Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Computing power of Turing machines based on quantum logic

Yun Shang AMSS, Chinese Academy of Sciences

UH-CAS Workshop on Mathematical Logic 2018-11-02 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Contents

1 Classical theory of computation

2 Theory of quantum automata 1.Quantum automata based on quantum mechanics 2.Quantum automata based on quantum logic

3 Theory of Turing machines based on quantum logic 1.Basic definitions 2.Languages of quantum Turing machines 3.Computing power of Turing machines based on quantum logic Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical theory of computation

1 Computability 2 Computational complexity 3 Algorithmic theory 4 Models of computation Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Probabilities representation of quantum mechanics

1 Quantum Turing machine: Feymann (1982), Deutsch(1985); 2 The universal quantum Turing machine: Bernstein and Vazirani (1997) 3 Quantum circuit families: Deutsch (1989), Yao(1993); 4 Quantum randomacess machine; Knill(1996)(classically controlled machine enriched with quantum device 5 Classical controlled quantum Turing machine: Perdrix and Jorrand (2007); Observable quantum Turing machine: Perdrix (2011); 6 Quantum finite state automata: Moore and Crutchfield (2000)(measure once); Kondacs and Watrous (1997)(measure many); Ambainis and R.Freivalds (1998) (one way) ; Yamasaki, Kobayashi, Imai (2001)(two way) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Logical foundation of quantum mechanics–Quantum logic

1 Classical mechanics: Classical logic 2 Closed quantum system: Von Neumann’s quantum logic (sharp quantum logic) → PV measurement 3 Open quantum system: unsharp quantum logic → POV measurement Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Algebraic model for quantum logic

1 sharp quantum logic → orthomodular lattice 2 unsharp quantum logic → effect algebra( Foulis,1994); Quantum MV algebra (Giuntini,1996) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Unsharp quantum logic model

Boolean difference poset - difference poset 6 (by F. Kˆopka) 6 (by F. Kˆopka) add distributivity ? ? ? MV algebra - effect algebra  QMV algebra  MV algebra Pi (by C. C. Chang) PP (by C. Giuntini) minus PP PP distributivity ?(by G. Georgescu) ?P P ? ? - pseudo MV algebra - pseudo effect algebra weak QMV algebra ? 6 6(by A. Dvureˇcenskij) 6 pseudo coupled semiring ? 6- ? ? weak pseudo boolean coupled semiring - pseudo difference poset difference poset Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Unsharp quantum logic models

Y. Shang, Y. M. Li,Int J Theort. Phys.2004,43(2) Y. Shang, Y. M. Li,and M.Y. Chen,Int J Theort. Phys.2004,43(5) Y. Shang, Y. M. Li,and M.Y. Chen,Int J Theort. Phys. 2003,42(12) Y. Shang, Li Y.,and M.Y. Chen,Int J Theort. Phys.2004,43(12) Y. Shang, and R. Q. Lu. Semirings and pseudo MV algebras. Soft Computing - A Fusion of Foundations, Methodologies and Applications (2007),11. X. Lu, Y. Shang, R. Q. Lu., J. Zhang, and F. F. Ma, Weak QMV algebras and some ring-like structures Soft Computing - A Fusion of Foundations, Methodologies and Applications(2017),21. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Quantum automata based on quantum logic

1 Logical proposition ⇒ element of orthomodular lattice 2 Mingsheng Ying and Daowen Qiu (finite state automata and pushdown automata based on sharp quantum logic)

3 Logical proposition ⇒ element of lattice ordered QMV algebra 4 Yun Shang, Xian Lu and Ruqian Lu (finite state automata and pushdown automata based on unsharp quantum logic) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

M. S. Ying, A theory of computation based on quantum logic(I). 75 pages, 2005, Theoretical computer science. D. W. Qiu, Automata theory based on quantum logic: some characterizations, Information and Computation, 109:2(2004) 179–195. Yun S., Xian L., Ruqian L.,Automata theory based on unsharp quantum logic, Mathematical structures in computer science, 19(2009),737-756. Yun S, Xian L, Ruqian L, A theory of computation based on unsharp quantum logic:finite state automata and pushdown automata,Theoretical computer science 434(2012),53-86. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Difference between classical computation and quantum computation

