TOWARDS A PROOF THEORY OF ANALOGICAL REASONING M. BAAZ VIENNA - PDF document

TOWARDS A PROOF THEORY OF ANALOGICAL REASONING M. BAAZ VIENNA UNIVERSITY OF TECHNOLOGY

Analogical Reasoning in Mathematics Euler became famous by deriving ∞ � n 2 = π 2 1 (1) 6 n =1

Let us consider Eulers reasoning. Consider the polynomial of even degree b 0 − b 1 x 2 + b 2 x 4 − · · · + ( − 1) n b n x 2 n (2) If it has the 2 n roots ± β 1 , . . . , ± β n � = 0 then (2) can be written as � � � � � � 1 − x 2 1 − x 2 1 − x 2 · · · b 0 (3) β 2 β 2 β 2 1 2 n By comparing coefficients in (2) and (3) one obtains that � 1 � + 1 + · · · + 1 b 1 = b 0 . (4) β 2 β 2 β 2 1 2 n Next Euler considers the Taylor series ∞ � x 2 n sin x ( − 1) n = (5) x (2 n + 1)! n =0 which has as roots ± π, ± 2 π, ± 3 π, . . . Now by way of analogy Euler assumes that the infinite degree polynomial (5) behaves in the same way as the finite polynomial (2). Hence in analogy to (3) he obtains � � � � � � 1 − x 2 1 − x 2 1 − x 2 sin x = · · · (6) π 2 4 π 2 9 π 2 x and in analogy to (4) he obtains � 1 � 1 π 2 + 1 4 π 2 + 1 9 π 2 + · · · 3! = (7) which immediately gives ∞ � n 2 = π 2 1 6 . (1) n =1

The structure of Eulers argument is the following. (a) (2) = (3) (mathematically derivable) (b) (2) = (3) ⊃ (4) (mathematically derivable) (c) (2) = (3) ⊃ (5) = (6) (analogical hypothesis) (d) (5) = (6) ⊃ (4) (modus ponens) (e) ((2) = (3) ⊃ (4)) ⊃ ((5) = (6) ⊃ (7)) (analogical hypothesis) (f) (5) = (6) ⊃ (7) (modus ponens) (g) (7) (modus ponens) (h) (7) ⊃ (1) (mathematically derivable) (i) (1) (modus ponens)

We will consider analogies based on 1 Generalizations wrt. invariant parts of the proofs (e.g., graphs of rule ap- plications) 1.1 Generalizations of conclusions 1.2 Generalizations of premises 2 Generalizations wrt. semantical features

The calculus LK logical axiom schema: A → A structural inferences: Γ 1 → ∆ 1 , A A, Γ 2 → ∆ 2 cut Γ 1 , Γ 2 → ∆ 1 , ∆ 2 Γ → ∆ Γ → ∆ weakening left A, Γ → ∆ weakening right Γ → ∆ , A Γ 1 , A, B, Γ 2 → ∆ Γ → ∆ 1 , A, B, ∆ 2 exchange left Γ 1 , B, A, Γ 2 → ∆ exchange right Γ → ∆ 1 , B, A, ∆ 2 A, A, Γ → ∆ Γ → ∆ , A, A contraction right Γ → ∆ , A contraction left A, Γ → ∆ logical inferences: Γ → ∆ , A A, Γ → ∆ ¬ :left ¬ A, Γ → ∆ ¬ :right Γ → ∆ , ¬ A A, Γ 1 ⊢ ∆ 1 B, Γ 2 ⊢ ∆ 2 Γ → ∆ , A ∨ :right 1 Γ → ∆ , A ∨ B ∨ :left A ∨ B, Γ 1 , Γ 2 ⊢ ∆ 1 , ∆ 2 A, Γ → ∆ Γ → ∆ , B ∨ :right 2 Γ → ∆ , A ∨ B ∧ :left 1 A ∧ B, Γ → ∆ B, Γ → ∆ Γ 1 ⊢ ∆ 1 , A Γ 2 ⊢ ∆ 2 , B ∧ :left 2 A ∧ B, Γ → ∆ ∧ :right Γ 1 , Γ 2 ⊢ ∆ 1 , ∆ 2 , A ∧ B Γ 1 ⊢ ∆ 1 , A B, Γ 2 ⊢ ∆ 2 A, Γ → ∆ , B ⊃ :left A ⊃ B, Γ 1 , Γ 2 ⊢ ∆ 1 , ∆ 2 ⊃ :right Γ → ∆ , A ⊃ B C ( e ) , Γ → ∆ Γ → ∆ , C ( r ) ∃ :right Γ → ∆ , ∃ αC ( α ) ∃ :left ∃ αC ( α ) , Γ → ∆ C ( r ) , Γ → ∆ Γ → ∆ , C ( e ) ∀ :left ∀ αC ( α ) , Γ → ∆ ∀ :right Γ → ∆ , ∀ αC ( α ) with the usual restrictions.

