logic is everywhere First-Order Logic la l ogica est a por todas - - PowerPoint PPT Presentation

logika je vˇ sude

logic is everywhere

logika je svuda Logika ada di mana-mana

Hikmat har Jaga Hai a l´

• gica est´

a em toda parte

Logik ist ¨ uberall

la logique est partout

Mantık her yerde la logica ` e dappertutto

la l´

• gica est´

a por todas partes

Logica este peste tot

• SHRUTI and Reflexive Reasoning

Steffen H¨

• lldobler

International Center for Computational Logic Technische Universit¨ at Dresden Germany

◮ First-Order Logic ◮ Reflexive Reasoning ◮ SHRUTI ◮ A Logical Reconstruction

Steffen H¨

• lldobler

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¨

• lldobler 1990, H¨
• lldobler, Kurfess 1992)

◮ First-Order Logic Programs and the Core Method ⊲ Initial Approach ⊲ Construction of Approximating Networks

Steffen H¨

• lldobler

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 108 rules and facts. ◮ Wanted: “Reflexive” decisions within sublinear time. ◮ Shastri, Ajjanagadde 1993: SHRUTI.

Steffen H¨

• lldobler

SHRUTI and Reflexive Reasoning 3

SHRUTI – Knowledge Base

◮ Finite set of constants C, finite set of variables V. ◮ Rules: ⊲ (∀X1 . . . Xm) (p1(. . .) ∧ . . . ∧ pn(. . .) → (∃Y1 . . . Yk p(. . .)). ⊲ p, pi, 1 ≤ i ≤ n, are multi-place predicate symbols. ⊲ Arguments of the pi: variables from {X1, . . . , Xm} ⊆ V. ⊲ Arguments of p are from {X1, . . . , Xm} ∪ {Y1, . . . , Yk} ∪ C. ⊲ {Y1, . . . , Yk} ⊆ V. ⊲ {X1, . . . , Xm} ∩ {Y1, . . . , Yk} = ∅. ◮ Facts and queries (goals): ⊲ (∃Z1 . . . Zl) q(. . .). ⊲ Multi-place predicate symbol q. ⊲ Arguments of q are from {Z1, . . . , Zl} ∪ C. ⊲ {Z1, . . . , Zl} ⊆ V.

Steffen H¨

• lldobler

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¨

• lldobler

SHRUTI and Reflexive Reasoning 5

SHRUTI – Example

◮ Rules P = {

• wns(Y, Z) ← gives(X, Y, Z),
• wns(X, Y ) ← buys(X, Y ),

can-sell(X, Y ) ← owns(X, Y ), gives(john, josephine, book), (∃X) buys(john, X),

• wns(josephine, ball)

}, ◮ Queries: can-sell(josephine, book) ❀ yes (∃X) owns(josephine, X) ❀ yes {X → book} {X → ball}

Steffen H¨

• lldobler

SHRUTI and Reflexive Reasoning 6

SHRUTI : The Network

✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆

gives

❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ♠ ♠ ♠ ♠ ♠

buys

✲ ✲ ✟ ✟ ✟ ✟ ❍ ❍ ❍ ❍ ✻ ✻ r r r r r r r r

from john from jos. from book

r

from john

• ✒

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ■

• ✠

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘

• ❄

❄ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆

• wns

can-sell

❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ♠ ♠ ♠ ♠ ❅ ❅ ❅ r r ❅ ❅ r r ❅ ❅ ❅ ❅ ❅ ❅ ✲✟ ✟ ❍ ❍ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ■ ✻ ❄ ❄ ❄

josephine john ball book

• ⑥

⑥

• ⑥

⑥

• ⑥

⑥

• ⑥

⑥

◮

• Steffen H¨
• lldobler

SHRUTI and Reflexive Reasoning 7

Solving the Variable Binding Problem

book john ball josephine can–sell △ can–sell ▽ can–sell 1st arg can–sell 2nd arg

• wns △
• wns ▽
• wns 1st arg
• wns 2nd arg
• wns ✄

gives △ gives ▽ gives 1st arg gives 2nd arg gives 3nd arg gives ✄ buys △ buys ▽ buys 1st arg buys 2nd arg buys ✄ Steffen H¨

• lldobler

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

• f 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 ◮ ROBIN (Lange, Dyer 1989): signatures instead of phases. ◮ Biological plausibility. ◮ Trading expressiveness for time and size.

Steffen H¨

• lldobler

SHRUTI and Reflexive Reasoning 9

A Logical Reconstruction of SHRUTI

◮ Beringer, H¨

• lldobler 1993

◮ The example revisited

← can-sell(josephine, book).

✁ ✁ ✁ ✁ ✁ ✁

can-sell(X, Y ) ← owns(X, Y ).

• ❍❍❍❍❍❍❍❍❍❍❍

❍ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭

• wns(Y, Z) ← gives(X, Y, Z).

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

• wns(josephine, ball).
• wns(X, Y ) ← buys(X, Y ).
• gives(john, josephine, book).

buys(john, c).

Steffen H¨

• lldobler

SHRUTI and Reflexive Reasoning 10

A Logical Reconstruction of SHRUTI

◮ Beringer, H¨

• lldobler 1993

◮ The example revisited

← can-sell(josephine, book).

✁ ✁ ✁ ✁ ✁ ✁

can-sell(X, Y ) ← owns(josephine, book).

• ❍❍❍❍❍❍❍❍❍❍❍

❍ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭

• wns(josephine, book)← gives(X, Y, Z).

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

• wns(josephine, ball).
• wns(john, c)← buys(X, Y ).
• gives(john, josephine, book).

buys(john, c).

◮ Reflexive reasoning is reasoning by reduction.

Steffen H¨

• lldobler

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¨

• lldobler

SHRUTI and Reflexive Reasoning 12

Literature

◮ Beringer, H¨

• lldobler 1993: On the Adequateness of the Connection Method.

In: Proceedings of the AAAI National Conference on Artificial Intelligence, 9-14. ◮ H¨

• lldobler 1990: A Structured Connectionist Unification Algorithm. In:

Proceedings of the AAAI National Conference on Artificial Intelligence, 587-593. ◮ H¨

• lldobler, 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¨

• lldobler

SHRUTI and Reflexive Reasoning 13