Interpretation between weak theories of concatenation and arithmetic Osam Yoshida and Yoshihiro Horihata (Tohoku university) Feb 23, 2012 Workshop on Proof Theory and Computability Theory 2012

Fundamental human abilities 読み（yomi）

書き（kaki）

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そろばん（soroban）

Fundamental human abilities 読み（yomi） 書き（kaki） そろばん（soroban） || || || Reading Writing Abacus (Arithmetic)

Figure 1: Reading and Writing

Figure 2: Abacus

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Fundamental human abilities Reading

Writing

Arithmetic ↑ Well-studied !

Example PA, IΣn , Q, R, second-order arithmetic, etc

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Fundamental human abilities Reading

Writing

Arithmetic

1930’s Tarski 1940’s Quine

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Fundamental human abilities Reading

Writing

Arithmetic

1930’s Tarski 1940’s Quine ↓ 2005 Grzegorczyk’s TC A Theory of Concatenation

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Back ground and known results C2

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5 TC

PA 5

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Q

TC : Theory of Concatenation In A. Grzegorczyk’s paper “Undecidability without arithmetization”(2005), he defined a (_ , ε , α , β )-theory TC of concatenation, 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 6= ε ∧ ∀x∀y(x_ y = α → x = ε ∨ y = ε ) (TC5) β 6= ε ∧ ∀x∀y(x_ y = β → x = ε ∨ y = ε ) (TC6) α 6= β 8

About (TC3); editors axiom If x_ y = u_ v,

y

x

v

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About (TC3); editors axiom If x_ y = u_ v,

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About (TC3); editors axiom If x_ y = u_ v,

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x w

w v

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TC : Theory of Concatenation  Definition



• x v y ≡ ∃k∃l(kxl = y) • x vini y ≡ ∃l(xl = y) • x vend y ≡ ∃k(kx = y) 



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What can TC prove?  Proposition



TC proves the following assertions: (1) ∀x(xα 6= ε ∧ α x 6= ε ) (2) ∀x∀y(xy = ε → x = ε ∧ y = ε ) (3) ∀x∀y(xα = yα ∨ α x = α y → x = y) Weak cancellation   Proposition

 

TC cannot prove the following assertions: • ∀x∀y∀z(xz = yz → x = y) 

cancellation 

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TC and undecidability  Theorem

[Grzegorczyk, 2005]

TC is undecidable.







Moreover,  Theorem [Grzegorczyk and Zdanowski, 2007]



TC is essentially undecidable. 



Before this, Yaegashi proved this fact in 2006 in his master’s thesis, by interpreting arithmetic R in TC. Grzegorczyk and Zdanowski conjectured that TC and Q are mutually interpretable. 14

Definition of interpretation L1 , L2 : languages of first order logic. A relative translation τ : L1 → L2 is a pair hδ , Fi such that • δ is an L2 -formula with one free variable. • F maps each relation-symbol R of L1 to an L2 -formula F(R). We translate L1 -formulas to L2 -formulas as follows: • (R(x1 , · · · , xn ))τ := F(R)(x1 , · · · , xn ); • (·)τ commutes with the propositional connectives; • (∀xϕ (x))τ := ∀x(δ (x) → ϕ τ ); • (∃xϕ (x))τ := ∃x(δ (x) ∧ ϕ τ ). 15

Definition of interpretation  Definition (relative interpretation)



L1 -theory T is (relatively) interpretable in L2 -theory S, denoted by S . T , iff there exists a relative translation τ : L1 → L2 such that 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 essentially undecidable. 



That is, the interpretability conserves the essential undecidability. 16

TC and Q In 2009, the following results were proved by three ways independently: ˇ Visser and Sterken, 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) 6= 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 6= 0 → ∃y(x = S(y))) Q is essentially undecidable and finitely axiomatizable. 17

Theory C2 and Peano arithmetic PA The theory C2 of concatenation consists of TC plus the following notation 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]



C2 and PA are mutually interpretable. 



