Two-way cost automata and cost logi c s o v er infinite trees Achim - PowerPoint PPT Presentation

Two-way cost automata and cost logi c s o v er infinite trees Achim Blumens a th 1 , T hom a s C ol c om b et 2 , D enis K uper b erg 3 , Pa wel Pa rys 3 , a nd M i c h a el Va nden B oom 4 1 TU Da rmst a dt, 2 U niversit ´ e Pa ris D iderot, 3 U niversity of Wa rs a w, 4 U niversity of O xford CSL - LICS 2014 V ienn a , A ustri a 1 / 14

B oundedness questions F inite power property [Simon ’78, Hashiguchi ’79] given regu l a r l a ng ua ge L of finite w ords, is there n ∈ N s uc h th a t L ∗ = { є } ∪ L 1 ∪ L 2 ∪ ⋯ ∪ L n ? S t a r-height pro b lem [Ha shig uc hi ’88, K irsten ’05 ] given regul a r l a ngu a ge L of finite words a nd n ∈ N , is there a regul a r expression for L with a t most n nestings of K leene st a r? F ixpoint c losure b oundedness [B lumens a th+ O tto+ W eyer ’09 ] given a n MSO formul a φ ( x , X ) positive in X , is there n ∈ N su c h th a t the le a st fixpoint of φ over finite words is a lw a ys re ac hed within n iter a tions? 2 / 14

B oundedness questions The theory of regul a r c ost fun c tions is an extension of the theor y of reg ul a r l a ngu a ges th a t ca n b e used to solve these b oundedness questions in a uniform w a y. 3 / 14

B oundedness questions The theory of regul a r c ost fun c tions is an extension of the theor y of reg ul a r l a ngu a ges th a t ca n b e used to solve these b oundedness questions in a uniform w a y. B oundedness pro b lem I nst a n c e: fun c tion f ∶ D → N ∪ { ∞ } ( D is set of words or trees over some fixed finite a lph ab et A ) Q uestion: I s there n ∈ N su c h th a t for a ll stru c tures s ∈ D , f ( s ) ≤ n ? 3 / 14

C ost fun c tions over finite words [C ol c om b et’09 ] Regul a r C ost F un c tions nondeterministic cost automata cost MSO BS expressions stab ili za tion monoids B oundedness de c id ab le [C ol c om b et’09, B oj a´ n c zyk+ C ol c om b et’06 ] 4 / 14

C ost fun c tions over finite words C ost mon a di c se c ond-order logi c ( CMSO ) a ( x ) ∣ X ∣ ≤ N ����������������������������������� Atomic formu l a s: x ∈ X m u st o ccu r positi v ely ��� ���� ������������������������������� C onstru c tors: ∧ , ∨ , ¬ ∃ x ∃ X first-order mon a di c B oole a n se c ond-order c onne c tives qu a ntifi ca tion qu a ntifi ca tion 5 / 14

C ost fun c tions over finite words C ost mon a di c se c ond-order logi c ( CMSO ) a ( x ) ∣ X ∣ ≤ N ����������������������������������� Atomic formu l a s: x ∈ X m u st o ccu r positi v ely ��� ���� ������������������������������� C onstru c tors: ∧ , ∨ , ¬ ∃ x ∃ X first-order mon a di c B oole a n se c ond-order c onne c tives qu a ntifi ca tion qu a ntifi ca tion S em a nti c s � φ � ∶ A ∗ → N ∪ { ∞ } � φ � ( u ) ∶ = inf { n ∶ u ⊧ φ [ n / N ]} 5 / 14

C ost fun c tions over finite words C ost mon a di c se c ond-order logi c ( CMSO ) a ( x ) ∣ X ∣ ≤ N ����������������������������������� Atomic formu l a s: x ∈ X m u st o ccu r positi v ely ��� ���� ������������������������������� C onstru c tors: ∧ , ∨ , ¬ ∃ x ∃ X mon a di c first-order B oole a n qu a ntifi ca tion se c ond-order c onne c tives qu a ntifi ca tion S em a nti c s � φ � ∶ A ∗ → N ∪ { ∞ } � φ � ( u ) ∶ = inf { n ∶ u ⊧ φ [ n / N ]} E x a mple I f φ is in MSO , then � φ � ( u ) ∶ = { 0 if u ⊧ φ ∞ otherwise 5 / 14

C ost fun c tions over finite words C ost mon a di c se c ond-order logi c ( CMSO ) a ( x ) ∣ X ∣ ≤ N ����������������������������������� Atomic formu l a s: x ∈ X m u st o ccu r positi v ely ��� ���� ������������������������������� C onstru c tors: ∧ , ∨ , ¬ ∃ x ∃ X first-order mon a di c B oole a n qu a ntifi ca tion se c ond-order c onne c tives qu a ntifi ca tion S em a nti c s � φ � ∶ A ∗ → N ∪ { ∞ } � φ � ( u ) ∶ = inf { n ∶ u ⊧ φ [ n / N ]} E x a mple Ma ximum length of a b lo c k of a ’s φ ∶ = ∀ X (( block ( X ) ∧ ∀ x ( x ∈ X → a ( x )) → ∣ X ∣ ≤ N ) 5 / 14

C ost fun c tions over finite words [C ol c om b et’09 ] Regu l a r C ost Fu n c tions nondeterministic cost automata cost MSO BS expressions stab ili za tion monoids B oundedness de c id ab le [C ol c om b et’09, B oj a´ n c zyk+ C ol c om b et’06 ] 6 / 14

