Effective interpol a tion for gu a rded logi c s Micha el B enedikt 1 - PowerPoint PPT Presentation

Effective interpol a tion for gu a rded logi c s Micha el B enedikt 1 , Ba lder ten Ca te 2 , M i c h a el Va nden B oom 1 1 U ni v ersit y of Ox ford 2 L ogi cB lo x a nd UC Sa nt a C r uz L og IC S emin a r a t I mperi a l C ollege L ondon D e c em b er 20 14 1 / 20

Some decidab le fr a gments of first-order logi c FO ML ML finite model propert y ✓ tree-like model propert y ✓ C r a ig interpol a tion ✓ 2 / 20

Some decidab le fr a gments of first-order logi c constrain number of variab les FO 2 FO ML FO 2 ML finite model propert y ✓ ✓ tree-like model propert y ✓ ✗ C r a ig interpol a tion ✓ ✗ 2 / 20

Some decidab le fr a gments of first-order logi c constrain number of variab les FO 2 c onstr a in q ua ntifi ca tion [A ndr ´ ek a , va n B enthem, FO N´ emeti ’95-’98 ] ML GF ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) FO 2 ML GF finite model propert y ✓ ✓ ✓ tree-like model propert y ✓ ✗ ✓ C r a ig interpol a tion ✓ ✗ ✗ 2 / 20

Some decidab le fr a gments of first-order logi c constrain number of variab les FO 2 c onstr a in q ua ntifi ca tion [A ndr ´ ek a , va n B enthem, FO N´ emeti ’95-’98 ] ML GF ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) UNF c onstr a in neg a tion [ ten Ca te, S ego u fin ’11 ] ∃ x ( ψ ( xy )) ¬ ψ ( x ) FO 2 ML GF UNF finite model propert y ✓ ✓ ✓ ✓ tree-like model propert y ✓ ✗ ✓ ✓ C r a ig interpol a tion ✓ ✗ ✗ ✓ 2 / 20

Some decidab le fr a gments of first-order logi c constrain number of variab les FO 2 c onstr a in q ua ntifi ca tion [A ndr ´ ek a , va n B enthem, FO N´ emeti ’95-’98 ] ML GF ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GNF UNF c onstr a in neg a tion [ ten Ca te, S ego u fin ’11 ] [B´ a r ´ a n y , ten Ca te, S ego u fin ’11 ] ∃ x ( ψ ( xy )) G ( xy ) ∧ ¬ ψ ( xy ) FO 2 ML GF UNF GNF finite model propert y ✓ ✓ ✓ ✓ ✓ tree-like model propert y ✓ ✗ ✓ ✓ ✓ C r a ig interpol a tion ✓ ✗ ✗ ✓ ✓ 2 / 20

Int erpol a tion φ ψ ⊧ 3 / 20

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ only uses rel a tions in b oth φ a nd ψ 3 / 20

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -gu a rded 3- c y c le using R ” 4 / 20

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -gu a rded 3- c y c le using R ” b a c 4 / 20

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -gu a rded 3- c y c le using R ” b a c 4 / 20

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -gu a rded 3- c y c le using R ” b a c interpol a nt χ ∶ = ∃ xyz ( Rxy ∧ Ryz ∧ Rzx ) “there is a 3- c y c le using R ” 4 / 20

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -gu a rded 3- c y c le using R ” b a c GNF interpol a nt χ ∶ = ∃ xyz ( Rxy ∧ Ryz ∧ Rzx ) “there is a 3- c y c le using R ” 4 / 20

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ only uses rel a tions in b oth φ a nd ψ T heorem ( B´ a r ´ a ny+ B enedikt+ten Ca te ’13) G iven GNF formul a s φ a nd ψ su c h th a t φ ⊧ ψ , there is a GNF interpol a nt χ ( b ut model theoreti c proof implies no b ound on size of χ ). 5 / 20

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ only uses rel a tions in b oth φ a nd ψ T heorem ( B´ a r ´ a ny+ B enedikt+ten Ca te ’13) G iven GNF formul a s φ a nd ψ su c h th a t φ ⊧ ψ , there is a GNF interpol a nt χ ( b ut model theoreti c proof implies no b ound on size of χ ). E ven when input is in GF , no ide a how to c omp u te interpol a nts (or other rewritings rel a ted to interpol a tion). 5 / 20

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ only uses rel a tions in b oth φ a nd ψ T heorem ( B´ a r ´ a ny+ B enedikt+ten Ca te ’13) G iven GNF formul a s φ a nd ψ su c h th a t φ ⊧ ψ , there is a GNF interpol a nt χ ( b ut model theoreti c proof implies no b ound on size of χ ). T heorem ( B enedikt+ten Ca te+ VB . ’14) G iven GNF formul a s φ a nd ψ su c h th a t φ ⊧ ψ , we ca n c onstr uc t a GNF interpol a nt χ of dou b ly exponenti a l DAG -size (in size of input). 5 / 20

Mosaics A mosaic τ ( a ) for φ is a c olle c tion of su b formul a s of φ over some gu a rded set a of p a r a meters. τ 1 ( ab ) τ 2 ( bc ) τ 3 ( d ) Raa Sb Sd ¬ Sa ¬ Rbb ¬ Sd ∃ z ( Rbz ∧ Sz ) ∃ yz ( Ryz ∧ Sz ) Rbc ∧ Sc ∀ z ( Rdz ) Sb Rcb Rba Sc Rdd ∨ Sd ⋯ ⋯ ⋯ 6 / 20

