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FO = FO 3 for Linear Orders with Monotone Binary Relations Marie - PowerPoint PPT Presentation

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FO = FO 3 for Linear Orders with Monotone Binary Relations Marie Fortin University of Liverpool YR-OWLS, June 16, 2020 1 / 18 The k -variable property How many variables are needed in first-order logic ? 2 / 18 The k -variable property How

  1. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. 7 / 18

  2. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) 7 / 18

  3. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) I 7 / 18

  4. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) I R R 7 / 18

  5. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) I R R Im ( R ) 7 / 18

  6. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) I R R R Im ( R ) 7 / 18

  7. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) ◮ R − 1 ( I ) is an interval of ( dom ( R ) , ≤ ) 7 / 18

  8. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) ◮ R − 1 ( I ) is an interval of ( dom ( R ) , ≤ ) I 7 / 18

  9. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) ◮ R − 1 ( I ) is an interval of ( dom ( R ) , ≤ ) dom ( R ) R R I 7 / 18

  10. A special case: monotone partial functions Any relation R corresponding to a monotone partial function is interval-preserving. ◮ R ( I ) is an interval of ( Im ( R ) , ≤ ) ◮ R − 1 ( I ) is an interval of ( dom ( R ) , ≤ ) dom ( R ) R R R I 7 / 18

  11. Applications FO = FO 3 over 1. Linear orders with partial non-decreasing or non-increasing functions (new) 8 / 18

  12. Applications FO = FO 3 over 1. Linear orders with partial non-decreasing or non-increasing functions (new) 2. Linear orders: finite or infinite words, R , Q , ordinals... 8 / 18

  13. Applications FO = FO 3 over 1. Linear orders with partial non-decreasing or non-increasing functions (new) 2. Linear orders: finite or infinite words, R , Q , ordinals... 3. ( R , ≤ , +1) , ( R , ≤ , (+ q ) q ∈ Q ) . . . 8 / 18

  14. Applications FO = FO 3 over 1. Linear orders with partial non-decreasing or non-increasing functions (new) 2. Linear orders: finite or infinite words, R , Q , ordinals... 3. ( R , ≤ , +1) , ( R , ≤ , (+ q ) q ∈ Q ) . . . 4. ( R , ≤ ) + polynomial functions (new) 8 / 18

  15. Applications 5. Message sequence charts (MSCs) 9 / 18

  16. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r 9 / 18

  17. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems 9 / 18

  18. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes 9 / 18

  19. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc 9 / 18

  20. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc ◮ Message relations ⊳ p,q 9 / 18

  21. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc ◮ Message relations ⊳ p,q 9 / 18

  22. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc Extended to a linear order ◮ Message relations ⊳ p,q 9 / 18

  23. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc Extended to a linear order ◮ Message relations ⊳ p,q FIFO → monotone 9 / 18

  24. Applications 5. Message sequence charts (MSCs) a a c a a a a a p a a a a a a a a a a q a a a c a a a b b r Executions of message-passing systems ◮ Fixed, finite set of processes ◮ Process order ≤ proc Extended to a linear order ◮ Message relations ⊳ p,q FIFO → monotone → Interval-preserving structure 9 / 18

  25. Applications FO = FO 3 over structures with ◮ one linear order ≤ , ◮ “interval-preserving” binary relations R 1 , R 2 , . . . , ◮ arbitrary unary predicates p, q, . . . 1. Linear orders with partial non-decreasing or non-increasing functions (new) 2. Linear orders: finite or infinite words, R , Q , ordinals... 3. ( R , ≤ , +1) , ( R , ≤ , (+ q ) q ∈ Q ) . . . 4. ( R , ≤ ) + polynomial functions (new) 5. MSCs 6. Mazurkiewicz traces 10 / 18

  26. How does the interval-preserving assumption help? 11 / 18

  27. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) x 2 R 2 R 1 x 1 y R 3 x 3 11 / 18

  28. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) x 2 R 2 R 1 x 1 y R 3 x 3 Equivalent FO 3 formula? 11 / 18

  29. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) R 1 ( x 1 ) R 3 ( x 3 ) R 2 ( x 2 ) Equivalent FO 3 formula? 11 / 18

  30. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) R 1 ( x 1 ) R 3 ( x 3 ) R 2 ( x 2 ) y Equivalent FO 3 formula? 11 / 18