We find the essential difference between automata theory based on quantum logic and classical automata theory. That is the universal validity of many fundamental properties of automata depend heavily not only on the distributive law but also on the non-contradiction law. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Sharp Unsharp Automata Quantum Automata Quantum Automata 6 6 6

Boolean Orthomodular a
a 6= -a Lattice-ordered Algebra Lattice QMV Algebra 6 6 (a
b) ∧ (a
c) 6= a
(b ∧ c) (a ∨ b) ∧ (a ∨ c) 6= a ∨ (b ∧ c)

Boolean a
a 6= -a Algebra MV Algebra Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Physical implication

For sharp quantum automata, in order to preserve the classical properties of automata, the underlying logic should degenerate to be a boolean algebra. It requires that the simple observables corresponding to projection operators are mutually commutative. For unsharp quantum automata, in order to preserve the classical properties of automata, the simple observables corresponding to effects need and only need to be coexistent. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

What about Turing machines based on quantum logic? Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Orthomodular lattices and boolean algebras

1 An orthcomplemented lattice is an orthomodular lattice if a ≤ c ⇒ c = a ∨ (a ∧ c0). (Orthomodular law) 2 An orthcomplemented lattice is a boolean algebra if a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). (Distributive law) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

S-algebras

A supplement algebra (S-algebra for short) is an algebraic structure 0 E = (E,
, , 0, 1) consisting of set M with two constant elements 0 0, 1, a unary operation and a binary operation
on M satisfying the following axioms: (S1) a
b = b
a. (S2) a
(b
c) = (a
b)
c. 0 (S3) a
a = 1. (S4) a
0 = a. (S5) a00 = a. (S6) a
1 = 1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

MV algebras and QMV algebras

An multiple-valued (MV) algebra (Chang, 1957) is an S-algebra that satisfies: 0 0 0 0 (MV) (a
b)
b = (a
b )
a A quantum MV (QMV) algebra (Giuntini,1996) is an S-algebra that satisfies: 0 0 (QMV) a
[(a e b) e (c e a )] = (a
b) e (a
c) where 0 0 0 a b = (a
b ) 0 a e b = (a ⊕ b ) b Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Turing machines

Definition A non-deterministic Turing machine (NTM) is a 7-tuple: M = (Q, Σ, Γ, δ, B, q0,F ), where 1. Q is a finite nonempty state set. 2. Σ is the finite set of input symbols. 3. Γ is the complete set of tape symbols; Σ ⊆ Γ/B. 4. F ⊆ Q is the set of final states. 5. δ ⊆ Q × Γ × Q × Γ × {R,L}. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Turing machines

Classical Turing machines

··· B a b c d B ···

q0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Turing machines

··· B a0 b c d B ···

q1 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Turing machines

··· B a0 b0 c d B ···

q2 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Classical Turing machines

··· B a0 b0 c0 d B ···

q3 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued Turing machines (LNTMs)

1 I(q0) = x

x ··· B a b c d B ··· z y

q0 0

Ref. Y.M. Li, P. Li, Turing machine based on quantum logic. Chinese Journal of Computers (35)2012 1407-1420. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued Turing machines (LNTMs)

0 1 δ(q0, a, q1, a ,R) = y

x ··· B a0 b c d B ··· z y

q1 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued Turing machines (LNTMs)

0 1 δ(q1, b, q2, b ,R) = z

x ··· B a0 b0 c d B ··· z y

q2 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued Turing machines (LNTMs)

0 1 δ(q2, c, q3, c ,R) = 1

x ··· B a0 b0 c0 d B ··· z y

q3 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued Turing machines (LNTMs)

1 T (q3) = z

x ··· B a0 b0 c0 d B ··· z y

q3 0 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

L-valued non-deterministic Turing machines (LNTMs)

Definition An L-valued non-deterministic Turing machines (LNTM) is a 7-tuple: M = (Q, Σ, Γ, δ, B, I, T ), where 1 Q is a finite nonempty set of state. 2 Σ is the input alphabet. 3 Γ is the tape alphabet; Σ ⊆ Γ/B. 4 δ : Q × Γ × Q × Γ × {L, S, R} −→ L is the transition function. L, R and S indicate that the head of the ENTM moves left, right or keep stationary. 5 B is the blank symbol. 6 I : Q −→ L is the initial state function. 7 T : Q −→ L is the final or accepting state function. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