Mosaics A mosaic τ ( a ) for φ is a c olle c tion of su b formul a s of φ over some gu a rded set a of p a r a meters. τ 1 ( ab ) τ 2 ( bc ) τ 3 ( d ) Raa Sb Interna ll y ¬ Sa ¬ Rbb ∃ z ( Rbz ∧ Sz ) Rbc ∧ Sc in c onsistent Sb Rcb Rba Sc (e.g. S d & ¬ S d) ⋯ ⋯ 6 / 20

Mosaics A mosaic τ ( a ) for φ is a c olle c tion of su b formul a s of φ over some gu a rded set a of p a r a meters. τ 1 ( ab ) τ 2 ( bc ) τ 3 ( d ) Interna ll y a c b b in c onsistent (e.g. S d & ¬ S d) I ntern a lly c onsistent mos a i c s a re windows into a (gu a rded) pie c e of a stru c ture. 6 / 20

Linking mosaics Mos a i c s ca n b e linked together to fulfill a n existenti a l requirement if they a gree on a ll formul a s th a t use only sh a red p a r a meters. τ 1 a b ∃ z ( R b z ∧ Sz ) 7 / 20

Linking mosaics Mos a i c s ca n b e linked together to fulfill a n existenti a l requirement if they a gree on a ll formul a s th a t use only sh a red p a r a meters. τ 1 τ 2 a c b b ∃ z ( R b z ∧ Sz ) 7 / 20

Linking mosaics Mos a i c s ca n b e linked together to fulfill a n existenti a l requirement if they a gree on a ll formul a s th a t use only sh a red p a r a meters. τ 1 τ 2 a c b b ∃ z ( R b z ∧ Sz ) W e s a y a set S of mos a i c s is s a t u r a ted if every existenti a l requirement in a mos a i c τ ∈ S is fulfilled in τ or in some linked τ ′ ∈ S . 7 / 20

Mosaics Fix some set P of size 2 ⋅ width ( φ ) a nd let M φ b e the set of mos a i c s for φ over p a r a meters P . T he size of M φ is dou b ly exponenti a l in the size of φ . Theorem φ is s a tisfi ab le iff there is a s a tur a ted set S of intern a lly c onsistent mos a i c s from M φ th a t c ont a ins some τ with φ ∈ τ . 8 / 20

Mosaics Fix some set P of size 2 ⋅ width ( φ ) a nd let M φ b e the set of mos a i c s for φ over p a r a meters P . T he size of M φ is dou b ly exponenti a l in the size of φ . Theorem φ is s a tisfi ab le iff there is a s a tur a ted set S of intern a lly c onsistent mos a i c s from M φ th a t c ont a ins some τ with φ ∈ τ . τ 3 { } S = , , , τ 1 τ 2 τ 3 τ 4 8 / 20

Mosaics Fix some set P of size 2 ⋅ width ( φ ) a nd let M φ b e the set of mos a i c s for φ over p a r a meters P . T he size of M φ is dou b ly exponenti a l in the size of φ . Theorem φ is s a tisfi ab le iff there is a s a tur a ted set S of intern a lly c onsistent mos a i c s from M φ th a t c ont a ins some τ with φ ∈ τ . τ 3 τ 4 { } S = , , , τ 1 τ 2 τ 3 τ 4 8 / 20

Mosaics Fix some set P of size 2 ⋅ width ( φ ) a nd let M φ b e the set of mos a i c s for φ over p a r a meters P . T he size of M φ is dou b ly exponenti a l in the size of φ . Theorem φ is s a tisfi ab le iff there is a s a tur a ted set S of intern a lly c onsistent mos a i c s from M φ th a t c ont a ins some τ with φ ∈ τ . τ 3 τ 1 τ 4 { } S = , , , τ 1 τ 2 τ 3 τ 4 8 / 20

Mosaics Fix some set P of size 2 ⋅ width ( φ ) a nd let M φ b e the set of mos a i c s for φ over p a r a meters P . T he size of M φ is dou b ly exponenti a l in the size of φ . Theorem φ is s a tisfi ab le iff there is a s a tur a ted set S of intern a lly c onsistent mos a i c s from M φ th a t c ont a ins some τ with φ ∈ τ . τ 3 τ 1 τ 4 τ 2 { } S = , , , τ 1 τ 2 τ 3 τ 4 ⋮ 8 / 20

Mosaic elimin a tion a lgorithm for s a tisfi ab ilit y testing M φ τ 5 τ 4 τ 2 τ 6 τ 3 τ 7 τ 1 9 / 20

Mosaic elimin a tion a lgorithm for s a tisfi ab ilit y testing S t a ge 1. M φ Elimin a te mos a i c s with intern a l τ 5 in c onsisten c ies. τ 4 τ 2 τ 6 τ 3 τ 7 τ 1 9 / 20

Mosaic elimin a tion a lgorithm for s a tisfi ab ilit y testing S t a ge 1. M φ Elimin a te mos a i c s with intern a l τ 5 in c onsisten c ies. τ 4 S t a ge i + 1 . E limin a te mos a i c s with existenti a l requirements th a t ca n only b e τ 2 τ 6 fulfilled using mos a i c s elimin a ted in e a rlier st a ges. τ 3 τ 7 τ 1 9 / 20