Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion On Local Domain Symmetry for Model Expansion Jo Devriendt, Bart Bogaerts, Maurice Bruynooghe, Marc Denecker University of Leuven / Aalto University October 21, 2016 1 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Intro General symmetry definition Given a vocabulary Σ, theory T , and domain D , a symmetry σ for T is a permutation on the set of D ,Σ-structures Γ D such that for all I ∈ Γ D : I | = T iff σ ( I ) | = T 2 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Intro General symmetry definition Given a vocabulary Σ, theory T , and domain D , a symmetry σ for T is a permutation on the set of D ,Σ-structures Γ D such that for all I ∈ Γ D : I | = T iff σ ( I ) | = T Why study symmetry? speeding up search – symmetry breaking avoid parts of the search space symmetrical to failed parts 3 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Outline Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion 4 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Prelims: First-Order Logic (FO) • vocabulary Σ of (function and predicate) symbols S / k • Σ-theory T • Σ-structure I • domain D • interpretations S I for all S ∈ Σ Semantics captured by satisfiability relation: I | = T 5 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Prelims: First-Order Logic (FO) • vocabulary Σ of (function and predicate) symbols S / k • Σ-theory T • Σ-structure I • domain D • interpretations S I for all S ∈ Σ Semantics captured by satisfiability relation: I | = T In ASP: program ↔ theory, set of facts ↔ structure. 6 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Prelims: First-Order Logic (FO) • vocabulary Σ of (function and predicate) symbols S / k • Σ-theory T • Σ-structure I • domain D • interpretations S I for all S ∈ Σ Semantics captured by satisfiability relation: I | = T In ASP: program ↔ theory, set of facts ↔ structure. For the rest of the talk: vocabulary and domain are implicit and fixed. 7 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Running example: graph coloring T gc : ∀ x 1 y 1 : Edge ( x 1 , y 1 ) ⇒ Color ( x 1 ) � = Color ( y 1 ) ∀ x 2 y 2 : Edge ( x 2 , y 2 ) ⇒ V ( x 2 ) ∧ V ( y 2 ) ∀ x 3 : C ( Color ( x 3 )) I gc : V I gc = { t , u , v , w } C I gc = { r , g , b } Edge I gc = { ( t , u ) , ( u , v ) , ( v , w ) , ( w , t ) } Color I gc = t �→ r , u �→ g , v �→ b , w �→ g , r �→ r , g �→ g , b �→ b u v g w r t b Note: I gc | = T gc 8 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Symmetry for a theory General symmetry definition Given a vocabulary Σ, theory T , and domain D , a symmetry σ for T is a permutation on the set of D ,Σ-structures Γ D such that for all I ∈ Γ D : I | = T iff σ ( I ) | = T Symmetries compose to symmetry groups 9 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Global Domain Symmetry Permutation π on D induces permutation σ π on Γ D : ( π ( d 1 ) , . . . , π ( d n )) ∈ P σ π ( I ) iff ( d 1 , . . . , d n ) ∈ P I f σ π ( I ) ( π ( d 1 ) , . . . , π ( d n )) = π ( d 0 ) iff f I ( d 1 , . . . , d n ) = d 0 Let’s call such induced σ π a Global Domain Symmetry for T . Intuitively, domain renaming π preserves satisfiability: σ π ( I ) | = T iff I | = T 10 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Graph coloring ctd. Let π = ( v r ), so σ π maps u v g w r t b to u r d g w v t d b which still models T gc . 11 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Connectively closed argument positions More precise notion of domain symmetry: apply π only on limited set of argument positions A . • Argument position S | n with S / k ∈ Σ and 1 ≤ n ≤ k denotes S ’s n th argument. f | 0 is output argument position. 12 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Connectively closed argument positions More precise notion of domain symmetry: apply π only on limited set of argument positions A . • Argument position S | n with S / k ∈ Σ and 1 ≤ n ≤ k denotes S ’s n th argument. f | 0 is output argument position. • Argument positions are connected under theory T if one occurs as subterm of the other, if they are connected by =, or if they are connected by quantified variables. 