Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Why machines can't think (logically)

Related Books

Free with a 30 day trial from Scribd

See all

Related Audiobooks

Free with a 30 day trial from Scribd

See all
  • Be the first to comment

Why machines can't think (logically)

  1. 1. Why Machines Can’t Think (logically) André Vellino Carleton University Cognitive Science Program
  2. 2. 2 Outline n General Question: What do Logic, Complexity Theory and Automated Theorem Proving have to say about the question “can machines think?” (or at least, “can machines reason?”) n The role of “Logics” in AI n Results in the Complexity of Automated Theorem Proving Procedures n Why Machines Can’t Think: The Argument n The Logicist Response
  3. 3. 3 Role of “Logics” in AI “[AI is] the study of the computations that make it possible to perceive, reason and act” Pat Winston Role of “Logics” is to: n (a) to provide a formal system powerful enough to model various representations of knowledge, belief and action; n (b) to characterize mechanisms that specify permissible (aka “valid”) inferences.
  4. 4. 4 Examples of “Logics” for AI n 2-valued Propositional Calculus n First Order Predicate Calculus n Modal Logic (possibility and necessity) n Deontic Logic (permissions and obligations) n Relevance Logic (logic of “relevant” implication) n Conditional Logic (counterfactuals) n Default Logic (“common sense” reasoning) n Epistemic Logic (beliefs and knowledge) n Description Logics (knowledge representation)
  5. 5. 5 Example: Defeasible Reasoning if the traffic light is red then stop (defeasible rule) [in the absence of any further information, i.e. under normal conditions] Red ⊃ Stop if the light for going straight is green, then go (straight) (absolute rule) Green → Go
  6. 6. 6 Expressive Power of a Logic n Depends on the complexity of the semantics. Expressive power of model theory Other 1st-order Theories 2-valued Propositional Calculus 1st-order Predicate Calculus Other Propositional Calculi
  7. 7. 7 Propositional Calculus (PC) PC is the language whose well-formed formulas are composed of a finite combination of: Logical constants: { ∨, &, ≡, → } An infinite set of atomic propositional variables: {a, b, c,..., a1, b1, c1, ....}. e.g. (p → (q → p)) & ((~a ∨ b) ≡ (a → b)) Without Loss of Generality, consider only formulas in Conjunctive Normal Form or “sets of clauses” (clauses are disjunctions) e.g. {((p ∨ q ∨ r), (~p ∨ s), (r ∨ t)}
  8. 8. 8 Satisfiability / Unsatisfiability a set of clauses Σ = {C1, C2, ...Cn} is satisfiable if ∃ an assignment of truth values to literals in Σ such that C1 & C2 & ...&Cn is true SAT a set of clauses Σ = {C1, C2, ...Cn} is unsatisfiable if no assignments of truth values to literals in Σ are such that C1 & C2 & ...&Cn is true Co-SAT
  9. 9. 9 Theorem Provers for co-SAT n To prove T is a tautology, assume ~T and prove that ∅ follows using a theorem prover such as: n Truth Tables (Wittgenstein / Frege / Carroll) n Semantic Tableaux (Beth) n Resolution (Robinson / Davis-Putnam) n Sequent Calculus (Gentzen Systems) n Axioms w/ substitution (Frege Systems)
  10. 10. 10 Example 1: Semantic Tableaux Simple example: prove the inconsistency of (a v b) & (e v f) & (~a v b) & ~b i.e. {ab, ef,~ab, ~b} b X ~a ~b X b X ~a ~a b X ~a X ~c X ~b ϑ a b ~b X fe
  11. 11. 11 Example 2: Resolution Resolution: a ∪ Β & ~a ∪ C ∴ Β ∪ C For the set of clauses {ab, ef,~ab,~b} 1) ab premise 2) ~ab premise 3) ~b premise 4) b by resolving on a in 1 & 2 5) ∅ by resolving on b in 4 & 3
  12. 12. 12 Computability, Decidability and Feasibility n Computable n There exists a Turing Machine (“decision procedure” / “algorithm”) that halts. n Decidable n Given {Σ, T} it is computable whether Σ |− T or whether Σ |− ∼ T n Feasibly Decidable n Decidable by a Turing Machine in polynomial time.
  13. 13. 13 Polynomial vs. Exponential n Polynomial complexity n Time (space) grows as a function nk where n is proportional to the size of the input and k is a constant n Exponential complexity n Time (space) grows as a function kn where n is proportional to the size of the input and k is a constant
  14. 14. 14 The Class P n P is the class of languages recognizable by a deterministic Turing Machine in polynomial time. Example: n tautology (falsifiability) of propositional biconditionals without negation ((a ≡ b) ≡ (c ≡ b)) ≡ (a ≡ c) n Integer divisibility (indivisibility) by 2 n co-P is the complement of P. P = co-P