  31. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) � � ≡ ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ ∧ � � ∃ y. R 1 ( x 1 , y ) ∧ R 3 ( x 3 , y ) ∧ ∧ � � ∃ y. R 2 ( x 2 , y ) ∧ R 2 ( x 3 , y ) ∧ R 1 ( x 1 ) R 3 ( x 3 ) R 2 ( x 2 ) y Equivalent FO 3 formula? 11 / 18

  32. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) � � ≡ ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ ∃ x. R 3 ( x, y ) ∧ � � ∃ y. R 1 ( x 1 , y ) ∧ R 3 ( x 3 , y ) ∧ ∃ x. R 2 ( x, y ) ∧ � � ∃ y. R 2 ( x 2 , y ) ∧ R 2 ( x 3 , y ) ∧ ∃ x. R 1 ( x, y ) R 1 ( x 1 ) R 3 ( x 3 ) R 2 ( x 2 ) y Equivalent FO 3 formula? 11 / 18

  33. How does the interval-preserving assumption help? ϕ ( x 1 , x 2 , x 3 ) = ∃ y. R 1 ( x 1 , y ) ∧ R 2 ( x 2 , y ) ∧ R 3 ( x 3 , y ) � � ≡ ∃ x 3 . R 1 ( x 1 , x 3 ) ∧ R 2 ( x 2 , x 3 ) ∧ ∃ x 1 . R 3 ( x 1 , x 3 ) ∧ � � ∃ x 2 . R 1 ( x 1 , x 2 ) ∧ R 3 ( x 3 , x 2 ) ∧ ∃ x 1 . R 2 ( x 1 , x 2 ) ∧ � � ∃ x 1 . R 2 ( x 2 , x 1 ) ∧ R 2 ( x 3 , x 1 ) ∧ ∃ x 2 . R 1 ( x 2 , x 1 ) R 1 ( x 1 ) R 3 ( x 3 ) R 2 ( x 2 ) y Equivalent FO 3 formula? 11 / 18

  34. The proof FO = FO 3 over structures with ◮ one linear order ≤ , ◮ “interval-preserving” binary relations R 1 , R 2 , . . . , ◮ arbitrary unary predicates p, q, . . . 12 / 18

  35. The proof FO = FO 3 over structures with ◮ one linear order ≤ , ◮ “interval-preserving” binary relations R 1 , R 2 , . . . , ◮ arbitrary unary predicates p, q, . . . Key idea: Go through an intermediate language: Star-free Propositional Dynamic Logic. FO FO 3 Star-free PDL 12 / 18

  36. Star-free Propositional Dynamic Logic Examples 13 / 18

  37. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q 13 / 18

  38. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ) ✗ ✓ ✗ ✓ ✓ ✓ ✓ ✗ ✓ 13 / 18

  39. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ) ✗ ✓ ✗ ✓ ✓ ✓ ✓ ✗ ✓ � R � q ✓ ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ 13 / 18

  40. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ) ✗ ✓ ✗ ✓ ✓ ✓ ✓ ✗ ✓ � R � q ✓ ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ �≤ · R − 1 � q ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 13 / 18

  41. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ) ✗ ✓ ✗ ✓ ✓ ✓ ✓ ✗ ✓ � R � q ✓ ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ �≤ · R − 1 � q ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ �≤ · {� R � q } ? · ≤� p ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✓ ✓ 13 / 18

  42. Star-free Propositional Dynamic Logic Examples R p q q p p p, q q ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ) ✗ ✓ ✗ ✓ ✓ ✓ ✓ ✗ ✓ � R � q ✓ ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ �≤ · R − 1 � q ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ �≤ · {� R � q } ? · ≤� p ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✓ ✓ � R c ∩ ≤� ( p ∧ q ) ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 13 / 18

  43. Star-free Propositional Dynamic Logic Examples Over ( R , <, { + q | q ∈ Q + } ) , ϕ U ( q,r ) ψ ≡ t + q t + r t ϕ ψ 14 / 18

  44. Star-free Propositional Dynamic Logic Examples Over ( R , <, { + q | q ∈ Q + } ) , � (+ q · < ) ∩ (+ r · < − 1 ) ∩ ( < · {¬ ϕ } ? · < ) c � ϕ U ( q,r ) ψ ≡ ψ + r + q < < ϕ ψ 14 / 18

  45. Star-free Propositional Dynamic Logic Syntax State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ ϕ | � π � ϕ PDL sf Path formulas: π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∪ π | π c 15 / 18

