Well-quasi-orders in Logic Sylvain Schmitz Panhellenic Logic - PowerPoint PPT Presentation

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-quasi-orders in Logic Sylvain Schmitz Panhellenic Logic Symposium, June 29, 2019 1/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Outline well-quasi-orders (wqo): ◮ robust notion ◮ selection of applications: verification algorithm termination proof theory relevance logic finite model theory preservation theorems database theory certain answers 2/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Outline well-quasi-orders (wqo): ◮ robust notion ◮ selection of applications: verification algorithm termination proof theory relevance logic finite model theory preservation theorems database theory certain answers 2/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? 3/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? 3/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory A One-Player Game ◮ over Q � 0 × Q � 0 ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? 3/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory If ( x 0 , y 0 ) � ( 0,0 ) , then choosing ( x j , y j ) = ( x 0 2 j , y 0 2 j ) wins. 4/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory A One-Player Game ◮ over N × N ( x 0, y 0 ) ◮ given initially ( x 0 , y 0 ) ( x 2, y 2 ) ◮ Eloise plays ( x j , y j ) s.t. ( x 1, y 1 ) ∀ 0 � i < j , x i > x j or y i > y j ◮ Can Eloise win, i.e. play indefinitely? 5/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . 6/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 3,4 ) ( 5,2 ) ( 2,3 ) 6/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 3,4 ) ( 5,2 ) ( 2,3 ) By the infinite Ramsey Theorem, there exists an infinite monochromatic subset of indices. 6/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Assume there exists an infinite sequence ( x j , y j ) j of moves over N 2 . Consider the pairs of indices i < j : color ( i , j ) purple if x i > x j but y i � y j , red if x i > x j and y i > y j , orange if y i > y j but x i � x j . ... ( 3,4 ) ( 5,2 ) ( 2,3 ) By the infinite Ramsey Theorem, there exists an infinite monochromatic subset of indices. In all cases, it implies the existence of an infinite decreasing sequence in N , a contradiction. 6/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions: ( X , � ) wqo i ff ◮ bad sequences are finite: x 0 , x 1 ,... is bad if ∀ i < j , x i � x j ◮ � is well-founded and has no infinite antichains ◮ finite basis property: ∅ � U ⊆ X has at least one and finitely many minimal elements ◮ ascending chain condition: any chain U 0 � U 1 � ··· of upwards-closed sets is finite ◮ etc. ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions: ( X , � ) wqo i ff ◮ bad sequences are finite: x 0 , x 1 ,... is bad if ∀ i < j , x i � x j ◮ � is well-founded and has no infinite antichains ◮ finite basis property: ∅ � U ⊆ X has at least one and finitely many minimal elements ◮ ascending chain condition: any chain U 0 � U 1 � ··· of upwards-closed sets is finite ◮ etc. ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions: ( X , � ) wqo i ff ◮ bad sequences are finite: x 0 , x 1 ,... is bad if ∀ i < j , x i � x j ◮ � is well-founded and has no infinite antichains ◮ finite basis property: ∅ � U ⊆ X has at least one and finitely many minimal elements ◮ ascending chain condition: any chain U 0 � U 1 � ··· of upwards-closed sets is finite ◮ etc. ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions: ( X , � ) wqo i ff ◮ bad sequences are finite: x 0 , x 1 ,... is bad if ∀ i < j , x i � x j ◮ � is well-founded and has no infinite antichains ◮ finite basis property: ∅ � U ⊆ X has at least one and finitely many minimal elements ◮ ascending chain condition: any chain U 0 � U 1 � ··· of upwards-closed sets is finite ◮ etc. ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions: ( X , � ) wqo i ff ◮ bad sequences are finite: x 0 , x 1 ,... is bad if ∀ i < j , x i � x j ◮ � is well-founded and has no infinite antichains ◮ finite basis property: ∅ � U ⊆ X has at least one and finitely many minimal elements ◮ ascending chain condition: any chain U 0 � U 1 � ··· of upwards-closed sets is finite ◮ etc. ◮ algebraic constructions 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Well-Quasi-Orders ◮ multiple equivalent definitions ◮ algebraic constructions ◮ Cartesian products (Dickson’s Lemma), ◮ finite sequences (Higman’s Lemma), ◮ disjoint sums, ◮ finite sets with Hoare’s quasi-ordering, ◮ finite trees (Kruskal’s Tree Theorem), ◮ graphs with minors (Robertson and Seymour’s Graph Minor Theorem), ◮ etc. 7/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Example: Ordinals ordinal: well-founded linear order bad sequences are descending sequences: α � � β i ff α > β 8/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Example: Dickson’s Lemma Lemma ( Dickson 1913) If ( X , � X ) and ( Y , � Y ) are two wqos, then ( X × Y , � × ) is a wqo, where � × is the product order- ing: ⇔ x � X x ′ ∧ y � Y y ′ . � x , y � � × � x ′ , y ′ � def Example ◮ ( N d , � × ) using the product ordering ◮ ( M ( X ) , � m ) for finite multiset embedding over finite X 9/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Example: Higman’s Lemma Lemma ( Higman 1952) If ( X , � ) is a wqo, then ( X ∗ , � ∗ ) is a wqo where � ∗ is the subword embedding ordering: � ∃ 1 � i 1 < ··· < i m � n , def a 1 ··· a m � ∗ b 1 ··· b n ⇔ � m j = 1 a j � A b i j . Example aba � ∗ baaacabbab 10/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Example: Bounded Tree-Depth Lemma ( Ding 1992) For all k , ( Graphs \ ↑ P k , ⊆ ) is wqo. Non-Examples ... ... 11/18

Well-Quasi-Orders Verification Proof Theory Finite Model Theory Database Theory Application: Algorithm Termination simple ( a , b ) � a 0 , b 0 , c 0 � c ← − 1 � a 1 , b 1 , c 1 � while a > 0 ∧ b > 0 . . . � a , b , c � ← − � a − 1, b ,2 c � � a i , b i , c i � or . . � � × � a , b , c � ← − � 2 c , b − 1,1 � . � a j , b j , c j � end ◮ in any execution, � a 0 , b 0 � ,..., � a n , b n � is a bad sequence over ( N 2 , � × ) , ◮ ( N 2 , � × ) is a wqo: all the runs are finite 12/18