Skeleton/proof analysis axiom axiom axiom ∀ :left ∀ :left ∀ :left ∧ :right ∀ :left ∀ :right contraction left cut ⊃ :right → ∀ x ∀ yP ( x, y ) ⊃ P (0 , a ) ∧ P ( S 0 , Sa ) The above analysis and sequent determine an (extended) proof matrix: P ( t 1 , t 2 ) → P ( t 1 , t 2 ) P ( r 3 , r 4 ) → P ( r 3 , r 4 ) P ( r 5 , r 6 ) → P ( r 5 , r 6 ) ∀ βP ( s 1 , s 2 ) → P ( t 1 , t 2 ) ∀ γP ( s, t ) → P ( r 3 , r 4 ) ∀ γP ( s, t ) → P ( r 5 , r 6 ) ∀ α ∀ βP ( r 1 , r 2 ) → P ( t 1 , t 2 ) ∀ γP ( s, t ) , ∀ γP ( s, t ) → P ( r 3 , r 4 ) ∧ P ( r 5 , r 6 ) ∀ α ∀ βP ( r 1 , r 2 ) → ∀ γP ( s, t ) ∀ γP ( s, t ) → P ( r 3 , r 4 ) ∧ P ( r 5 , r 6 ) ∀ α ∀ βP ( r 1 , r 2 ) → P ( r 3 , r 4 ) ∧ P ( r 5 , r 6 )

Then all substitutions producing proofs can be characterized by the following equations r 1 ( α ) = s 1 = t 1 = s ( e ) ∧ r 2 ( β ) = s 2 ( β ) = t 2 = t ( e ) ∧ ∧ s ( γ 1 ) = r 3 ∧ t ( γ 1 ) = r 4 ∧ s ( γ 2 ) = r 5 ∧ t ( γ 2 ) = r 6 (i.e., the do not occur in the matrix). If we take into account the term structure of the end sequent, we can get rid of the restrictions and can substitute the above equations with r 1 ( α ) = α r 2 ( β ) = β r 3 = 0 r 4 = a r 5 = S 0 r 6 = Sa. Then, using the validity of ∃ α ∃ β ( α = s ( e ) ∧ β = t ( e )), the condition on deriv- ability of our formula with the proof analysis reduces to ∃ st ∃ γ 1 γ 2 [ s ( γ 1 ) = 0 ∧ t ( γ 1 ) = a ∧ s ( γ 2 ) = S 0 ∧ t ( γ 2 ) = Sa ] . If the above holds then s ( γ ) = γ and St (0) = t ( S 0) = Sa . Thus, our formula is derivable with the considered analysis iff a is S n 0 for some n .

Theorem (Orevkov, Krajicek and Pudlak) . For every r.e. set E , there is a skeleton S E with universal or existential cuts and a sequent Π E → Γ E , A E ( a ) such that Π E → Γ E , A ( s n ( a )) is provable with S E � n ∈ E.

