Theories of concatenation, arithmetic, and undecidability Yoshihiro - PowerPoint PPT Presentation

Theories of concatenation, arithmetic, and undecidability Yoshihiro Horihata Yonago National College of Technology Feb 19, 2013 Computability Theory and Foundations of Mathematics

Contents • An introduction for Theories of Concatenation • Weak theories of concatenation and arithmetic • Minimal essential undecidability 2

����������������������������� Back ground and known results C 2 PA ⊲⊳ ▽ ▽ TC Q ⊲⊳ 3

TC : Theory of Concatenation In A. Grzegorczyk’s paper “Undecidability without arith- metization”(2005), he defined a ( ⌢ , ε , α , β ) -theory TC of con- catenation, whose axioms are: (TC1) ∀ x ( x ⌢ ε = ε ⌢ x = x ) Axiom for identity (TC2) ∀ x ∀ y ∀ z ( x ⌢ ( y ⌢ z ) = ( x ⌢ y ) ⌢ z ) Associativity (TC3) Editors Axiom: ∀ x ∀ y ∀ u ∀ v ( x ⌢ y = u ⌢ v → ∃ w (( x ⌢ w = u ∧ y = w ⌢ v ) ∨ ( x = u ⌢ w ∧ w ⌢ y = v ))) (TC4) α � = ε ∧∀ x ∀ y ( x ⌢ y = α → x = ε ∨ y = ε ) (TC5) β � = ε ∧∀ x ∀ y ( x ⌢ y = β → x = ε ∨ y = ε ) (TC6) α � = β 4

About (TC3); editors axiom If x ⌢ y = u ⌢ v , y x v u 5

About (TC3); editors axiom If x ⌢ y = u ⌢ v , y x v u 6

About (TC3); editors axiom If x ⌢ y = u ⌢ v , y x w w v u 7

TC : Theory of Concatenation ✓ ✏ Definition • x ⊑ y ≡ ∃ k ∃ l (( k ⌢ x ) ⌢ l = y ) • x ⊑ ini y ≡ ∃ l ( x ⌢ l = y ) • x ⊑ end y ≡ ∃ k ( k ⌢ x = y ) ✒ ✑ 8

What can TC prove? ✓ ✏ Proposition TC proves the following assertions: ( 1 ) ∀ x ( x α � = ε ∧ α x � = ε ) ( 2 ) ∀ x ∀ y ( xy = ε → x = ε ∧ y = ε ) ( 3 ) ∀ x ∀ y ( x α = y α ∨ α x = α y → x = y ) Weak cancellation ✒ ✑ ✓ ✏ Proposition TC cannot prove the following assertions: • ∀ x ∀ y ∀ z ( xz = yz → x = y ) cancellation ✒ ✑ 9

TC and undecidability ✓ ✏ Theorem [Grzegorczyk, 2005] TC is undecidable. ✒ ✑ Moreover, ✓ ✏ Theorem [Grzegorczyk and Zdanowski, 2007] TC is essentially undecidable. ✒ ✑ Grzegorczyk and Zdanowski conjectured that (i) TC and Q are mutually interpretable; (ii) TC is minimal essentially undecidable theory. 10

Definition of interpretation L 1 , L 2 : languages of first order logic. A relative translation τ : L 1 → L 2 is a pair � δ , F � such that • δ is an L 2 -formula with one free variable. • F maps each relation-symbol R of L 1 to an L 2 -formula F ( R ) . We translate L 1 -formulas to L 2 -formulas as follows: • ( R ( x 1 , ··· , x n )) τ : = F ( R )( x 1 , ··· , x n ) ; • ( · ) τ commutes with the propositional connectives; • ( ∀ x ϕ ( x )) τ : = ∀ x ( δ ( x ) → ϕ τ ) ; • ( ∃ x ϕ ( x )) τ : = ∃ x ( δ ( x ) ∧ ϕ τ ) . 11

Definition of interpretation ✓ ✏ Definition (relative interpretation) L 1 -theory T is (relatively) interpretable in L 2 -theory S , de- noted by S ⊲ T , iff there exists a relative translation τ : L 1 → L 2 such that (i) S ⊢ ∃ x δ ( x ) and (ii) for each axiom σ of T , S ⊢ σ τ . ✒ ✑ ✓ ✏ Proposition Let S be a consistent theory. If S ⊲ T and T is essentially undecidable, then S is also es- sentially undecidable. ✒ ✑ The interpretability conserves the essential undecidability. 12

