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Why Machines Can’t Think
(logically)
André Vellino
vellino@sympatico.ca
Carleton University
Cognitive Science Program
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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
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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.
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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)
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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
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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
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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)}
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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
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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)
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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
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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
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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.
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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
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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
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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.
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P
NP
NP-complete
Open Problem: is P =NP ?
n Steve Cook (1971)
P
NP
NP-complete
NP-I
P = NP
?oror
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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.
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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
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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
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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
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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
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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
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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;
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“Exponential” isn’t bad if
exponent is small
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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”
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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
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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.
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