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GPU-accelerated End-to-end Differentiable Planning and Reasoning Tim Rockt aschel Whiteson Research Lab, University of Oxford http://rockt.github.com Twitter: @ rockt tim.rocktaschel@cs.ox.ac.uk Talk at GTC Europe, ID 23372 12th of

  1. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  2. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Rule : head :– body . head : atom body : (possibly empty) list of atoms representing conjunction grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  3. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Rule : head :– body . head : atom body : (possibly empty) list of atoms representing conjunction grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Fact : ground rule (no free variables) with empty body, e.g., parentOf ( homer , bart ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  4. Example Knowledge Base 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , lisa ) . 3. parentOf ( homer , bart ) . 4. grandpaOf ( abe , lisa ) . 5. grandfatherOf ( abe , maggie ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 10/39

  5. Example Knowledge Base 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , lisa ) . 3. parentOf ( homer , bart ) . 4. grandpaOf ( abe , lisa ) . 5. grandfatherOf ( abe , maggie ) . 6. grandfatherOf ( X 1 , Y 1 ) :– fatherOf ( X 1 , Z 1 ) , parentOf ( Z 1 , Y 1 ) . 7. grandparentOf ( X 2 , Y 2 ) :– grandfatherOf ( X 2 , Y 2 ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 10/39

  6. Prolog Backward Chaining Example Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  7. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  8. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  9. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  10. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? 1 2 3 failure success failure { X / abe , Y / bart , Z / homer } Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  11. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? 1 2 3 failure success failure { X / abe , Y / bart , Z / homer } What about grandpaOf ( abe , bart )? Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  12. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  13. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  14. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit ( apple ), apple ∼ organge , isFruit ( orange )? Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  15. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit ( apple ), apple ∼ organge , isFruit ( orange )? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’ ( X , Y ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  16. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit ( apple ), apple ∼ organge , isFruit ( orange )? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’ ( X , Y ) But... leads to powerful inference mechanisms and proofs for predictions: fatherOf ( abe , homer ) . parentOf ( homer , lisa ) . parentOf ( homer , bart ) . grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . grandfatherOf ( abe , Q )? { Q / lisa } , { Q / bart } Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  17. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit ( apple ), apple ∼ organge , isFruit ( orange )? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’ ( X , Y ) But... leads to powerful inference mechanisms and proofs for predictions: fatherOf ( abe , homer ) . parentOf ( homer , lisa ) . parentOf ( homer , bart ) . grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . grandfatherOf ( abe , Q )? { Q / lisa } , { Q / bart } Fairly easy to debug and trivial to incorporate domain knowledge: Show to domain expert and let her change/add rules and facts Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  18. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  19. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  20. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  21. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  22. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) But... need large amount of training data Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  23. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) But... need large amount of training data No direct way of incorporating prior knowledge v grandfatherOf ( X , Y ) :– v fatherOf ( X , Z ) , v parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  24. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  25. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  26. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  27. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  28. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  29. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  30. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007) Neural-symbolic Connectionism Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  31. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007) Neural-symbolic Connectionism Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (Garcez and Zaverucha, 1999) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  32. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Inductive Logic Programming, e.g., Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007) Neural-symbolic Connectionism Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (Garcez and Zaverucha, 1999) First-order inference (no training of symbol representations): Unification Neural Networks (Holld¨ obler, 1990; Komendantskaya 2011), SHRUTI (Shastri, 1992), Neural Prolog (Ding, 1995), CLIP++ (Franca et al. 2014), Lifted Relational Networks (Sourek et al. 2015) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  33. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  34. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) v s , v i , v j ∈ R k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  35. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) v s , v i , v j ∈ R k f ( v s , v i , v j ) = v ⊤ s ( v i ⊙ v j ) � = v sk v ik v jk k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  36. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ s ( v i ⊙ v j ) � = v sk v ik v jk k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  37. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ f ( v s , v i , v j ) = s ( v i ⊙ v j ) � real( v s ) ⊤ (real( v i ) ⊙ real( v j )) = v sk v ik v jk + real( v s ) ⊤ (imag( v i ) ⊙ imag( v j )) k + imag( v s ) ⊤ (real( v i ) ⊙ imag( v j )) − imag( v s ) ⊤ (imag( v i ) ⊙ real( v j )) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  38. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ f ( v s , v i , v j ) = s ( v i ⊙ v j ) � real( v s ) ⊤ (real( v i ) ⊙ real( v j )) = v sk v ik v jk + real( v s ) ⊤ (imag( v i ) ⊙ imag( v j )) k + imag( v s ) ⊤ (real( v i ) ⊙ imag( v j )) − imag( v s ) ⊤ (imag( v i ) ⊙ real( v j )) Training Loss � L = − y log ( σ ( f ( v s , v i , v j ))) − (1 − y ) log (1 − σ ( f ( v s , v i , v j ))) r s ( e i , e j ) , y ∈ T Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  39. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  40. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  41. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , ∅ parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  42. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  43. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  44. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  45. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  46. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) Y / bart Z / homer · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  47. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  48. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · parentOf homer bart Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  49. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · parentOf homer bart 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) · · · 1. fatherOf ( abe , homer ) X / abe Y / bart X / abe Z / homer Y / bart Z / homer Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  50. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  51. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  52. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  53. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score X / Q Y / bart S Ψ S ρ Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  54. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

  55. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Discrete objects (rules, atoms etc.) are used to instantiate proof modules Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

  56. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Discrete objects (rules, atoms etc.) are used to instantiate proof modules Modules transform proof states into new proof states Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

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