This is a positive answer for Yaegashi’s question raised in his master thesis. 18

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

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Arithmetic R (MRT, 1953)  (+, ·, 0, 1, ≤)-theory

R



· · + 1} ) For each n, m ∈ ω , ( n represents |1 + ·{z n

(R1) (R2) (R3) (R4) (R5)

n+m = n+m n·m = n·m n 6= m (if n 6= m)
∀x x ≤ n → x = 0 ∨ x = 1 ∨ · · · ∨ x = n ∀x(x ≤ n ∨ n ≤ x)





* R is Σ1 -complete and essentially undecidable. * R6  Q, since Q is finitely axiomatizable. 20

Arithmetic R0 (Cobham, 1960’s)  (+, ·, 0, 1, ≤)-theory

R0



For each n, m ∈ ω , (R1) n + m = n + m (R2) n · m = n · m (R3) n 6= m (if n 6= m)
(R4’) ∀x x ≤ n↔x = 0 ∨ x = 1 ∨ · · · ∨ x = n 



* R0 interprets R by translating ‘ ≤ ’ by ‘ l ’ as follows: x l y ≡ [0 ≤ y ∧ ∀u (u ≤ y ∧ u 6= y → u + 1 ≤ y)] → x ≤ y. * R0 is minimal theory which is Σ1 -complete and essentially undecidable. 21

WTC: Weak Theory of Concatenation (_ , ε , α , β )-theory WTC has the following axioms: for each u ∈ {α , β }∗ , (WTC1) ∀x v u (x_ ε = ε _ x = x); (WTC2) ∀x ∀y ∀z [[x_ (y_ z)v u ∨ (x_ y)_ zv u] → x_ (y_ z) = (x_ y)_ z]; (WTC3) ∀x ∀y ∀s ∀t [(x_ y = s_t ∧ x_ yv u) → ∃w ((x_ w = s ∧ y = w_t) ∨ (x = s_ w ∧ w_ y = t))]; (WTC4) α 6= ε ∧ ∀x ∀y (x_ y = α → x = ε ∨ y = ε ); (WTC5) β 6= ε ∧ ∀x ∀y (x_ y = β → x = ε ∨ y = ε ); (WTC6) α 6= β .

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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, ααβ α = ((αα )β )α . We call each u (∈ {α , β }∗ ) standard string.   Definition • x v y ≡ (x = y) ∨ ∃k ∃l [kx = y ∨ xl = y ∨ (kx)l = y ∨ k(xl) = y] • x vini y ≡ (x = y) ∨ ∃l (xl = y) • x vend y ≡ (x = y) ∨ ∃k (kx = y) 



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Σ1 -completeness of WTC  Lemma



WTC proves the following assertion: ∀x (x v u ↔

_

x = v).

vvu   Theorem

 

WTC is Σ1 -complete, that is, for each Σ1 -sentence ϕ , if {α , β }∗  ϕ then WTC ` ϕ . 



{α , β }∗ is a standard model of TC. 24

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”. 25

WTC interprets R  An idea for the definition of witness



Witness w for 2 × 3 is as follows: w = β β β β β αβ ααβ β ααβ (αα )(αα )β β αααβ (αα )(αα )(αα )β β

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 relation “ w is a witness for product of x and y ” by a formula PWitn(x, y, w). 26

WTC interprets R  Translation of product



We translate the multiplication “x × y = z” by (∃!w PWitn(x, y, w) ∧ β β zβ vend 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) ∧ ∀w0 (PWitn(u, v, w0 ) → w = w0 ).   Theorem

 

WTC interprets R. 

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R interprets WTC Conversely, we can prove that R interprets WTC, by applying 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 subtheory of T has a finite model. Since WTC is locally finitely satisfiable, we can get the following result:   Corollary R interprets WTC. 



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Conclusion of part I  Theorem



WTC and R are mutually interpretable.   Corollary

 

(1) WTC is essentially undecidable. (2) WTC interprets T iff T is locally finitely satisfiable. (3) WTC cannot interpret TC. (4) WTC2 and WTCn (n ≥ 2) are mutually interpretable. 



