Computing power of Turing machines based on quantum logicDi erence between classical computation and quantum computation We nd the essential di erence between automata theory based

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

Computing power of Turing machines based onquantum logic

Yun ShangAMSS, Chinese Academy of Sciences

UH-CAS Workshop on Mathematical Logic2018-11-02

Contents

1 Classical theory of computation

2 Theory of quantum automata1.Quantum automata based on quantum mechanics2.Quantum automata based on quantum logic

3 Theory of Turing machines based on quantum logic1.Basic definitions2.Languages of quantum Turing machines3.Computing power of Turing machines based on quantum logic

Classical theory of computation

1 Computability

2 Computational complexity

3 Algorithmic theory

4 Models of computation

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 controlledmachine 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)(measuremany); Ambainis and R.Freivalds (1998) (one way) ; Yamasaki,Kobayashi, Imai (2001)(two way)

Logical foundation of quantum mechanics–Quantum logic

1 Classical mechanics: Classical logic

2 Closed quantum system: Von Neumann’s quantum logic (sharpquantum logic) → PV measurement

3 Open quantum system: unsharp quantum logic → POVmeasurement

Algebraic model for quantum logic

1 sharp quantum logic → orthomodular lattice

2 unsharp quantum logic → effect algebra( Foulis,1994); QuantumMV algebra (Giuntini,1996)

Unsharp quantum logic model

pseudo booleandifference poset

pseudo difference poset

weakcoupled semiring

pseudocoupled semiring

pseudo MV algebra

(by G. Georgescu)

pseudo effect algebra

(by A. Dvurecenskij)

weak QMV algebra

MV algebra

(by C. C. Chang)

effect algebra QMV algebra

(by C. Giuntini)

MV algebra

Boolean difference poset

(by F. Kopka)

difference poset

(by F. Kopka)

? ? PPPP

minusdistributivity

- � �?

adddistributivity

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. SoftComputing - A Fusion of Foundations, Methodologies andApplications (2007),11.

X. Lu, Y. Shang, R. Q. Lu., J. Zhang, and F. F. Ma, Weak QMValgebras and some ring-like structures Soft Computing - A Fusion ofFoundations, Methodologies and Applications(2017),21.

Quantum automata based on quantum logic

1 Logical proposition ⇒ element of orthomodular lattice

2 Mingsheng Ying and Daowen Qiu (finite state automata andpushdown 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 andpushdown automata based on unsharp quantum logic)

M. S. Ying, A theory of computation based on quantum logic(I). 75pages, 2005, Theoretical computer science.

D. W. Qiu, Automata theory based on quantum logic: somecharacterizations, Information and Computation, 109:2(2004)179–195.

Yun S., Xian L., Ruqian L.,Automata theory based on unsharpquantum logic, Mathematical structures in computer science,19(2009),737-756.

Yun S, Xian L, Ruqian L, A theory of computation based on unsharpquantum logic:finite state automata and pushdownautomata,Theoretical computer science 434(2012),53-86.

Difference between classical computation and quantumcomputation

We find the essential difference between automata theory based onquantum logic and classical automata theory. That is the universalvalidity of many fundamental properties of automata depend heavily notonly on the distributive law but also on the non-contradiction law.

BooleanAlgebra MV Algebra

BooleanAlgebra

OrthomodularLattice

Lattice-orderedQMV Algebra

ClassicalAutomata

SharpQuantum Automata

UnsharpQuantum Automata

a� a 6= a

(a ∨ b) ∧ (a ∨ c) 6= a ∨ (b ∧ c)(a� b) ∧ (a� c) 6= a� (b ∧ c)

Physical implication

For sharp quantum automata, in order to preserve the classicalproperties of automata, the underlying logic should degenerate to bea boolean algebra. It requires that the simple observablescorresponding to projection operators are mutually commutative.

For unsharp quantum automata, in order to preserve the classicalproperties of automata, the simple observables corresponding toeffects need and only need to be coexistent.

What about Turing machines based on

quantum logic?