E-valued non-deterministic Turing machines (ENTMs)

Definition An E-valued non-deterministic Turing machine (ENTM) is a 7-tuple: M = (Q, Σ, Γ, δ, B, I, T ), where 1 Q is a finite nonempty set of state. 2 Σ is the input alphabet. 3 Γ is the tape alphabet; Σ ⊆ Γ/B. 4 δ : Q × Γ × Q × Γ × {L, S, R} −→ E is the transition function. L, R and S indicate that the head of the ENTM moves left, right or keep stationary. 5 B is the blank symbol. 6 I : Q −→ E is the initial state function. 7 T : Q −→ E is the final or accepting state function. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Languages of ENTMs

A language accepted by an ENTM in depth-first model:

^ ^ ^ † † |M|d(s) = I(q0)
δ (q0s, C1)
δ (C1,C2)
··· n≥1 Ci q0∈Q †
δ (Cn−1,Cn)
T (St(Cn)) A language accepted by an ENTM in width-first model: " ! ! ^ ^ ^ ^ ^ † † |M|w(s) = ··· I(q0)
δ (q0s, C1)
δ (C1,C2) n≥1 Cn C2 C1 q0 ! ! # †
δ (C2,C3) ···
T (St(Cn)) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

E-valued non-deterministic Turing machine (ENTM)

Theorem

(i) |M|w ≤ |M|d for any ENTM M.

(ii) |M|w = |M|d for any ENTM M iff E is an MV algebra. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Comparision between |M|w and |M|d

|M|d(xy) = (a
b) ∧ (a
c) |M|w(xy) = a
(b ∧ c)

q0

δ(q0, x, q1) = a ? q1 @ δ(q1, y, q2) = b @ δ(q1, y, q3) = c © @R q2 q3 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Definition A partial function L :Σ+ → E is called an E-valued d-recursively enumerable (d-R.E.) language or E-valued w-recursively enumerable T T (w-R.E.) language if L ∈ Ld (E, Σ) or L ∈ Lw(E, Σ).

Definition A function L :Σ+ → E is called a E-valued d-recursive (d-R.) language or E-valued w-recursive (w-R.) language if L = |M|d (L = |M|w) for some M ∈ NTM(E, Σ), where M could halt for any input. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Definition Letf, g be E-valued languages. 1 The intersection of two E-valued languages f and g, denoted by f ∧ g, is defined as (f ∧ g)(s) = f(s) ∧ g(s) for any s ∈ Σ∗. 2 The sum of E-valued languages f and g, denoted by f
g, is ∗ defined as (f
g)(s) = f(s)
g(s) for any s ∈ Σ . 3 The concatenation of two E-valued languages f and g, denoted by V ∗ f · g, is defined as (f · g)(s) = [f(s1)
g(s2)] for any s ∈ Σ . s1s2=s Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Closure properties in unsharp quantum Turing languages

QMV algebra intersection disjoint sum concatenation Kleene * depth-first X × × × width-first X × × ×

If QMV satisfies distributive law: (a
b) ∧ (a
c) = a
(b ∧ c) then

MV algebra intersection disjoint sum concatenation Kleene * depth-first X X X X width-first X X X X Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Closure properties in unsharp quantum Turing languages

Logical implication: all elements are compatible.

Physical implication: observables are coexistent.