Here, WTCn is WTC with n-th single-letters. (4) is from WTC2  R  WTCn  WTC2 . 29

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Part II Minimal essential undecidability and variations of WTC

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Minimal essential undecidability  Question



Is WTC minimal essentially undecidable ? 



Here, minimal essentially undecidable means if one omits one axiom from WTC, then the resulting theory is no longer essentially undecidable. Again, WTC is: for each u ∈ {α , β }∗ (WTC1) ∀xv u (x_ ε = ε _ x = x); (WTC2) ∀x∀y∀z[[x_ (y_ z)v u ∨ (x_ y)_ zv u] → x_ (y_ z) = (x_ y)_ z]; (WTC3) ∀x∀y∀s∀t[(x_ y = s_t ∧ x_ yv u) → ∃w((x_ w = s ∧ y = w_t) ∨ (x = s_ w ∧ w_ y = t))]; (WTC4) α 6= ε ∧ ∀x∀y(x_ y = α → x = ε ∨ y = ε ); (WTC5) β 6= ε ∧ ∀x∀y(x_ y = β → x = ε ∨ y = ε ); (WTC6) α 6= β . 31

Minimal essential undecidability  Proposition



WTC−(WTC k) (k = 3, 4, 5, 6) is not essentially undecidable.





We can find a decidable consistent extension of each WTC−(WTC k) (k = 3, 4, 5, 6). Hence remaining question is WTC−(WTC k) (k = 1, 2) is essentially undecidable ? Recently, we proved the following:  Theorem



WTC−(WTC1) can interpret WTC. Hence, WTC−(WTC1) is still essentially undecidable. 



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WTC−(WTC1) 
WTC This is proved as follows.  Lemma



For each u ∈ {α , β }∗ , WTC - (WTC1) proves uε = ε u = u. 



⇒ Without (WTC1), axiom for identity, we can prove that the empty string works well, as an identity element, for at least all standard strings.   Main Lemma W 0 0 WTC - (WTC1) ` ∀x (x v u∧∃x (x = (ε x )ε ) → vvu x = v).





Although weWdo not know whether WTC−(WTC1) can prove ∀x (x v u → vvu x = v) or not, the above corollary is strong enough to interpret WTC into WTC−(WTC1). 33

WTC−(WTC1) 
WTC Then, we interpret WTC in WTC - (WTC1) as follows: Domain δ (x) ≡ x = α ∨ ∃x0 (x = (β x0 )ε ). Remark that if (β x0 )ε is standard, then (β x0 )ε = β ((ε x0 )ε ). Constants ε ⇒ β , α ⇒ β α , β ⇒ β β . x_ y = z Let Ω(x, y) ≡ ∃!x0 ∃!y0 (x = (β x0 )ε ∧ y = (β y0 )ε ). Then we translate concatenation as Conc(x, y, z) ≡ x = α ∨y = α → z = α ∧ Ω(x, y) → ∃x0 ∃y0 [x = (β x0 )ε ∧ y = (β y0 )ε ∧ z = (β ((x0 ε )y0 ))ε ] ∧ o.w. → z = α .   Lemma For each w ∈ {α , β }∗ , WTC - (WTC1) can prove that if Conc(x, y, β w), then x and y are also standard. 



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WTC−(WTC1) 
WTC By this lemma, we can prove WTC−(WTC1)  WTC.   Question Is WTC−(WTC1) minimal essentially undecidable ?



