≻ t u [ s ] ≻ u [ t ] for every context u [ ] implies Albert Rubio - UPC RTA 2008– p.10/31

  21. Reduction Orderings (cont.) ≻ is a reduction ordering if it’s well-founded � ∃ t 1 ≻ t 2 ≻ t 3 ≻ . . . stable under sustitution s ≻ t s σ ≻ t σ implies for every substitution σ monotonic s ≻ t u [ s ] ≻ u [ t ] for every context u [ ] implies l ≻ r for all l → r ∈ R R terminates if and only if for some reduction ordering ≻ . Albert Rubio - UPC RTA 2008– p.10/31

  22. Reduction Orderings (cont.) ≻ is a higher-order reduction ordering if it’s well-founded � ∃ t 1 ≻ t 2 ≻ t 3 ≻ . . . stable under substitution s ≻ t s σ ≻ t σ implies for all substitution σ monotonic s ≻ t u [ s ] ≻ u [ t ] for all context u [ ] implies Albert Rubio - UPC RTA 2008– p.11/31

  23. Reduction Orderings (cont.) ≻ is a higher-order reduction ordering if it’s well-founded � ∃ t 1 ≻ t 2 ≻ t 3 ≻ . . . stable under substitution s ≻ t s σ ≻ t σ implies for all substitution σ monotonic s ≻ t u [ s ] ≻ u [ t ] for all context u [ ] implies functional s → β t s ≻ t implies Albert Rubio - UPC RTA 2008– p.11/31

  24. Reduction Orderings (cont.) ≻ is a higher-order reduction ordering if it’s well-founded � ∃ t 1 ≻ t 2 ≻ t 3 ≻ . . . stable under substitution s ≻ t s σ ≻ t σ implies for all substitution σ monotonic s ≻ t u [ s ] ≻ u [ t ] for all context u [ ] implies functional s → β t s ≻ t implies l ≻ r for all l → r ∈ R R terminates if and only if for some higher-order reduction ordering ≻ . Albert Rubio - UPC RTA 2008– p.11/31

  25. Polynomial Interpretations Albert Rubio - UPC RTA 2008– p.12/31

  26. Polynomial Interpretations Interprets every function symbol as a (linear) polynomial: [ f ] = A k − → A where A is often the positive integers or the reals. Albert Rubio - UPC RTA 2008– p.12/31

  27. Polynomial Interpretations Interprets every function symbol as a (linear) polynomial: [ f ] = A k − → A where A is often the positive integers or the reals. Interpret terms by: Pol ( f ( t 1 , . . . , t n )) = [ f ]( Pol ( t 1 ) , . . . , Pol ( t n )) Albert Rubio - UPC RTA 2008– p.12/31

  28. Polynomial Interpretations Interprets every function symbol as a (linear) polynomial: [ f ] = A k − → A where A is often the positive integers or the reals. Interpret terms by: Pol ( f ( t 1 , . . . , t n )) = [ f ]( Pol ( t 1 ) , . . . , Pol ( t n )) Then a reduction ordering is obtained by s ≻ t if Pol ( s ) − Pol ( t ) > 0 if all coeficients are strictly positive. If there are zero coeficients then it is just weakly monotonic. Albert Rubio - UPC RTA 2008– p.12/31

  29. The recursive path ordering Albert Rubio - UPC RTA 2008– p.13/31

  30. The recursive path ordering compares terms by extending an ordering on function symbols. Albert Rubio - UPC RTA 2008– p.13/31

  31. The recursive path ordering compares terms by extending an ordering on function symbols. Let ≻ F be a (well-founded) ordering on the function symbols For instance: prod ≻ F + ≻ F s ≻ F 0 Albert Rubio - UPC RTA 2008– p.13/31

  32. The recursive path ordering compares terms by extending an ordering on function symbols. Let ≻ F be a (well-founded) ordering on the function symbols For instance: prod ≻ F + ≻ F s ≻ F 0 s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g s ≻ rpo t j for all 1 ≤ j ≤ m . and 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Albert Rubio - UPC RTA 2008– p.13/31

  33. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 Albert Rubio - UPC RTA 2008– p.14/31

  34. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Albert Rubio - UPC RTA 2008– p.14/31