TC and Q In 2009, the following results were proved by three ways independently: Visser and Sterken, ˇ Svejdar, and Ganea. ✓ ✏ Theorem [2009] TC interprets Q . (Hence TC ✄✁ Q .) ✒ ✑ Here, Q is Robinson’s arithmetic, whose language is (+ , · , 0 , S ) ( Q1 ) ∀ x ∀ y ( S ( x ) = S ( y ) → x = y ) ( Q2 ) ∀ x ( S ( x ) � = 0 ) ( Q3 ) ∀ x ( x + 0 = x ) ( Q4 ) ∀ x ∀ y ( x + S ( y ) = S ( x + y )) ( Q5 ) ∀ x ( x · 0 = 0 ) ( Q6 ) ∀ x ∀ y ( x · S ( y ) = x · y + x ) ( Q7 ) ∀ x ( x � = 0 → ∃ y ( x = S ( y ))) Q is essentially undecidable and finitely axiomatizable. 13

Theory C 2 and Peano arithmetic PA The theory C 2 of concatenation consists of TC plus the following induction: ϕ ( ε ) ∧∀ x ( ϕ ( x ) → ϕ ( x ⌢ α ) ∧ ϕ ( x ⌢ β )) → ∀ x ϕ ( x ) . Here, ϕ is a ( ⌢ , ε , α , β ) -formula. Then, Ganea proved that ✓ ✏ Theorem [Ganea, 2009] C 2 and PA are mutually interpretable. ✒ ✑ 14

����������������������������� Part I A weak theory WTC of concatenation and mutual interpretability with R 15

Arithmetic R ( Mostowski-Robinson-Tarski, 1953 ) ✓ ✏ (+ , · , 0 , 1 , ≤ ) -theory R For each n , m ∈ ω , ( n represents 1 + ··· + 1 ) � �� � n (R1) n + m = n + m (R2) n · m = n · m (R3) n � = m ( if n � = m ) � � (R4) ∀ x x ≤ n → x = 0 ∨ x = 1 ∨···∨ x = n (R5) ∀ x ( x ≤ n ∨ n ≤ x ) ✒ ✑ * R is Σ 1 -complete and essentially undecidable. * R � ✄ Q , since Q is finitely axiomatizable. 16

Arithmetic R 0 (Cobham, 1960’s) ✓ ✏ (+ , · , 0 , 1 , ≤ ) -theory R 0 For each n , m ∈ ω , (R1) n + m = n + m (R2) n · m = n · m (R3) n � = m ( if n � = m ) � � (R4’) ∀ x x ≤ n ↔ x = 0 ∨ x = 1 ∨···∨ x = n ✒ ✑ * R 0 interprets R by translating ‘ ≤ ’ by ‘ ⋖ ’ as follows: x ⋖ y ≡ [ 0 ≤ y ∧∀ u ( u ≤ y ∧ u � = y → u + 1 ≤ y )] → x ≤ y . * R 0 is minimal theory which is Σ 1 -complete and essentially undecidable. 17

Arithmetic R 1 (Jones and Shepherdson, 1983) ✓ ✏ (+ , · , 0 , 1 , ≤ ) -theory R 1 For each n , m ∈ ω , (R2) n · m = n · m (R3) n � = m ( if n � = m ) � � (R4’) ∀ x x ≤ n ↔ x = 0 ∨ x = 1 ∨···∨ x = n ✒ ✑ * R 1 interprets R 0 by J. Robinson’s definition of ad- dition in terms of multiplication. * R 1 is minimal theory which is essentially undecid- able. 18