  15. 15. 15 The Class NP / NP-Complete NP is the class of languages recognizable by a non-deterministic Turing machine in polynomial time e.g.: all problems in P all "guess and verify" problems such as SAT, 3-SAT Traveling Salesman, Subgraph Isomorphism co-NP is the class of languages in the complement of NP e.g.: co-SAT L is in NP-complete if, for every problem L' in NP there exists a polynomial time transformation from L' to L.
  16. 16. 16 P NP NP-complete Open Problem: is P =NP ? n Steve Cook (1971) P NP NP-complete NP-I P = NP ?oror
  17. 17. 17 Strategy for Proof that P ≠ NP if P = NP then co-NP = NP (since co-P = P ) ∴ co-NP ≠ NP implies P ≠ NP ∃ an efficient proof method for TAUT iff co-NP = NP. ∴ if no theorem proving procedure can produce proofs for all tautologies that are a polynomial function of the length of the tautology (i.e. the lengths of all proofs for theorems are exponentially long), then P ≠ NP.
  18. 18. 18 Summary n Verify SAT P p ∨ q & r ∨ ~q T F T T n Find SAT NP p ∨ q & r ∨ ~q ? ? ? ? n Prove UNSAT co-NP a ∨ b & ~a ∨ b & ~b
  19. 19. 19 Complexity vs. AI n Complexity Game (co-NP=NP?) n To find “hard examples” for increasingly general propositional theorem proving procedures. n AI Reasoning Game n To find “efficient” and practical theorem- proving procedures in Logics for AI
  20. 20. 20 Hard Problems for Resolution n Pigeon Hole Clauses (Haken ‘85) n balls can't fit into n-1 holes ~ball_1_is_in_hole_1 v ~ball_2_is_in_hole_1 ~ball_1_is_in_hole_1 v ~ball_3_is_in_hole_1 ~ball_2_is_in_hole_1 v ~ball_3_is_in_hole_1 ~ball_1_is_in_hole_2 v ~ball_2_is_in_hole_2 ~ball_1_is_in_hole_2 v ~ball_3_is_in_hole_2 ~ball_2_is_in_hole_2 v ~ball_3_is_in_hole_2 each hole can fit only one ball n x (n-1)2 clauses ball_1_is_in_hole_1 v ball_1_is_in_hole_2 ball_2_is_in_hole_1 v ball_2_is_in_hole_2 ball_3_is_in_hole_1 v ball_3_is_in_hole_2 3 balls can fit into 2 holes n clauses
  21. 21. 22 Search-Space vs. Proof Length n For problems in NP (SAT), the search space is exponentially large but the proof is polynomial n For problems in Co-NP (co-SAT), the minimal length proof is exponential and the search space even larger
  22. 22. 23 Why Machines Can’t Think n If (any) “reasoning” is done by “logical rule- following” and n If any problems that people solve (feasibly) can’t be solved (feasibly) by following rules of logic Then, either n people don't reason logically or n logic is no foundation for artificial intelligence
  23. 23. 24 A Few Responses 1) Worst-case complexity is irrelevant because average-case complexity is what matters in practice; 2) Exponential growth is irrelevant if the exponent is small for all realistic inputs 3) There are efficient theorem proving methods that are sound but incomplete; 4) Computational complexity can be overcome by increasing the power of the logic;
  24. 24. Selman, Mitchell & Levesque ‘96
  25. 25. 26 “Exponential” isn’t bad if exponent is small
  26. 26. 27 Devise Sound, Tractable but Incomplete ATPs n Vivid Reasoning (Levesque) n Wants to make “believers out of computers” and devise incomplete but tractable logics that are psychologically realistic (e.g. capture the logic of “mental models” theory – Johnson-Laird) n Bounded Rationality (Cherniak) n “Rational agents” need to use “a better than random, but not perfect, gambling strategy for identifying sound inferences”
  27. 27. 28 Use “Stronger” Logics n People don’t map ordinary problems (e.g. pigeon-hole problem) into languages (PC) that are computationally hard n Use a different, more powerful logic in which propositionally-hard-to-prove formulae are easy to prove (e.g. extended resolution) n Problem: punt the exponential-length-of-proof constraint to a search-for-a-short-proof problem
  28. 28. 29 Concluding Remarks n If “language of thought” has a structure that can represented as or even modeled by a logic then you need to characterize what is “infeasibly computable” about it and why; n If you can understand what inferences are “cognitively hard” for people experimentally, then you can test hypotheses about what “logics” are being used in people to draw inferences.

    Be the first to comment

    Login to see the comments

  • dmitriijanouski

    Dec. 3, 2014


Total views


On Slideshare


From embeds


Number of embeds