  35. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Case 2 prod ( s ( n ) , m ) ≻ rpo prod ( n , m ) Albert Rubio - UPC RTA 2008– p.14/31

  36. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Case 2 prod ( s ( n ) , m ) ≻ rpo prod ( n , m ) Case 3 { s ( n ) , m } ≻≻ rpo { n , m } Albert Rubio - UPC RTA 2008– p.14/31

  37. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Case 2 prod ( s ( n ) , m ) ≻ rpo prod ( n , m ) Case 3 { s ( n ) , � m } ≻≻ rpo { n , � m } multiset comparison Albert Rubio - UPC RTA 2008– p.14/31

  38. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Case 2 prod ( s ( n ) , m ) ≻ rpo prod ( n , m ) Case 3 { s ( n ) , � m } ≻≻ rpo { n , � m } multiset comparison s ( n ) ≻ rpo n Albert Rubio - UPC RTA 2008– p.14/31

  39. The recursive path ordering (cont.) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Let prod ≻ F + ≻ F s ≻ F 0 prod ( s ( n ) , m ) ≻ rpo s ( prod ( n , m )) Case 2 prod ( s ( n ) , m ) ≻ rpo prod ( n , m ) Case 3 { s ( n ) , � m } ≻≻ rpo { n , � m } multiset comparison s ( n ) ≻ rpo n Case 1 Albert Rubio - UPC RTA 2008– p.14/31

  40. Powerful automatable methods Methods that can be fully automated Can handle hard examples. Albert Rubio - UPC RTA 2008– p.15/31

  41. Powerful automatable methods Methods that can be fully automated Can handle hard examples. Only two will be considered in this talk: The Dependency Pair Method The Monotonic Semantic Path Ordering Albert Rubio - UPC RTA 2008– p.15/31

  42. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. Albert Rubio - UPC RTA 2008– p.16/31

  43. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) Albert Rubio - UPC RTA 2008– p.16/31

  44. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(PROD(n,m)) SUM(n,m) Albert Rubio - UPC RTA 2008– p.16/31

  45. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → prod ( 0, m ) ⇒ 0 ZERO prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) SUM ( s ( n ) , m ) ⇒ SUC ( n + m ) Albert Rubio - UPC RTA 2008– p.16/31

  46. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → prod ( 0, m ) ⇒ 0 ZERO prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) SUM ( s ( n ) , m ) ⇒ SUC ( n + m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(prod(n,m)) SUM(n,m) SUM(s(n),m) PROD(s(n),m) SUC(n+m) ZERO Does not change the potential cycles!!! Albert Rubio - UPC RTA 2008– p.16/31

  47. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → prod ( 0, m ) ⇒ 0 ZERO prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) SUM ( s ( n ) , m ) ⇒ SUC ( n + m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(prod(n,m)) SUM(n,m) SUM(s(n),m) PROD(s(n),m) SUC(n+m) ZERO Albert Rubio - UPC RTA 2008– p.16/31

  48. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → prod ( 0, m ) ⇒ 0 ZERO prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) SUM ( s ( n ) , m ) ⇒ SUC ( n + m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(prod(n,m)) SUM(n,m) SUM(s(n),m) PROD(s(n),m) SUC(n+m) ZERO Albert Rubio - UPC RTA 2008– p.16/31

  49. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → prod ( 0, m ) ⇒ 0 ZERO prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) SUM ( s ( n ) , m ) ⇒ SUC ( n + m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(PROD(n,m)) SUM(n,m) Albert Rubio - UPC RTA 2008– p.16/31

  50. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(PROD(n,m)) SUM(n,m) Albert Rubio - UPC RTA 2008– p.16/31

  51. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) PROD(s(n),m) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(prod(n,m)) SUM(n,m) Albert Rubio - UPC RTA 2008– p.16/31

  52. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) PROD(s(n),m) SUM(s(n),m) PROD(n,m) SUM(n,m) Albert Rubio - UPC RTA 2008– p.16/31

