On Local Domain Symmetry for Model Expansion Jo Devriendt, Bart - PowerPoint PPT Presentation

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