13 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Connectively closed argument positions More precise notion of domain symmetry: apply π only on limited set of argument positions A . • Argument position S | n with S / k ∈ Σ and 1 ≤ n ≤ k denotes S ’s n th argument. f | 0 is output argument position. • Argument positions are connected under theory T if one occurs as subterm of the other, if they are connected by =, or if they are connected by quantified variables. • A set of argument positions A is connectively closed under T if no other argument positions of Σ are connected to A under T . Intuitively, a partition of connectively closed argument positions under T corresponds to a well-defined typing of T . 14 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Graph coloring ctd. T gc : ∀ x 1 y 1 : Edge ( x 1 , y 1 ) ⇒ Color ( x 1 ) � = Color ( y 1 ) ∀ x 2 y 2 : Edge ( x 2 , y 2 ) ⇒ V ( x 2 ) ∧ V ( y 2 ) ∀ x 3 : C ( Color ( x 3 )) 15 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Graph coloring ctd. T gc : ∀ x 1 y 1 : Edge ( x 1 , y 1 ) ⇒ Color ( x 1 ) � = Color ( y 1 ) ∀ x 2 y 2 : Edge ( x 2 , y 2 ) ⇒ V ( x 2 ) ∧ V ( y 2 ) ∀ x 3 : C ( Color ( x 3 )) Connectively closed argument position partition under T gc : • { V | 1 , Edge | 1 , Edge | 2 , Color | 1 } • { C | 1 , Color | 0 } 16 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Local Domain Symmetry Permutation π on D and argument position set A induce permutation σ A π on Γ D : π ( I ) iff ( d 1 , . . . , d n ) ∈ P I ( π A ( d 1 ) , . . . , π A ( d n )) ∈ P σ A f σ A π ( I ) ( π A ( d 1 ) , . . . , π A ( d n )) = π A ( d 0 ) iff f I ( d 1 , . . . , d n ) = d 0 where π A applies π only on argument positions in A . 17 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Local Domain Symmetry Permutation π on D and argument position set A induce permutation σ A π on Γ D : π ( I ) iff ( d 1 , . . . , d n ) ∈ P I ( π A ( d 1 ) , . . . , π A ( d n )) ∈ P σ A f σ A π ( I ) ( π A ( d 1 ) , . . . , π A ( d n )) = π A ( d 0 ) iff f I ( d 1 , . . . , d n ) = d 0 where π A applies π only on argument positions in A . If A is connectively closed under T , σ A π is a local domain symmetry of T . Intuitively, σ A π permutes domain element tuples according to some type A of T . 18 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Graph coloring ctd. Let π = ( v r ) and A = { V | 1 , Edge | 1 , Edge | 2 , Color | 1 } , so σ A π maps u v g w r t b to u r d g t w r b which still models T gc . 19 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Local Domain Symmetry • So far, so good: more or less known in literature (MACE, SEM, Paradox, ...) 20 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Local Domain Symmetry • So far, so good: more or less known in literature (MACE, SEM, Paradox, ...) • What if we have to take pre-interpreted symbols into account? → Model eXpansion 21 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion First-Order Model Expansion (MX) In: • vocabulary Σ = Σ in ∪ Σ out (Σ in ∩ Σ out = ∅ ) • Σ-theory T • Σ in -structure I in • domain D • interpretations S I in to S ∈ Σ in Out: • Σ out -structure I out such that I in ⊔ I out | = T • same domain D • I in ⊔ I out merges both structures to a Σ-structure • or ”unsat” 22 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion First-Order Model Expansion (MX) In: • vocabulary Σ = Σ in ∪ Σ out (Σ in ∩ Σ out = ∅ ) • Σ-theory T • Σ in -structure I in • domain D • interpretations S I in to S ∈ Σ in Out: • Σ out -structure I out such that I in ⊔ I out | = T • same domain D • I in ⊔ I out merges both structures to a Σ-structure • or ”unsat” Shortened as MX ( T, I in ). 23 / 43
Intro Theory symmetry MX symmetry Efficient breaking More symmetry Conclusion Graph coloring ctd. T gc : ∀ x 1 y 1 : Edge ( x 1 , y 1 ) ⇒ Color ( x 1 ) � = Color ( y 1 ) ∀ x 2 y 2 : Edge ( x 2 , y 2 ) ⇒ V ( x 2 ) ∧ V ( y 2 ) ∀ x 3 : C ( Color ( x 3 )) I gcin : V I gcin = { t , u , v , w } C I gcin = { r , g , b } Edge I gcin = { ( t , u ) , ( u , v ) , ( v , w ) , ( w , t ) } u v g w r t b 24 / 43
Recommend
More recommend