  53. The Dependency Pair Method (cont.) (Arts and Giesl 1997) Basically, it transforms the rewrite system into a new reduction system which is simpler to analyze and (dis)prove termination. prod ( 0, m ) → 0 prod ( s ( n ) , m ) → prod ( n , m ) + m PROD ( s ( n ) , m ) ⇒ SUM ( prod ( n , m ) , m ) PROD ( s ( n ) , m ) ⇒ PROD ( n , m ) 0 + m → m s ( n ) + m → s ( n + m ) SUM ( s ( n ) , m ) ⇒ SUM ( n , m ) The first phase its a simple rewriting trace analysis. This method was a breakthrough in the development of automated termination provers for term rewriting Albert Rubio - UPC RTA 2008– p.16/31

  54. The Dependency Pair Method (cont.) After simplifying the graph we have to show that all pontential cycles are finite. Solve the ordering constraints ensuring that all cycles in the graph are strictly decreasing. Find an ordering satisfying the constraint: Polynomial Interpretations. RPO-like orderings with argument filterings. ... Albert Rubio - UPC RTA 2008– p.17/31

  55. The (Monotonic) Semantic Path Ordering (Borralleras, Ferreira and Rubio 2000) Albert Rubio - UPC RTA 2008– p.18/31

  56. The (Monotonic) Semantic Path Ordering (Borralleras, Ferreira and Rubio 2000) s = f ( s 1 , . . . , s n ) ≻ rpo t if and only if 1. s i � rpo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , f ≻ F g and s ≻ rpo t j for all 1 ≤ j ≤ m . 3. t = f ( t 1 , . . . , t m ) { s 1 , . . . , s n } ≻≻ rpo { t 1 , . . . , t m } and Albert Rubio - UPC RTA 2008– p.18/31

  57. The (Monotonic) Semantic Path Ordering (Borralleras, Ferreira and Rubio 2000) s = f ( s 1 , . . . , s n ) ≻ spo t if and only if 1. s i � spo t for some 1 ≤ i ≤ n . 2. t = g ( t 1 , . . . , t m ) , s ❂ t and s ≻ spo t j for all 1 ≤ j ≤ m . 3. t = g ( t 1 , . . . , t m ) , s ⊒ t and { s 1 , . . . , s n } ≻≻ spo { t 1 , . . . , t m } in addition you need s ⊒ t to ensure monotonicity, where ⊒ is monotonic. Albert Rubio - UPC RTA 2008– p.18/31

Recommend

Certification of proving termination of term rewriting by matrix interpretations Adam Koprowski

Certification of proving termination of term rewriting by matrix interpretations Adam Koprowski

Certification of proving termination of term rewriting by matrix interpretations Adam Koprowski and Hans Zantema Eindhoven University of Technology Department of Mathematics and Computer Science 21 January 2008 SOFSEM08 Nov Smokovec,

1.06k views • 78 slides
Semantic Labelling for Proving Termination of Combinatory Reduction Systems Makoto Hamana

Semantic Labelling for Proving Termination of Combinatory Reduction Systems Makoto Hamana

Semantic Labelling for Proving Termination of Combinatory Reduction Systems Makoto Hamana Department of Computer Science, Gunma University, Japan WFLP09 June, 2009. 1 Intro: Termination Proof by RPO Term Rewriting System (TRS) R : fact

540 views • 17 slides
. Needs effective (finitary) representation .. Failed Termination Proof vs. Non- TNT:

. Needs effective (finitary) representation .. Failed Termination Proof vs. Non- TNT:

Runtime Errors Proving Non-Termination Ashutosh K. Gupta Thomas A. Henzinger Rupak Majumdar MPI-SWS EPFL UCLA Andrey Rybalchenko Ru-Gang Xu MPI-SWS UCLA 2 Non-Termination Errors Proving Non-Termination Search for infinite

402 views • 6 slides
Rewriting Part 4. Termination of Term Rewriting Systems Temur Kutsia RISC, JKU Linz Termination

Rewriting Part 4. Termination of Term Rewriting Systems Temur Kutsia RISC, JKU Linz Termination

Rewriting Part 4. Termination of Term Rewriting Systems Temur Kutsia RISC, JKU Linz Termination Definition 4.1 A term rewriting system R is terminating iff R is terminating, i.e., there is no infinite reduction chain t 0 R t 1 R t

1.22k views • 94 slides
C ONTENT A PPLYING A R EWRITE R ULE l Intro & motivation, getting started