Yun S., Xian L., Ruqian L.,Closure properties of quantum Turing languages.Cie 2012, how the worlds computes(UK, Cambridge, 2012-06-18/06-23),125. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

E-valued deterministic Turing machine(EDTM )

Definition An E-valued deterministic Turing machine (EDTM) is an ENTM whose transition function δ satisfies that, for any p ∈ Q and a ∈ Γ there exists at most one set {q, b, D} such that δ(p, a, q, b, D) 6= 1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM with classical characters

Definition Let M = (Q, Σ, Γ, δ, B, I, T ) ∈ ENTM. We call δ to be classical if δ(p, a, q, b, D) = 0 or 1, ∀p, q ∈ Q, ∀a, b ∈ Γ and ∀D ∈ {L, S, R}. Similarly we call I (T ) to be classical if I(p) = 0 or 1 (T (p) = 0 or 1), ∀p ∈ Q. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM with classical characters

Lemma

For any M ∈ NTM(E, Σ) there exists MI ∈ NTMI (E, Σ) such that |M|d = |MI |d and |M|w = |MI |w. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM with classical characters

Lemma

For any M ∈ NTM(E, Σ) there exists MT ∈ NTMT (E, Σ) such that |M|d = |MT |d. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Definition A QMV algebra is said to be locally finite iff ∀a ∈ E s.t. a 6= 0, ∃n ∈ N s.t. n · a = 1. Let M be an ENTM and
RM = {a1
a2
···
an : ai ∈ RM , n ∈ N} ∪ {0}.
If E is locally finite then RM is also finite. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM with classical characters

Lemma Let M be an ENTM. If E is locally finite, there exists some ENTM M c with classical transitions which accepts the same E-valued language. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM is not equivalent to EDTM

Theorem ENTMs are not equivalent to EDTMs and ENTMs have more computational power than EDTMs.

There exists an EDTM that can be simulated by a classical Turing machine. There exists an ENTM that cannot be simulated by any classical Turing machine. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

ENTM is more powerful than EDTM

Example

Let Mu = (Qu, Σ, Γ, δu, B, pI ,QT ) be the universal Turing machine accepting Lu. Construct an ENTM M = (Q, Σ, Γ, δ, B, qI ,T ) such that, for any given 0 < x < 1,

Q = Qu ∪ {qI , qT } where qI , qT ∈/ Qu.

δ(qI , a, pI , a, S) = δ(qI , a, qT , a, S) = 0 for ∀a ∈ Σ.

δ(p, a, q, b, D) = 0 if and only (q, b, D) ∈ δu(p, a), and δ = 1 for the others.

T (p) = 0 for p ∈ QT , and T = x for the others. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Continue

Obviously M is an ENTM and its language is: |M|d(s) = 0 for ∀s ∈ Lu and |M|d(s) = x for ∀s∈ / Lu. If there exists some EDTM M 0 simulating M,then M 0 can be simulated by some classical turing machine M 00. ∗ 0 ∗ The classic language {s ∈ Σ : |M |d(s) = x} = Σ − Lu must be r.e., which contradicts that Lu is r.e.. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Computing power of Turing machines based on quantum logic

Some notations

Computing levels in the Arithmetical Hierarchy: 0 0 0 0 Σ0 = Π0, Σ1, Π1, Σ2, Π2 ··· where 0 Σn = {E : E = Q1x1...QnxnF } for some primitive recursive formula F , where Qj = ∃, if i is odd, and Qj = ∀, if i is even,n ≥ 0; 0 Πn = {E : E = Q1x1...QnxnF } for some primitive recursive formula F , where Qj = ∀, if i is odd, and Qj = ∃, if i is even,n ≥ 0.( C.Smorynski,1977) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Notations about our languages class: T Ld (E, Σ) = {|M|d : M is an ENTM}. T Lw(E, Σ) = {|M|w : M is an ENTM}. T Ld (L, Σ) = {|M|d : M is an LNTM}. T Lw(L, Σ) = {|M|w : M is an LNTM}. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

The computing power of classical Turing machines and Wiedermann’s Turing machines

1 0 The languages class of classical Turing machines is Σ1; 2 0 0 The languages class of Wiedermann’s Turing machines is Σ1 ∪ Π1( Theoretical computer science, 2004(317)) Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Main results for ENTMs

Theorem 0 0 T Σ1 ∪ Π1 ⊆ Ld (E, Σ); 0 0 T Σ1 ∪ Π1 ⊆ Lw(E, Σ).

Proof. 0 0 0 Let L ∈ Σ1 ∪ Π1. If L ∈ Σ1, the classical Turing machine accepting L is 0 0 Mu = (Q , Σ, Γ, δ , q0,B,F ). Construct the ENTM M = (Q, Σ, Γ, δ, B, I, T ) as: 0 Q = Q ∪ {q1}.