Orthomodular lattices and boolean algebras

1 An orthcomplemented lattice is an orthomodular lattice ifa ≤ c⇒ c = a ∨ (a ∧ c′). (Orthomodular law)

2 An orthcomplemented lattice is a boolean algebra ifa ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c). (Distributive law)

S-algebras

A supplement algebra (S-algebra for short) is an algebraic structureE = (E,�,′ ,0,1) consisting of set M with two constant elements0,1, a unary operation ′ and a binary operation � on M satisfyingthe following axioms:

(S1) a� b = b� a.(S2) a� (b� c) = (a� b)� c.(S3) a� a′ = 1.(S4) a� 0 = a.(S5) a′′ = a.(S6) a� 1 = 1.

MV algebras and QMV algebras

An multiple-valued (MV) algebra (Chang, 1957) is an S-algebra thatsatisfies:(MV) (a′ � b)′ � b = (a� b′)′ � a

A quantum MV (QMV) algebra (Giuntini,1996) is an S-algebra thatsatisfies:(QMV) a� [(a′ e b) e (c e a′)] = (a� b) e (a� c)where

a� b = (a′ � b′)′

a e b = (a⊕ b′)� b

Classical Turing machines

Definition

A non-deterministic Turing machine (NTM) is a 7-tuple:M = (Q,Σ,Γ, δ, B, q0, F ), where1. 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}.

· · · B a b c d B · · ·

· · · B a′ b c d B · · ·

· · · B a′ b′ c d B · · ·

· · · B a′ b′ c′ d B · · ·

L-valued Turing machines (LNTMs)

· · · B a b c d B · · ·

I(q0) = x

Ref. Y.M. Li, P. Li, Turing machine based on quantum logic. ChineseJournal of Computers (35)2012 1407-1420.

· · · B a′ b c d B · · ·

δ(q0, a, q1, a′, R) = y

· · · B a′ b′ c d B · · ·

δ(q1, b, q2, b′, R) = z

· · · B a′ b′ c′ d B · · ·

δ(q2, c, q3, c′, R) = 1

· · · B a′ b′ c′ d B · · ·

T (q3) = z

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,Rand S indicate that the head of the ENTM moves left, right or keepstationary.

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.

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,Rand S indicate that the head of the ENTM moves left, right or keepstationary.

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.

Languages of ENTMs

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

|M |d(s) =∧n≥1

∧q0∈Q

I(q0)� δ†(q0s, C1)� δ†(C1, C2)� · · ·

� δ†(Cn−1, Cn)� T (St(Cn))

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

|M |w(s) =∧n≥1

[∧Cn

(· · ·

(∧C2

(∧C1

(∧q0

I(q0)� δ†(q0s, C1)

)� δ†(C1, C2)

� δ†(C2, C3)

)· · ·

)� T (St(Cn))

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.

Comparision between |M |w and |M |d

��

δ(q1, y, q2) = b δ(q1, y, q3) = c

δ(q0, x, q1) = a

|M |d(xy) = (a� b) ∧ (a� c)

|M |w(xy) = a� (b ∧ c)

Definition

A partial function L : Σ+ → E is called an E-valued d-recursivelyenumerable (d-R.E.) language or E-valued w-recursively enumerable(w-R.E.) language if L ∈ LT

d (E ,Σ) or L ∈ LTw(E ,Σ).

Definition

A function L : Σ+ → E is called a E-valued d-recursive (d-R.) languageor E-valued w-recursive (w-R.) language if L = |M |d (L = |M |w) forsome M ∈ NTM(E ,Σ), where M could halt for any input.

DefinitionLetf, g be E-valued languages.

1 The intersection of two E-valued languages f and g, denoted byf ∧ 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, isdefined as (f � g)(s) = f(s)� g(s) for any s ∈ Σ∗.

3 The concatenation of two E-valued languages f and g, denoted byf · g, is defined as (f · g)(s) =

∧s1s2=s

[f(s1)� g(s2)] for any s ∈ Σ∗.

