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Hikmat har Jaga Hai SHRUTI and Reflexive Reasoning logika je svuda � � ��� � � �� ���� � Steffen H¨ olldobler � � �� Mantık her yerde International Center for Computational Logic logika je vˇ sude Technische Universit¨ at Dresden la logica ` e dappertutto Germany logic is everywhere ◮ First-Order Logic la l´ ogica est´ a por todas partes ◮ Reflexive Reasoning ��� � �� ��� ◮ SHRUTI � ��� � � �� ◮ A Logical Reconstruction Logik ist ¨ uberall Logika ada di mana-mana Logica este peste tot ��� � ���� ��� �������� �� � �� � a l´ ogica est´ a em toda parte la logique est partout Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 1

First-Order Logic ◮ Some Existing Approaches ⊲ Reflexive Reasoning and SHRUTI (Shastri, Ajjanagadde 1993) ⊲ Connectionist Term Representations ◮ Holographic Reduced Representations (Plate 1991) ◮ ◮ Recursive Auto-Associative Memory (Pollack 1988) ◮ ⊲ Horn logic and CHCL (H¨ olldobler 1990, H¨ olldobler, Kurfess 1992) ◮ First-Order Logic Programs and the Core Method ⊲ Initial Approach ⊲ Construction of Approximating Networks Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 2

Reflexive Reasoning ◮ Humans are capable of performing a wide variety of cognitive tasks with extreme ease and efficiency. ◮ For traditional AI systems, the same problems turn out to be intractable. ◮ Human consensus knowledge: about 10 8 rules and facts. ◮ Wanted: “Reflexive” decisions within sublinear time. ◮ Shastri, Ajjanagadde 1993: SHRUTI . Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 3

SHRUTI – Knowledge Base ◮ Finite set of constants C , finite set of variables V . ◮ Rules: ⊲ ( ∀ X 1 . . . X m ) ( p 1 ( . . . ) ∧ . . . ∧ p n ( . . . ) → ( ∃ Y 1 . . . Y k p ( . . . )) . ⊲ p, p i , 1 ≤ i ≤ n , are multi-place predicate symbols. ⊲ Arguments of the p i : variables from { X 1 , . . . , X m } ⊆ V . ⊲ Arguments of p are from { X 1 , . . . , X m } ∪ { Y 1 , . . . , Y k } ∪ C . ⊲ { Y 1 , . . . , Y k } ⊆ V . ⊲ { X 1 , . . . , X m } ∩ { Y 1 , . . . , Y k } = ∅ . ◮ Facts and queries (goals): ⊲ ( ∃ Z 1 . . . Z l ) q ( . . . ) . ⊲ Multi-place predicate symbol q . ⊲ Arguments of q are from { Z 1 , . . . , Z l } ∪ C . ⊲ { Z 1 , . . . , Z l } ⊆ V . Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 4

Further Restrictions ◮ Restrictions to rules, facts, and goals: ⊲ No function symbols except constants. ⊲ Only universally bound variables may occur as arguments in the conditions of a rule. ⊲ All variables occurring in a fact or goal occur only once and are existentially bound. ⊲ An existentially quantified variable is only unified with variables. ⊲ A variable which occurs more than once in the conditions of a rule must occur in the conclusion of the rule and must be bound when the conclusion is unified with a goal. ⊲ A rule is used only a fixed number of times. � Incompleteness. Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 5

SHRUTI – Example ◮ Rules P = { owns ( Y, Z ) ← gives ( X, Y, Z ) , owns ( X, Y ) ← buys ( X, Y ) , can-sell ( X, Y ) ← owns ( X, Y ) , gives ( john , josephine , book ) , ( ∃ X ) buys ( john , X ) , } , owns ( josephine , ball ) ◮ Queries: can-sell ( josephine , book ) yes ❀ ( ∃ X ) owns ( josephine , X ) yes { X �→ book } ❀ { X �→ ball } Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 6

SHRUTI : The Network josephine book john ball ✁ ❆ ❆ ✁ ⑥ ♠ ⑥ ♠ � � can-sell ✁ ❆ ❆ ✁ ✻ ❄ ❄ ❄ owns � ✁ ❆ ❆ ✁ ♠ ⑥ ♠ ⑥ � ✁ ❆ ❆ ✁ ❅ ✒ � ■ ❅ ❅ ■ � ❅ � ❅ ❅ � ❅ ❅ ❅ ❅ ❅ � � ❅ � ❅ � ❅ ❅ ❅ ❅ ❅ ❅ � � ❅ � ❅ � ❅ ❅ ❅ r ❅ � � � ❅ � ❅ ❅ ❅ ❅ r ❅ � � � � ❅ ❅ ❅ ❅ ❅ ❍ ❅ ✲✟ ❍ � � � � ❅ ❅ ❅ ❅ r r ❅ ✟ � ❅ � � � ❅ ❅ ❅ � � ❅ � ❅ � ❅ ❅ ❅ � ❅ � ❅ ❅ ❅ � ✠ � ❄ ❄ ❅ ❘ ❅ ❅ ❘ ❅ ❘ ✁ ❆ ❆ ✁ ♠ ⑥ ♠ ⑥ ♠ ✁ ❆ ❆ ✁ ⑥ ♠ ♠ ⑥ gives � � � buys ✁ ❆ ❆ ✁ ✁ ❆ ❆ ✁ ✻ ✻ r from john r r from jos. from john r from book ❍ ❍ ✲ ❍ ✲ ❍ r r r r r ◮ ✟ ✟ ✟ ✟ Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 7