I(q0) = 0 and I(q) = 1 otherwise.

T (q) = 0 if q ∈ F , T (q1) = 0 and T (q) = 1 otherwise. δ(p, a, q, b, D) = 0 if (p, a, q, b, D) is a transition of M 0 and δ = 1 otherwise. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Easy to check that |M|d(s) = 0 for any s ∈ L and |M|d(s) = 1 for all 0 T s∈ / L. So Σ1 ⊆ Ld (E, Σ). 0 If L ∈ Π1, classical Turing machine accepting L (the complement of L) is Mu. As above, there is an ENTM M such that |M|d(s) = 0 for any 0 T s∈ / L and |M|d(s) = 1 for all s ∈ L. So Π1 ⊆ Ld (E, Σ).

q0

Mu q1 Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Main results for ENTMs

Theorem T 0 If E is locally finite, then Ld (E, Σ) ⊆ Π2. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Proof. T Let M be an ENTM. Here we treat any E-valued language L ∈ Ld (E, Σ) equivalently as its graph {s#L(s): s ∈ Σ∗}. Define

L1 ={s#e : ∀C ∈ ID(s),T (St(C)) ≥ e} L2 ={s#e : ∃ε ∈ E, ∀C ∈ ID(s),T (St(C)) ≥ e
ε and ε 6= 0}.

For ∀s#e ∈ L1, e is a lower bound of |M|d(s); for ∀s#e ∈ L2, e is not the greatest lower bound of the set {T (St(C)) : C ∈ ID(s)}. Since |M|d(s) should be the greatest lower bound of the E-value of all paths accepting s, further the greatest lower bound of {T (St(C)) : C ∈ ID(s)} by Lemma. Therefore the language of M is L1 − L2. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

0 0 Now we will prove that L1 − L2 = L1 ∩ L2 ∈ Π2. First, consider a nondeterministic Turing machine M1 that guesses C to see whether C ∈ ID(s). Then, since ∧ is computable, M1 can decide the order relation of T (St(C)) and e as follows: 1 computing T (St(C)) ∧ e; 2 compare T (St(C)) ∧ e with T (St(C)), and with e: 1 If T (St(C)) ∧ e = T (St(C)), then e ≥ T (St(C)); 2 If T (St(C)) ∧ e = e, then T (St(C)) ≥ e; 3 If T (St(C)) ∧ e 6= T (St(C)) nor e, then T (St(C)) and e are not compatible. 0 0 That is, if ∧ is computable, so is the order ≥. Thus L1 ∈ Π1, L2 ∈ Σ2 0 0 and L1 − L2 = L1 ∩ L2 ∈ Π2. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

E valued Turing machines are more powerful than classical Turing machines and Wiedermann’s Turing machines.

0 0 0 There exists a language in Π2 but not in Σ1 ∪ Π1 that could be accepted by our ENTM. Example Suppose that Σ is {0, 1}, z is any binary integer. Let L be some non 0 0 0 recursively language in Σ1 and its complement L ∈ Π1. Obviously, the language {z|“z is even and z∈ / L” or “z is odd and z ∈ L”} is in 0 0 0 Π2, but not in Σ1 and not in Π1. First, we construct an ENTM that can recognize the language “z is even and z ∈ L”. It is easy to construct a new E-valued Turing machine Mˆ such that |Mˆ |d(z) = 0 if “z is even and z ∈ L”, and |Mˆ |d(z) = 1 otherwise. Second,we construct an ENTM that can recognize the language “z is odd and z∈ / L”. Let 0 < c < a be elements of E. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Continue Construct an E-valued Turing machine M˜ = (Q,˜ Σ, Γ, δ,˜ B, I,˜ T˜) as follows: ˜ 0 0 0 0 Q = {qinit} ∪ Q ∪ {qf } ∪ Q0, where qinit, qf ∈/ Q ∪ Q0.

I˜(qinit) = 0 and I˜(q) = 1 otherwise. T˜(q) = c for any q ∈ F , and T˜(q) = 1 if q ∈ Q0 − F .