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 Xwidth-first X X X X

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 Turinglanguages.Cie 2012, how the worlds computes(UK, Cambridge,2012-06-18/06-23),125.

E-valued deterministic Turing machine(EDTM )

Definition

An E-valued deterministic Turing machine (EDTM) is an ENTM whosetransition function δ satisfies that, for any p ∈ Q and a ∈ Γ there existsat most one set {q, b,D} such that δ(p, a, q, b,D) 6= 1.

ENTM with classical characters

Definition

Let M = (Q,Σ,Γ, δ, B, I, T ) ∈ ENTM. We call δ to be classical ifδ(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.

For any M ∈ NTM(E ,Σ) there exists MI ∈ NTMI(E ,Σ) such that|M |d = |MI |d and |M |w = |MI |w.

For any M ∈ NTM(E ,Σ) there exists MT ∈ NTMT (E ,Σ) such that|M |d = |MT |d.

DefinitionA 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 andR�

M = {a1 � a2 � · · ·� an : ai ∈ RM , n ∈ N} ∪ {0}.If E is locally finite then R�

M is also finite.

LemmaLet 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.

ENTM is not equivalent to EDTM

TheoremENTMs are not equivalent to EDTMs and ENTMs have morecomputational power than EDTMs.

There exists an EDTM that can be simulated by a classical Turingmachine.

There exists an ENTM that cannot be simulated by any classicalTuring machine.

ENTM is more powerful than EDTM

Example

Let Mu = (Qu,Σ,Γ, δu, B, pI , QT ) be the universal Turing machineaccepting 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 theothers.

T (p) = 0 for p ∈ QT , and T = x for the others.

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 ′ simulating M ,then M ′ can be simulatedby some classical turing machine M ′′.The classic language {s ∈ Σ∗ : |M ′|d(s) = x} = Σ∗ − Lu must be r.e.,which contradicts that Lu is r.e..

Computing power of Turing machines based on quantumlogic

Some notations

Computing levels in the Arithmetical Hierarchy:Σ0 = Π0, Σ0

1, Π01, Σ0

2, Π02 · · ·

Σ0n = {E : E = Q1x1...QnxnF} for some primitive recursive

formula F , where Qj = ∃, if i is odd, and Qj = ∀, if i is even,n ≥ 0;

Π0n = {E : E = Q1x1...QnxnF} for some primitive recursive

formula F , where Qj = ∀, if i is odd, and Qj = ∃, if i iseven,n ≥ 0.( C.Smorynski,1977)

Notations about our languages class:

LTd (E ,Σ) = {|M |d : M is an ENTM}.

LTw(E ,Σ) = {|M |w : M is an ENTM}.

LTd (L,Σ) = {|M |d : M is an LNTM}.

LTw(L,Σ) = {|M |w : M is an LNTM}.

The computing power of classical Turing machines andWiedermann’s Turing machines

1 The languages class of classical Turing machines is Σ01;

2 The languages class of Wiedermann’s Turing machines is Σ01 ∪Π0

1(Theoretical computer science, 2004(317))

Main results for ENTMs

Theorem

Σ01 ∪Π0

1 ⊆ LTd (E ,Σ);

Σ01 ∪Π0

1 ⊆ LTw(E ,Σ).

Proof.

Let L ∈ Σ01 ∪Π0

1. If L ∈ Σ01, the classical Turing machine accepting L is

Mu = (Q′,Σ,Γ, δ′, q0, B, F ). Construct the ENTMM = (Q,Σ,Γ, δ, B, I, T ) as:

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 ′ and δ = 1otherwise.

Easy to check that |M |d(s) = 0 for any s ∈ L and |M |d(s) = 1 for alls /∈ L. So Σ0

1 ⊆ LTd (E ,Σ).

If L ∈ Π01, 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 anys /∈ L and |M |d(s) = 1 for all s ∈ L. So Π0

1 ⊆ LTd (E ,Σ).

Main results for ENTMs

Theorem

If E is locally finite, then LTd (E ,Σ) ⊆ Π0

Proof.