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TC−ε

On the other hand, we can consider the theory of concatenation without empty string: (_ , α , β )theory TC−ε has the following axioms: (TC−ε 1) ∀x∀y∀z(x_ (y_ z) = (x_ y)_ z) Associativity (TC−ε 2) Editors Axiom: ∀x ∀y ∀s ∀t (x_ y = s_t → (x = s ∧ y = t)∨ ∃w ((x_ w = s ∧ y = w_t) ∨ (x = s_ w ∧ w_ y = t))) (TC−ε 3) ∀x ∀y (α 6= x_ y) (TC−ε 4) ∀x ∀y (β 6= x_ y) (TC−ε 5) α 6= β

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WTC−ε A weak version WTC−ε of TC−ε has the following axioms: for each u ∈ {α , β }+ , (WTC−ε 1) ∀x ∀y ∀z [[x_ (y_ z)v u ∨ (x_ y)_ zv u] → x_ (y_ z) = (x_ y)_ z]; (WTC−ε 2) ∀x ∀y ∀s ∀t [(x_ y = s_t ∧ x_ yv u) → (x = y) ∧ (s = t)∨ ∃w ((x_ w = s ∧ y = w_t) ∨ (x = s_ w ∧ w_ y = t))]; (WTC−ε 3) ∀x ∀y (x_ y 6= α ); (WTC−ε 4) ∀x ∀y (x_ y 6= β ); (WTC−ε 5) α 6= β .

For this theory, we proved the following:

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WTC−ε 
WTC  Proposition



WTC−ε and WTC are mutually interpretable. Hence WTC−ε is essentially undecidable. 

WTC  WTC−ε is easy. We interpret WTC in WTC−ε as: Domain δ (x) ≡ x = α ∨ x = β ∨ ∃x0 (x = β x0 ). Constants ε ⇒ β , α ⇒ β α , β ⇒ β β . x_ y = z Let Ω(x, y) ≡ ∃!x0 ∃!y0 (x = β x0 ∧ y = β y0 ), and translate the concatenation by Conc(x, y, z) ≡ [x = α ∨ y = α → z = α ] ∧ [x = β → z = y] ∧ [y = β → z = x]∧ [Ω(x, y) → ∃x0 ∃y0 (x = β x0 ∧ y = β y0 ∧ z = β (x0 y0 ))]∧ [o.w. → z = α ].

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WTC−ε is minimal essentially undecidable  Theorem



WTC−ε is minimal essentially undecidable. 



The essential part of the above theorem is to prove “WTC−ε −(WTC−ε 1) is not essentially undecidable”. This is proved by showing the followings (the proof is due to K. Higuchi):   Theorem (K. Higuchi) WTC−ε −(WTC−ε 1) is interpretable in S2S. 



Here, S2S is a monadic second-order logic whose language is L = {S0 , S1 , (Pa )a∈A }. S0 , S1 are two successors and Pa ’s are unary predicates. Then, S2S := {ϕ | ϕ is an L-sentence & {0, 1}∗  ϕ }. S2S is proved to be decidable by M. O. Rabin (1969). 39

Minimal essential undecidability

This result partially contributes the following question by Grzegorczyk and Zdanowski:   Question Is TC−ε minimal essentially undecidable ? 



The remaining part of the question is the essential undecidability of TC−ε −(TC−ε 1), that is, TC without associative law. We can easily find an decidable extension of each TC−ε −(TC−ε k), (k = 2, 3, 4, 5).

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Variations of WTC: WTC+(TC1) + (TC2) 
WTC Recall that (TC1) ∀x (x_ ε = ε _ x = x) (TC2) ∀x ∀y ∀z (x_ (y_ z) = x_ (y_ z)) (TC3) ∀x∀y∀s∀t[(x_ y = s_t) → ∃w((x_ w = s ∧ y = w_t) ∨ (x = s_ w ∧ w_ y = t))]   Proposition WTC interprets WTC+(TC1) + (TC2) 



Because WTC+(TC1) + (TC2) is locally finitely satisfiable.  Proposition



WTC can not interpret WTC+(TC3). 



Because WTC+(TC3) is not locally finitely satisfiable. 41

Conclusion of Part II

The following are mutually interpretable (n ≥ 2): WTCn + (Identity) + (Assoc) WTCn + (Identity) WTCn + (Assoc) WTCn−ε + (Assoc) ε WTCn WTC− n WTCn −(Identity)  Theorem



WTC−ε is minimal essentially undecidable. 