  46. Star-free Propositional Dynamic Logic Syntax State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ ϕ | � π � ϕ PDL sf Path formulas: π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∪ π | π c Combines features from ◮ Propositional Dynamic Logic [Fisher-Ladner 1979] ◮ Star-free regular expressions ◮ The calculus of relations 15 / 18

  47. Star-free Propositional Dynamic Logic Syntax State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ ϕ | � π � ϕ PDL sf Path formulas: π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∪ π | π c Combines features from ◮ Propositional Dynamic Logic [Fisher-Ladner 1979] ◮ Star-free regular expressions ◮ The calculus of relations Theorem: [Tarski-Givant 1987 (calculus of relations)] PDL sf and FO 3 are expressively equivalent 15 / 18

  48. A fragment of Star-free PDL 16 / 18

  49. A fragment of Star-free PDL State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ ϕ | � π � ϕ PDL sf Path formulas: π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∪ π | π c π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∩ π | ( ≤ · π · ≤ ) c | ( ≤ · π · ≥ ) c | PDL int sf ( ≥ · π · ≤ ) c | ( ≥ · π · ≥ ) c 16 / 18

  50. A fragment of Star-free PDL State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ ϕ | � π � ϕ PDL sf Path formulas: π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∪ π | π c π ::= ≤ | R | { ϕ } ? | π − 1 | π · π | π ∩ π | ( ≤ · π · ≤ ) c | ( ≤ · π · ≥ ) c | PDL int sf ( ≥ · π · ≤ ) c | ( ≥ · π · ≥ ) c Lemma : ∀ π ∈ PDL int sf , � π � is interval-preserving 16 / 18

  51. Equivalences over interval-preserving structures PDL int FO sf FO 3 PDL sf 17 / 18

  52. Equivalences over interval-preserving structures PDL int FO sf def. def. FO 3 PDL sf 17 / 18

  53. Equivalences over interval-preserving structures PDL int FO sf def. def. FO 3 PDL sf trivial induction 17 / 18

  54. Equivalences over interval-preserving structures PDL int FO sf def. def. FO 3 PDL sf trivial induction ◮ State formula ϕ ∈ PDL sf ϕ FO ( x ) ∈ FO � ◮ Path formula π ∈ PDL sf π FO ( x, y ) ∈ FO � 17 / 18

  55. Equivalences over interval-preserving structures PDL int FO sf def. def. FO 3 PDL sf trivial induction ◮ State formula ϕ ∈ PDL sf ϕ FO ( x ) ∈ FO � ∃ y.π FO ( x, y ) ∧ ϕ FO ( y ) � π � ϕ � ◮ Path formula π ∈ PDL sf π FO ( x, y ) ∈ FO � ∃ z.π FO 1 ( x, z ) ∧ π FO π 1 · π 2 2 ( z, y ) � 17 / 18

  56. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . 17 / 18

  57. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . 17 / 18

  58. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Atomic formulas, disjunction: easy 17 / 18

  59. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Negation: Express π c using ( ≤ · π · ≤ ) c , ( ≤ · π · ≥ ) c , ( ≥ · π · ≤ ) c , ( ≥ · π · ≥ ) c . 17 / 18

  60. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Existential quantification: Similar to the example before. 17 / 18

  61. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Existential quantification: Similar to the example before. ∃ x. � i π FO i ( x i , x ) 17 / 18

  62. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Existential quantification: Similar to the example before. ∃ x. � i π FO i ( x i , x ) � �� � intersection of n intervals 17 / 18

  63. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Existential quantification: Similar to the example before. ∃ x. � i π FO i ( x i , x ) � �� � intersection of n intervals π j π i ϕ x j x i ∃ x 17 / 18

  64. Equivalences over interval-preserving structures ? PDL int FO sf Any FO formula Φ( x 1 , . . . , x n ) is equivalent to a finite positive boolean combination of formulas of the form π FO ( x i , x j ) , where π ∈ PDL int sf . Proof: by induction on Φ . ◮ Existential quantification: Similar to the example before. ∃ x. � ≡ � i,j ( π i · { ϕ } ? · π − 1 i π FO j ) FO ( x i , x j ) i ( x i , x ) � �� � � �� � intersection of n intervals pairwise intersections π j π i ϕ x j x i ∃ x 17 / 18

  65. Conclusion ◮ Over linearly ordered structures with interval-preserving binary relations, FO = PDL sf = FO 3 18 / 18

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