Let M be an ENTM. Here we treat any E-valued language L ∈ LTd (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 notthe 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 pathsaccepting s, further the greatest lower bound of {T (St(C)) : C ∈ ID(s)}by Lemma. Therefore the language of M is L1 − L2.

Now we will prove that L1 − L2 = L1 ∩ L′2 ∈ Π02.

First, consider a nondeterministic Turing machine M1 that guesses C tosee whether C ∈ ID(s). Then, since ∧ is computable, M1 can decide theorder 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.

That is, if ∧ is computable, so is the order ≥. Thus L1 ∈ Π01, L2 ∈ Σ0

and L1 − L2 = L1 ∩ L′2 ∈ Π02.

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

There exists a language in Π02 but not in Σ0

1 ∪Π01 that could be accepted

by our ENTM.

Example

Suppose that Σ is {0, 1}, z is any binary integer. Let L be some nonrecursively language in Σ0

1 and its complement L′ ∈ Π01. Obviously,

the language {z|“z is even and z /∈ L” or “z is odd and z ∈ L”} is inΠ0

2, but not in Σ01 and not in Π0

First, we construct an ENTM that can recognize the language “zis even and z ∈ L”. It is easy to construct a new E-valued Turingmachine M such that |M |d(z) = 0 if “z is even and z ∈ L”, and|M |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 .

ContinueConstruct an E-valued Turing machine M = (Q,Σ,Γ, δ, B, I, T ) asfollows:

Q = {qinit} ∪Q′ ∪ {qf} ∪Q′0, where qinit, qf /∈ Q′ ∪Q′0.

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

T (q) = c for any q ∈ F , and T (q) = 1 if q ∈ Q′ − F .

T (qinit) = 1 and T (qf ) = a.

T (q) = T ′0(q) for q ∈ Q′0.

δ(qinit, a, q, a, S) = 0 for q ∈ {q0, qf}, and

δ(qinit, a, q′0, a, S) = I ′0(q′0).

δ(p, a, q, b,D) = 0 if (p, a, q, b,D) is a transition of M ′.

δ(qinit, a, qf , a,D) = 0.

δ(p, a, q, b,D) = δ′0(p, a, q, b,D) for any p, q ∈ Q′0.

δ = 1 otherwise.

Easy to see that |M |d(z) = a if “z is odd and z /∈ L” and |M |d(z) = cotherwise.

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 andz /∈ L”, and |M |d(z) = c otherwise.

Obviously, for the set of {z ∈ Σ∗|“z is even and z /∈ L” or “z is odd andz ∈ L”}, the recognized degree by M is c. Since the language{z|“z is even and z /∈ L” or “z is odd and z ∈ L”} is in Π0

2, but not inΣ0

1 and not in Π01, so the languages of E-valued Turing machines exceed

Σ01 ∪Π0

1. So our E valued Turing machines are more powerful thanclassical Turing machines and Wiedermann’s Turing machines.

Main Results for ENTMs

Theorem

If E is locally finite and linear, then LTd (E ,Σ) = LT

w(E ,Σ) = Σ01 ∪Π0

Proof.

We only need to prove LTd (E ,Σ) = LT

w(E ,Σ) ⊆ Σ01 ∪Π0

1.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 thatguesses C and simulates M on s to check whether C ∈ ID(s) andT (St(C)) = e. So L1 ∈ Σ0

1 and similarly L2 ∈ Π01, the language is

L1 ∩ L2 ∈ Σ01 ∪Π0

1. Since E is linear, for any s ∈ Σ∗ there iss#|M |d(s) ∈ L1. Otherwise there may be L1 ∩ L2 = ∅. That isLTd (E ,Σ) ⊆ Σ0

1 ∪Π01, and therefore LT

d (E ,Σ) = Σ01 ∪Π0

1.Since E will degenerate to be an MV algebra when it is linear, it followsthat LT

d (E ,Σ) = LTw(E ,Σ) = Σ0

1 ∪Π01.

Computing power of Turing machines based on quantum logicDi erence between classical computation and quantum computation We nd the essential di erence between automata theory based

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