C ONTENT A PPLYING A R EWRITE R ULE l Intro & motivation, getting started

L AST T IME Introducing new Types Equations and Term Rewriting Confluence and Termination of reduction systems NICTA Advanced Course Term Rewriting in Isabelle Slide 1 Slide 3 Theorem Proving Principles, Techniques, Applications

374 views • 8 slides
Term rewriting Equational logic Term rewriting systems Termination Confluence

Term rewriting Equational logic Term rewriting systems Termination Confluence

Term rewriting Equational logic Term rewriting systems Termination Confluence Rapid prototyping Summary and Exercises Equational logic Reflexivity t = t t 1 = t 2 Symmetry t 2 = t 1 t 1 = t 2 t 2 = t 3 Transitivity t 1 =

769 views • 52 slides
Stratification Criteria and Rewriting Techniques for Checking Chase Termination S. Greco, F .

Stratification Criteria and Rewriting Techniques for Checking Chase Termination S. Greco, F .

Stratification Criteria and Rewriting Techniques for Checking Chase Termination Stratification Criteria and Rewriting Techniques for Checking Chase Termination S. Greco, F . Spezzano and I. Trubitsyna DEIS, Universit` a della Calabria - Italy

1.51k views • 112 slides
Proving Termination of Imperative Programs using Max-SMT Daniel Larraz, Albert Oliveras, Enric

Proving Termination of Imperative Programs using Max-SMT Daniel Larraz, Albert Oliveras, Enric

Introduction SMT/Max-SMT solving Invariant generation Termination analysis Further work Proving Termination of Imperative Programs using Max-SMT Daniel Larraz, Albert Oliveras, Enric Rodr guez-Carbonell and Albert Rubio Universitat

1.18k views • 73 slides
Better termination proving through cooperation Marc Brockschmidt 1 Byron Cook 2 , 3 Carsten Fuhs 3

Better termination proving through cooperation Marc Brockschmidt 1 Byron Cook 2 , 3 Carsten Fuhs 3

Better termination proving through cooperation Marc Brockschmidt 1 Byron Cook 2 , 3 Carsten Fuhs 3 1 RWTH Aachen University 2 Microsoft Research Cambridge 3 University College London Deduktionstreffen 2013 Termination Analysis: Invariants and Rank

538 views • 37 slides
On Proving Almost-Sure Termination Joost-Pieter Katoen Talk 2019 Meeting IFIP WG 2.2, Vienna

On Proving Almost-Sure Termination Joost-Pieter Katoen Talk 2019 Meeting IFIP WG 2.2, Vienna

IFIP WG 2.2, 2019 On Proving Almost-Sure Termination Joost-Pieter Katoen Talk 2019 Meeting IFIP WG 2.2, Vienna Joost-Pieter Katoen On Proving Almost-Sure Termination 1/30 IFIP WG 2.2, 2019 Termination of programs that roll dice?

631 views • 34 slides
Compatible rewriting of noncommutative polynomials for proving operator identities C.Chenavier

Compatible rewriting of noncommutative polynomials for proving operator identities C.Chenavier

Compatible rewriting of noncommutative polynomials for proving operator identities C.Chenavier C.Hofstadler C.G.Raab G.Regensburger Johannes Kepler Universitt, Linz, Austria 45th ISSAC Kalamata, Greece, July 20-23, 2020 Chenavier,

697 views • 59 slides
Compatible rewriting systems and completion for proving operator identities (j.w. Clemens

Compatible rewriting systems and completion for proving operator identities (j.w. Clemens

Compatible rewriting systems and completion for proving operator identities (j.w. Clemens Hofstadler, Clemens G. Raab, and Georg Regensburger) Cyrille Chenavier Johannes Kepler University, Institute for Algebra Journes Nationales du Calcul

1.03k views • 58 slides
CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki

CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki

CO3 a COnverter for proving COnfluence of COnditional term rewriting systems Ver. 1.1 Naoki Nishida Makishi Yanagisawa Karl Gmeiner Nagoya University UAS Technikum Wien CoCo 2014 Vienna, July 13, 2014 Target and Main

230 views • 5 slides
Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University,

Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University,

Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University, Germany VTSA 12, Saarbr ucken, Germany Overview I. Termination of Term Rewriting 1 Termination of Term Rewrite Systems 2 Non-Termination of Term

1.93k views • 148 slides
Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University,

Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University,

Automated Termination Analysis J urgen Giesl LuFG Informatik 2, RWTH Aachen University, Germany VTSA 12, Saarbr ucken, Germany Overview I. Termination of Term Rewriting 1 Termination of Term Rewrite Systems 2 Non-Termination of Term

1.47k views • 133 slides
Proving Equivalence of Imperative Programs via Constrained Rewriting Induction Carsten Fuhs

Proving Equivalence of Imperative Programs via Constrained Rewriting Induction Carsten Fuhs

Proving Equivalence of Imperative Programs via Constrained Rewriting Induction Carsten Fuhs Birkbeck, University of London joint work with Cynthia Kop (U Copenhagen) and Naoki Nishida (U Nagoya) Workshop on Program Equivalence, London, UK 11

2.1k views • 197 slides
Structural counter abstraction Proving fair-termination of depth bounded systems Kshitij Bansal 1

Structural counter abstraction Proving fair-termination of depth bounded systems Kshitij Bansal 1

Structural counter abstraction Proving fair-termination of depth bounded systems Kshitij Bansal 1 with Eric Koskinen 1 , Thomas Wies 1 , Damien Zufferey 2 1 New York University 2 IST Austria March 18, 2013 TACAS, Rome, Italy Introduction

714 views • 32 slides
Proving termination using dependent types: the case of xor-terms J.-F. Monin J. Courant VERIMAG

Proving termination using dependent types: the case of xor-terms J.-F. Monin J. Courant VERIMAG

Proving termination using dependent types: the case of xor-terms J.-F. Monin J. Courant VERIMAG Grenoble, France GDR LAC, Chambery, 2007 Outline Motivation The case of cryptographic systems State of the art Back to cryptographic systems

1.46k views • 117 slides
Disproving Confluence of Term Rewriting Systems by Interpretation and Ordering FroCoS 2013

Disproving Confluence of Term Rewriting Systems by Interpretation and Ordering FroCoS 2013

Disproving Confluence of Term Rewriting Systems by Interpretation and Ordering FroCoS 2013 Takahito Aoto (Tohoku University) Outline 1. Backgrounds: TRS and Confluence 2. Backgrounds: Proving (Non)-Confluence 3. Proving Non-Joinability by

684 views • 40 slides
PERSONNEL ISSUES: DISCIPLINE AND TERMINATION Personnel Issues: Discipline and Termination

PERSONNEL ISSUES: DISCIPLINE AND TERMINATION Personnel Issues: Discipline and Termination

PERSONNEL ISSUES: DISCIPLINE AND TERMINATION Personnel Issues: Discipline and Termination EMPLOYMENT LIABILITIES I. The Litigious Workforce A. Employment related lawsuits, while once very rare, have become commonplace in our present society.

678 views • 8 slides
Automatic Cyclic Termination Proofs for Recursive Procedures in Separation Logic Reuben Rowe and

Automatic Cyclic Termination Proofs for Recursive Procedures in Separation Logic Reuben Rowe and

Automatic Cyclic Termination Proofs for Recursive Procedures in Separation Logic Reuben Rowe and James Brotherston University College London TAPAS, Edinburgh, Wednesday 7 th September 2016 Automatically Proving Termination: Challenges proc

815 views • 57 slides
On Theorem Proving for Program Checking Historical perspective and recent developments Maria

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

Outline Introduction Where is theorem proving in program checking Inside theorem proving Big and little engines together: a new theorem proving style Decision procedures with speculative inferences Current and future challenges On Theorem

944 views • 69 slides
The Dependency Pair Technique Proving Termination of Term Rewrite Systems Hannes Saffrich

The Dependency Pair Technique Proving Termination of Term Rewrite Systems Hannes Saffrich

Program Analysis Seminar The Dependency Pair Technique Proving Termination of Term Rewrite Systems Hannes Saffrich University of Freiburg Department of Computer Science June 30, 2015 Hannes Saffrich The Dependency Pair Technique Program

441 views • 23 slides
Types of Termination Mutual With Consent Termination Automatic Termination

Types of Termination Mutual With Consent Termination Automatic Termination

Types of Termination Mutual With Consent Termination Automatic Termination Termination Termination for Convenience Without Consent Frustration /Force Majeure Contractual Breach Breach Common Law Breach

444 views • 29 slides

More recommend