WTC : Weak Theory of Concatenation ( ⌢ , ε , α , β ) -theory WTC has the following axioms: for each u ∈ { α , β } ∗ , (WTC1) ∀ x ⊑ u ( x ⌢ ε = ε ⌢ x = x ) ; (WTC2) ∀ x ∀ y ∀ z [[ x ⌢ ( y ⌢ z ) ⊑ u ∨ ( x ⌢ y ) ⌢ z ⊑ u ] → x ⌢ ( y ⌢ z ) = ( x ⌢ y ) ⌢ z ] ; (WTC3) ∀ x ∀ y ∀ s ∀ t [( x ⌢ y = s ⌢ t ∧ x ⌢ y ⊑ u ) → ∃ w (( x ⌢ w = s ∧ y = w ⌢ t ) ∨ ( x = s ⌢ w ∧ w ⌢ y = t ))] ; (WTC4) α � = ε ∧∀ x ∀ y ( x ⌢ y = α → x = ε ∨ y = ε ) ; (WTC5) β � = ε ∧∀ x ∀ y ( x ⌢ y = β → x = ε ∨ y = ε ) ; (WTC6) α � = β . 19

WTC : Weak Theory of Concatenation Here, { α , β } ∗ is a set of finite strings over { α , β } , including empty string ε . Let { α , β } + : = { α , β } ∗ \{ ε } . For each u ∈ { α , β } ∗ , we represent u in theories as u by adding parentheses from left . For example, ααβα = (( αα ) β ) α . We call each u ( ∈ { α , β } ∗ ) standard string. ✓ ✏ Definition • x ⊑ y ≡ ( x = y ) ∨∃ k ∃ l [ kx = y ∨ xl = y ∨ ( kx ) l = y ∨ k ( xl ) = y ] • x ⊑ ini y ≡ ( x = y ) ∨∃ l ( xl = y ) • x ⊑ end y ≡ ( x = y ) ∨∃ k ( kx = y ) ✒ ✑ 20

Σ 1 -completeness of WTC ✓ ✏ Lemma WTC proves the following assertion: � ∀ x ( x ⊑ u ↔ x = v ) . v ⊑ u ✒ ✑ ✓ ✏ Theorem WTC is Σ 1 -complete, that is, for each Σ 1 -sentence ϕ , if { α , β } ∗ � ϕ then WTC ⊢ ϕ . ✒ ✑ { α , β } ∗ is a standard model of TC . 21

WTC interprets R From now on, we consider the translation of R into WTC . ✓ ✏ translation of 0 , 1 , + We translate 0 , 1 , + as follows: • 0 ⇒ ε ; • 1 ⇒ α ; • x + y ⇒ x ⌢ y ; • x ≤ y ⇒ ∃ z ( x ⌢ z = y ) . ✒ ✑ To translate the product, we have to make it total on ω . To do this, we consider notion, “witness for product”. 22

WTC interprets R ✓ ✏ An idea for the definition of witness Witness w for 2 × 3 is as follows: w = βββββαβααββααβ ( αα )( αα ) ββαααβ ( αα )( αα )( αα ) ββ This is from the following interpretation of 2 × 3 : ( 0 , 0 ) → ( 1 , 2 ) → ( 2 , 2 + 2 ) → ( 3 , 2 + 2 + 2 ) . That is, 2 × 3 is interpreted as adding 2 three times. ✒ ✑ By the help of above idea, we can represent the re- lation “ w is a witness for product of x and y ” by a formula PWitn ( x , y , w ) . 23

WTC interprets R ✓ ✏ Translation of product We translate the multiplication “ x × y = z ” by ( ∃ ! w PWitn ( x , y , w ) ∧ ββ y β z ββ ⊑ end w ) ∨ ( ¬ ( ∃ ! w PWitn ( x , y , w ))) ∧ z = 0 . ✒ ✑ ✓ ✏ Lemma (uniqueness of the witness on ω ) For each u , v ∈ { α } ∗ , there exists w ∈ { α , β } ∗ such that WTC proves PWitn ( u , v , w ) ∧∀ w ′ ( PWitn ( u , v , w ′ ) → w = w ′ ) . ✒ ✑ ✓ ✏ Theorem WTC interprets R . ✒ ✑ 24

R interprets WTC Conversely, we can prove that R interprets WTC , by apply- ing the Visser’s following theorem: ✓ ✏ Visser’s theorem (2009) T is interpretable in R iff T is locally finitely satisfiable ✒ ✑ Here, a theory T is locally finitely satisfiable iff any finite sub- theory of T has a finite model. Since WTC is locally finitely satisfiable, we can get the follow- ing result: ✓ ✏ Corollary R interprets WTC . ✒ ✑ 25