C ost fun c tions over finite words [C ol c om b et’09 ] Regu l a r C ost Fu n c tions nondeterministic cost automata cost MSO BS expressions stab ili za tion monoids B oundedness de c id ab le [C ol c om b et’09, B oj a´ n c zyk+ C ol c om b et’06 ] l a ngu a ge univers a lity, in c lusion, a nd emptiness de c id ab le 6 / 14

C ost fun c tions over finite words [C ol c om b et’09 ] Regu l a r C ost Fu n c tions nondeterministic cost automata cost MSO BS expressions stab ili za tion monoids B oundedness de c id ab le [C ol c om b et’09, B oj a´ n c zyk+ C ol c om b et’06 ] l a ngu a ge univers a lity, finite power property, in c lusion, a nd emptiness st a r height pro b lem, de c id ab le fixpoint c losure b oundedness, ... de c id ab le 6 / 14

T heory of regul a r c ost fun c tions The theor y of reg ul a r c ost fun c tions is a ro b ust de c id ab le extension of the theory of regul a r l a ngu a ges over: ✓ finite words [C ol c om b et ’09, B oj a n c zyk+ C ol c om b et ’06 ] ✓ infinite words [K uper b erg+ VB ’12, C ol c om b et unpu b lished ] ✓ finite trees [C ol c om b et+ L ¨ oding ’10 ] 7 / 14

T heor y of reg u l a r c ost f u n c tions The theor y of reg ul a r c ost fun c tions is a ro b ust de c id ab le extension of the theory of regul a r l a ngu a ges over: ✓ finite w ords [C ol c om b et ’09, B oj a n c zyk+ C ol c om b et ’06 ] ✓ infinite w ords [K uper b erg+ VB ’12, C ol c om b et unpu b lished ] ✓ finite trees [C ol c om b et+ L ¨ oding ’10 ] ? infinite trees 7 / 14

M otiv a ting open pro b lem M ostowski index pro b lem I nst a n c e: regu l a r l a ng ua ge L of infinite trees, a nd set { i , i + 1, . . . , j } Q uestion: I s there a nondeterministi c p a rity a utom a ton A using only priorities { i , i + 1, . . . , j } su c h th a t L = L ( A ) ? 8 / 14

M otiv a ting open pro b lem M ostowski index pro b lem I nst a n c e: regu l a r l a ng ua ge L of infinite trees, a nd set { i , i + 1, . . . , j } Q uestion: I s there a nondeterministi c p a rity a utom a ton A using only priorities { i , i + 1, . . . , j } su c h th a t L = L ( A ) ? R edu c ed to de c iding b oundedness for c ert a in c ost fun c tions over infinite trees [C ol c om b et+ L ¨ oding ’08 ] 8 / 14

C ost fun c tions over infinite trees Regu l a r C ost Fu n c tions a ltern a ting c ost-p a rity a utom a t a QW C ost Fu n c tions qu a si-we a k c ost a utom a t a B oundedness de c id ab le [K uper b erg+ VB ’11 ] we a k c ost a utom a t a spe c i a l ca se WCMSO of M ostowski index pro b lem 9 / 14

C ost fun c tions over infinite trees Regu l a r C ost Fu n c tions a ltern a ting c ost-p a rity a utom a t a QW C ost Fu n c tions qu a si-we a k c ost a utom a t a QWCMSO B oundedness de c id ab le [K uper b erg+ VB ’11 ] we a k c ost a utom a t a spe c i a l ca se WCMSO of M ostowski index pro b lem 9 / 14

C ost fun c tions over infinite trees Regu l a r C ost Fu n c tions a ltern a ting 2 - wa y/1-w a y c ost-p a rity a utom a t a QW C ost Fu n c tions 2-w a y/1-w a y qw c ost a utom a t a QWCMSO B oundedness de c id ab le [K uper b erg+ VB ’11 ] we a k c ost a utom a t a spe c i a l ca se WCMSO of M ostowski index pro b lem 9 / 14

C ost p a rit y a u tom a t a on infinite trees A = ⟨ A , Q , q 0 , δ , Ω ⟩ δ describes possib le moves Ω ∶ Q → P for E ve a nd A d a m, for a finite set of a nd a sso c i a ted c ounter ac tions priorities P (in c rement, reset, le a ve un c h a nged) n - acc ept a n c e g a me A × t ▶ P ositions in the g a me a re Q × dom ( t ) . ▶ E ve a nd A d a m sele c t the next position in the pl a y ba sed on δ . ▶ E ve is trying to ensure the pl a y h a s c ounter v a lue a t most n a nd the m a ximum priority o cc urring infinitely often in the pl a y is even. S em a nti c s � A � ( t ) ∶ = inf { n ∶ E ve wins the n - acc ept a n c e g a me A × t } 1 0 / 14

W e a k c ost a utom a t a a nd logi c over infinite trees W e a k c ost a utom a ton a ltern a ting c ost-p a rity a utom a ton su c h th a t 1 2 no c y c le visits b oth even a nd odd priorities 11 / 14

W e a k c ost a utom a t a a nd logi c over infinite trees W e a k c ost a utom a ton a ltern a ting c ost-p a rity a utom a ton su c h th a t 1 2 no c y c le visits b oth even a nd odd priorities W e a k c ost mon a di c se c ond-order logi c ( WCMSO ) S ynt a x like CMSO , b ut interpret se c ond-order qu a ntifi ca tion over finite sets 11 / 14