computing power of turing machines based on quantum logicdi erence between classical computation and...
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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
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 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 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 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)
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 (sharpquantum logic) → PV measurement
3 Open quantum system: unsharp quantum logic → POVmeasurement
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); QuantumMV algebra (Giuntini,1996)
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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)
--6
-
? 6
?
6
?
-
6
?
? ? PPPP
PPPP
Pi
? ?
minusdistributivity
- � �?
adddistributivity
6
?
6
?
-
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. 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.
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 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)
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). 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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
BooleanAlgebra MV Algebra
BooleanAlgebra
OrthomodularLattice
Lattice-orderedQMV Algebra
ClassicalAutomata
SharpQuantum Automata
UnsharpQuantum Automata
-
6 6
6 6 6
-
a� a 6= a
a� a 6= a
(a ∨ b) ∧ (a ∨ c) 6= a ∨ (b ∧ c)(a� b) ∧ (a� c) 6= a� (b ∧ c)
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 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.
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 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)
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 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.
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 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 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 ), 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}.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Classical Turing machines
Classical Turing machines
q0
· · · B a b c d B · · ·
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Classical Turing machines
q1
· · · B a′ b c d B · · ·
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Classical Turing machines
q2
· · · B a′ b′ c d B · · ·
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Classical Turing machines
q3
· · · B a′ b′ c′ d B · · ·
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
L-valued Turing machines (LNTMs)
q0
· · · B a b c d B · · ·
0
y
x
1
z
I(q0) = x
Ref. Y.M. Li, P. Li, Turing machine based on quantum logic. ChineseJournal 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)
q1
· · · B a′ b c d B · · ·
0
y
x
1
z
δ(q0, a, q1, a′, R) = y
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
L-valued Turing machines (LNTMs)
q2
· · · B a′ b′ c d B · · ·
0
y
x
1
z
δ(q1, b, q2, b′, R) = z
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
L-valued Turing machines (LNTMs)
q3
· · · B a′ b′ c′ d B · · ·
0
y
x
1
z
δ(q2, c, q3, c′, R) = 1
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
L-valued Turing machines (LNTMs)
q3
· · · B a′ b′ c′ d B · · ·
0
y
x
1
z
T (q3) = z
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,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.
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,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.
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) =∧n≥1
∧Ci
∧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))
]
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
q2 q3
q1
q0
@@@R
��
�
?
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)
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-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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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∗.
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 Xwidth-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 Turinglanguages.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 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.
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 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.
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
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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
ENTM with classical characters
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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
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 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.
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 ′ 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..
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Computing power of Turing machines based on quantumlogic
Some notations
Computing levels in the Arithmetical Hierarchy:Σ0 = Π0, Σ0
1, Π01, Σ0
2, Π02 · · ·
where
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)
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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}.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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))
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
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 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).
q0
Mu q1
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Main results for ENTMs
Theorem
If E is locally finite, then LTd (E ,Σ) ⊆ Π0
2.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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
2
and L1 − L2 = L1 ∩ L′2 ∈ Π02.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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
1.
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 .
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
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 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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Main Results for ENTMs
Theorem
If E is locally finite and linear, then LTd (E ,Σ) = LT
w(E ,Σ) = Σ01 ∪Π0
1.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
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.
Classical theory of computation Theory of quantum automata Theory of Turing machines based on quantum logic
Similar Results for LNTMs
1 Σ01 ∪Π0
1 ⊆ LTd (L,Σ).
2 Σ01 ∪Π0
1 ⊆ LTw(L,Σ).
3 LTd (L,Σ) ⊆ Π0
2.
4 If L is finite, then LTw(L,Σ) ⊆ Π0
2.
5 If L is linear, then LTd (L,Σ) = LT
w(L,Σ) = Σ01 ∪Π0
1.
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-Turingcomputational power. It is more powerful than Wiedermann’s Turingmachines and classical Turing machines.
2 Physical system determines the logical structure, conversely, logicalstructures 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 quantumlogic, Proceeding of Turing 100-Alan Turing Centenary Conference,278-288.
Yun S., Xian L., Ruqian L.,Turing machines based on unsharpquantum logic, 8th quantum physics and logic, 251-261.
Yun S., Xian L., Ruqian L.,Computing power of Turing machinesbased 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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