Solving the Variable Binding Problem buys ✄ buys 2nd arg buys 1st arg buys ▽ buys △ gives ✄ gives 3nd arg gives 2nd arg gives 1st arg gives ▽ gives △ owns ✄ owns 2nd arg owns 1st arg owns ▽ owns △ can–sell 2nd arg can–sell 1st arg can–sell ▽ can–sell △ josephine ball john book Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 8

SHRUTI – Some Remarks ◮ Answers are derived in time proportional to depth of search space. ◮ Number of units as well as of connections is linear in the size of the knowledge base. ◮ Extensions: ⊲ compute answer substitutions ⊲ allow a fixed number of copies of rules ⊲ allow multiple literals in the body of a rule ⊲ built in a taxonomy ⊲ support of negation and inconsistency ⊲ simple learning using Hebbian learning ◮ R OBIN (Lange, Dyer 1989): signatures instead of phases. ◮ Biological plausibility. ◮ Trading expressiveness for time and size. Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 9

A Logical Reconstruction of SHRUTI ◮ Beringer, H¨ olldobler 1993 ◮ The example revisited ← can-sell ( josephine , book ) . ✁ ✁ ✁ ✁ ✁ ✁ can-sell ( X, Y ) ← owns ( X, Y ) . ✭ ✭ ✭ ❍❍❍❍❍❍❍❍❍❍❍ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ � ❍ owns ( Y, Z ) ← gives ( X, Y, Z ) . owns ( josephine , ball ) . owns ( X, Y ) ← buys ( X, Y ) . ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � gives ( john , josephine , book ) . buys ( john , c ) . Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 10

A Logical Reconstruction of SHRUTI ◮ Beringer, H¨ olldobler 1993 ◮ The example revisited ← can-sell ( josephine , book ) . ✁ ✁ ✁ ✁ ✁ ✁ can-sell ( X, Y ) ← owns ( josephine , book ) . ✭ ✭ ✭ ❍❍❍❍❍❍❍❍❍❍❍ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ ✭ � ✭ ✭ ✭ ✭ ✭ � ❍ owns ( josephine , book ) ← gives ( X, Y, Z ) . owns ( josephine , ball ) . owns ( john , c ) ← buys ( X, Y ) . ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � ✟ ✟ � gives ( john , josephine , book ) . buys ( john , c ) . ◮ Reflexive reasoning is reasoning by reduction. Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 11

Influence of Restrictions in SHRUTI ◮ Only constants � no complex data structures by unification. ◮ Only universally quantified variables in conditions of rules. ◮ All variables in a fact are existentially bound and removed by skolemization. � All facts are ground. ◮ Existentially bound variables in the head of a rule are replaced by Skolem functions. � They can only be unified with variables; moreover, such bindings are not propagated. ◮ Variables which occur more than once in the conditions of a rule must ⊲ also occur in the head of a rule ⊲ be bound to a constant when the head is unified with a goal � Subgoals in conditions can be solved independently and in parallel. ◮ Rules are used only a fixed number of times � Logic becomes decidable. � The underlying logic is decidable in linear time and linear space. Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 12

Literature ◮ Beringer, H¨ olldobler 1993: On the Adequateness of the Connection Method. In: Proceedings of the AAAI National Conference on Artificial Intelligence , 9-14. ◮ H¨ olldobler 1990: A Structured Connectionist Unification Algorithm. In: Proceedings of the AAAI National Conference on Artificial Intelligence , 587-593. ◮ H¨ olldobler, Kurfess 1992: CHCL – A Connectionist Inference System. In: Parallelization in Inference Systems , Lecture Notes in Artificial Intelligence, 590, 318-342. ◮ Lange, Dyer 1989: High-Level Inferencing in a Connectionist Network. Connection Science 1, 181-217. ◮ Plate 1991: Holographic Reduced Representations. In Proceedings of the International Joint Conference on Artificial Intelligence , 30-35. ◮ Pollack 1988: Recursive auto-associative memory: Devising compositional distributed representations. In: Proceedings of the Annual Conference of the Cognitive Science Society , 33-39. ◮ Shastri, Ajjanagadde 1993: From Associations to Systematic Reasoning: A Connectionist Representation of Rules, Variables and Dynamic Bindings using Temporal Synchrony. Behavioural and Brain Sciences 16, 417-494. Steffen H¨ olldobler SHRUTI and Reflexive Reasoning 13