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Questions (1) Is WTC-(Identity)-(Assoc) essentially undecidable ? ⇒ Our conjecture is NO. (2) Is WTC-(Identity) Σ1 -complete ? ⇒ Our conjecture is NO. (3) WTC+ (Editors Axiom) . TC ? ⇒ Our conjecture is YES. (4) Are there some natural theory T such that TC  T  WTC and WTC 6 T and T 6 TC ? 43

References [1] M. Ganea. Arithmetic on semigroups. The Journal of Symbolic Logic, 74(1):265–278, 2009. [2] A. Grzegorczyk. Undecidability without arithmetization. Studia Logica, 79(1):163–230, 2005. [3] A. Grzegorczyk and K. Zdanowski. Undecidability and concatenation. In V. W. Marek A. Ehrenfeucht and M. Srebrny, editors, Andrzej Mostowski and foudational studies, pages 72–91. IOS Press, 2008. [4] Y. Horihata. Weak theories of concatenation and arithmetic. to appear in Notre Dame Journal of Formal Logic. [5] A. Tarski, A. Mostowski, and R. M. Robinson. Undecidable theories. North-Holland, 1953. [6] A. Visser. Why the theory R is special. August 10, 2009.

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WTC interprets R  Definition of “Good”



We define the formula Good(x) as follows: Good(x) ≡ ID(x) ∧ AS(x) ∧ EA(x), where • ID(x) ≡ ∀s v x(s_ ε = ε _ s = s); • AS(x) ≡ ∀s0 ∀s1 ∀s2 [[s0 _ (s1 _ s2 ) v x ∨ (s0 _ s1 )_ s2 v x] → s0 _ (s1 _ s2 ) = (s0 _ s1 )_ s2 ] • EA(x) ≡ ∀s0 ∀s1 ∀t0 ∀t1 [(s0 _ s1 = t0 _t1 ∧ s0 _ s1 v x) → ∃w((s0 _ w = t0 ∧ s1 = w_t1 ) ∨ (s0 = t0 _ w ∧ w_ s1 = t1 ))] 



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WTC interprets R  Properties



of Good

(1) For each u ∈ {α , β , γ }∗ , WTC ` Good(u); WTC proves the following assertions: (2) ∀x(Good(x) → ∀y v x Good(y)), that is Good is closed under taking substrings. 



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WTC interprets R To translate the product, we define “witness for product”. First, we define a notion “number strings” as follows:   Definition of “Num” We define the formula Num(x) as follows: Num(x) ≡ ∀y((y v x ∧ y 6= ε ) → α vend y).   Fact

 

For each u ∈ {α }∗ , WTC ` Num(u). 

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 Definition



of PWitn

We define a formula PWitn(x, y, w) as follows: (i) Num(x) ∧ Num(y) ∧ Good(w); (ii) β γβ vini w; (iii) ∃z(Num(z) ∧ β yγ zβ vend w); (iv) ∀p∀z(Num(z) ∧ pβ yγ zβ = w → ∀z0 (Num(z0 ) → ¬(β yγ z0 β v pβ )); (v) ∀p∀q∀s2 ∀t2 [(Num(s2 ) ∧ Num(t2 ) ∧ pβ s2 γ t2 β q = w ∧ p 6= ε ) → (∃s1 ∃t1 (Num(s1 ) ∧ Num(t1 ) ∧ s2 = s1 α ∧ t2 = t1 x ∧ β s1 γ t1 β vend pβ ))]; (vi) ∀p∀q∀s∀t((Num(s1 ) ∧ Num(t1 ) ∧ pβ sγ t β q = w ∧ q 6= ε ) → β sαγ txβ vini β q). 

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WTC interprets R PWitn(x, y, w)

w

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WTC interprets R PWitn(x, y, w)

condition (ii)

β γβ w

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WTC interprets R PWitn(x, y, w)

condition (iii)

β yγ zβ

for some z

w

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WTC interprets R PWitn(x, y, w)

condition (iv) β yγ does not appear

β y γ zβ w

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