Theorem. Let S cutfree and Π( a ) → Γ( a ) be given. If there is a proof at all, there is a most general proof T Π( t ) → Γ( t ) such that any other proof has the form Tσ Π( tσ ) → Γ( tσ ) .

Second Order Unification Let L be a set of function symbols, a 1 , . . . , a m variables. Let T = ( T, Sub 1 , . . . , Sub m ) be the algebra of terms where T is the set of terms in L , a 1 , . . . , a m and for i = 1 , . . . , m Sub i ( δ, σ ) := δ { σ/a i } are substitutions as binary operations on T . A second order unification is a finite set of equations in the language T ∪ { Sub 1 , . . . , Sub m } plus free variables for elements of T . Theorem. Let L contain a unary function symbol S , a constant 0 , and a binary function symbol. Let τ 0 be a term variable. Then for every recursively enumerable set E there exists a second order unification problem Ω E such that Ω E ∪ { τ 0 = S n (0) } has a solution iff n ∈ E .

Idea T Π → Γ , P ( a ) ∨ P ( t ) , P ( u ) ∨ P ( v ) A, Π → Γ ∃ -r exchange ∃ -r such that a is not free anymore contraction cut with weakened formula ⇒ v = t { u/a }

LK B Replace Π → Γ , A ( a ) Π → Γ , A ( a 1 , . . . , a n ) by Π → Γ , ∀ xA ( x ) Π → Γ , ∀ x 1 , . . . , x n A ( x 1 , . . . , x n ) A ( t ) , Π → Γ A ( t 1 , . . . , t n ) , Π → Γ by ∀ xA ( x ) , Π → Γ ∀ x 1 , . . . , x n A ( x 1 , . . . , x n ) , Π → Γ Π → Γ , A ( t ) Π → Γ , A ( t 1 , . . . , t n ) by Π → Γ , ∃ xA ( x ) Π → Γ , ∃ x 1 , . . . , x n A ( x 1 , . . . , x n ) A ( a ) , Π → Γ A ( a 1 , . . . , a n ) , Π → Γ by ∃ xA ( x ) , Π → Γ ∃ x 1 , . . . , x n A ( x 1 , . . . , x n ) , Π → Γ n arbitrary

Theorem. Let S contain only existential and universal cuts and let Π( a ) → Γ( a ) be given. If there is a proof at all, there is a most general proof T Π( t ) → Γ( t ) such that any other proof has the form Tσ Π( tσ ) → Γ( tσ ) .

Semiunification A semiunification problem is given by a set of pairs of terms ( s 1 , t 1 ) , . . . , ( s n , t n ). A solution to the semiunification problem is a substitution δ such that there exist substitutions σ 1 , . . . , σ n such that s 1 δ = t 1 δσ 1 , . . . , s n = t n δσ n ; a solutions will be also called a semiunifier. A most general semiunifier is a semiunifier δ 0 such that for every semiunifier δ there exists a substitution δ ′ such that δ = δ 0 δ ′ (i.e., δ ( x ) = δ ′ ( δ 0 ( x ))). Theorem. If a semiunification problem has a semiunifier then it has a most general one. Example. • { ( x, s ( x )) } unsolvable • { ( y, s ( x )) } solution e.g., δ = { s ( s ( x )) /y } most general solution δ mg = { s ( x ′ ) /y } Semiunification is undecidable!

� � � � ∀ xA ( x ) ⊃ A ( t ) A ′ A ′′ σ A ( a ) A ( t ) ( t arbitrary) Second-order unification problem A ′′ = A ′ { x/a } σ solution ∀ x 1 , . . . , x n A ( x 1 , . . . , x n ) ⊃ A ( t 1 , . . . , t n ) A ′ A ′′ σ A ( a 1 , . . . , a n ) A ( t 1 , . . . , t n ) Semiunification problem { ( A ′′ , A ′ ) } σ solution

Theorem. It is undecidable whether there is a proof with (non tree-like) LK B -skeleton S with universal/existential cuts for any instance of Π( a ) → Γ( a ) .