T˜(qinit) = 1 and T˜(qf ) = a. ˜ 0 0 T (q) = T0(q) for q ∈ Q0. ˜ δ(qinit, a, q, a, S) = 0 for q ∈ {q0, qf }, and ˜ 0 0 0 δ(qinit, a, q0, a, S) = I0(q0). δ˜(p, a, q, b, D) = 0 if (p, a, q, b, D) is a transition of M 0. ˜ δ(qinit, a, qf , a, D) = 0. ˜ 0 0 δ(p, a, q, b, D) = δ0(p, a, q, b, D) for any p, q ∈ Q0. δ˜ = 1 otherwise.

Easy to see that |M˜ |d(z) = a if “z is odd and z∈ / L” and |M˜ |d(z) = c otherwise. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Continue

Since languages of ENTM are closed under the ∧ operation. Therefore there is an E-valued Turing machine M¯ such that |M¯ |d(z) = 0 if “z is even and z ∈ L”, |M¯ |d(z) = a if “z is odd and z∈ / L”, and |M¯ |d(z) = c otherwise. Obviously, for the set of {z ∈ Σ∗|“z is even and z∈ / L” or “z is odd and z ∈ L”}, the recognized degree by M¯ is c. Since the language 0 {z|“z is even and z∈ / L” or “z is odd and z ∈ L”} is in Π2, but not in 0 0 Σ1 and not in Π1, so the languages of E-valued Turing machines exceed 0 0 Σ1 ∪ Π1. So our E valued Turing machines are more powerful than classical Turing machines and Wiedermann’s Turing machines. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Main Results for ENTMs

Theorem T T 0 0 If E is locally finite and linear, then Ld (E, Σ) = Lw(E, Σ) = Σ1 ∪ Π1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Proof. T T 0 0 We only need to prove Ld (E, Σ) = Lw(E, Σ) ⊆ Σ1 ∪ Π1. Let M = (Q, Σ, Γ, δ, B, pI ,T ) be an ENTM, and assume the operation ∧ is computable. Define

L1 ={s#e : ∃C ∈ ID(s),T (St(C)) = e}

L2 ={s#e : ∀C ∈ ID(s),T (St(C)) ≥ e}.

Then there is a classical nondeterministic Turing machine M1 that guesses C and simulates M on s to check whether C ∈ ID(s) and 0 0 T (St(C)) = e. So L1 ∈ Σ1 and similarly L2 ∈ Π1, the language is 0 0 ∗ L1 ∩ L2 ∈ Σ1 ∪ Π1. Since E is linear, for any s ∈ Σ there is s#|M|d(s) ∈ L1. Otherwise there may be L1 ∩ L2 = ∅. That is T 0 0 T 0 0 Ld (E, Σ) ⊆ Σ1 ∪ Π1, and therefore Ld (E, Σ) = Σ1 ∪ Π1. Since E will degenerate to be an MV algebra when it is linear, it follows T T 0 0 that Ld (E, Σ) = Lw(E, Σ) = Σ1 ∪ Π1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Similar Results for LNTMs

1 0 0 T Σ1 ∪ Π1 ⊆ Ld (L, Σ). 2 0 0 T Σ1 ∪ Π1 ⊆ Lw(L, Σ). 3 T 0 Ld (L, Σ) ⊆ Π2. 4 T 0 If L is finite, then Lw(L, Σ) ⊆ Π2. 5 T T 0 0 If L is linear, then Ld (L, Σ) = Lw(L, Σ) = Σ1 ∪ Π1. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Conclusions

1 Turing machines based on quantum logic has some super-Turing computational power. It is more powerful than Wiedermann’s Turing machines and classical Turing machines. 2 Physical system determines the logical structure, conversely, logical structures affect properties of physical machines. Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

Reference

Yun S., Xian L., Ruqian L.,Computing theory based on quantum logic, Proceeding of Turing 100-Alan Turing Centenary Conference, 278-288. Yun S., Xian L., Ruqian L.,Turing machines based on unsharp quantum logic, 8th quantum physics and logic, 251-261. Yun S., Xian L., Ruqian L.,Computing power of Turing machines based on quantum logic,(Theoretical computer science, 2015,598,2-14). Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic

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