Turing machines based on unsharp quantum logic
BBart Jacobs, Peter Selinger, and Bas Spitters (Eds.):8th International Workshop on Quantum Physics and Logic (QPL 2011)EPTCS 95, 2012, pp. 251–261, doi:10.4204/EPTCS.95.17 c (cid:13)
Y. Shang, X. Lu& R.Q. LuThis work is licensed under theCreative Commons Attribution License.
Turing machines based on unsharp quantum logic ∗ Yun Shang
Institute of Mathematics,AMSS, CASBeijing, P.R.China [email protected]
Xian Lu
Institute of Mathematics,AMSS, CASBeijing, P.R.China [email protected]
Ruqian Lu
Institute of Mathematics,AMSS, CASBeijing, P.R.ChinaCAS Key Lab of IIPInstitute of Computing Technology, CAS.Beijing, P.R.China [email protected]
In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-orderedquantum multiple-valued (MV) algebra E , we introduce E -valued non-deterministic Turing ma-chines ( E NTMs) and E -valued deterministic Turing machines ( E DTMs). We discuss different E -valued recursively enumerable languages from width-first and depth-first recognition. We find thatwidth-first recognition is equal to or less than depth-first recognition in general. The equivalencerequires an underlying E value lattice to degenerate into an MV algebra. We also study variants of E NTMs. E NTMs with a classical initial state and E NTMs with a classical final state have the samepower as E NTMs with quantum initial and final states. In particular, the latter can be simulatedby E NTMs with classical transitions under a certain condition. Using these findings, we prove that E NTMs are not equivalent to E DTMs and that E NTMs are more powerful than E DTMs. This is anotable difference from the classical Turing machines.
In traditional von Neumann quantum logic, P ( H ) (the set of all projection operators of a Hilbert space H ) is regarded as a set of quantum events. It constitutes an orthomodular lattice, which is the mainalgebraic model in quantum logic. However, since the set of projection operators is not the maximalset of possible events according to the statistical rules of open quantum systems, E ( H ) (the set of allpositive operators dominated by the identity on H ) becomes a new quantum event set. Since any eventin P ( H ) always satisfies the non-contradiction law, traditional quantum logic is called sharp quantumlogic. Quantum events represented by E ( H ) do not satisfy the non-contradiction law, and the quantumlogic corresponding to E ( H ) is called unsharp quantum logic. Many algebraic structures have beenproposed to characterize unsharp quantum events, and effect algebras [4] are the main model for unsharpquantum logic. Multiple-valued (MV) algebras, as algebraic models of multiple-valued logic, play ananalogous role to that of Boolean algebras in sharp quantum logic [3]. Quantum MV (QMV) algebrasare another important type of unsharp quantum structure [5].For abstract mathematical machines, automata theory is one of the main branches in classical com-puting theory. It mainly consists of finite-state automata, pushdown automata, and Turing machines.Although classical computing theory can be regarded as part of classical mathematical theory, the log-ical foundation of automata theory is still Boolean logic. Quantum logic differs from classical logicand quantum devices should obey their own logic. Hence, an interesting question arises: can we set ∗ This work was supported by an NSFC project under Grant No. 61073023, by a 973-Project under Grant No.2009CB320701, by the K. C. Wong Education Foundation, Hong Kong, and by the National Center for Mathematics andInterdisciplinary Sciences, CAS.
52 Turing machines based on unsharp quantum logicup a quantum computation theory based on quantum logic? Ying et al. set up finite-state automata andpushdown automata theories based on sharp quantum logic [15, 11]. They found that some importantproperties similar to classical automata are universally valid if and only if the underlying truth value lat-tice degenerates to a Boolean algebra. Li proved that deterministic finite automata and non-deterministicfinite automata based on sharp quantum logic are equivalent, independent of the distributive law [8].Since unsharp quantum logic is more universal than sharp quantum logic, Shang et al. set up finite-stateautomata and pushdown automata theories based on unsharp quantum logic. They found that some im-portant properties similar to classical automata are universally valid if and only if the underlying truthvalue lattice degenerates to an MV algebra [13, 12].Since Turing machines are a core concept in the study of computing theory, we continue to studyTuring machines based on unsharp quantum logic. Deutsch proposed quantum Turing machines froma quantum mechanics point of view [2] and Perdrix generalized this to observable quantum Turing ma-chines [10]. Perdrix and Jorrand introduced classically controlled Turing machines [9]. Bernstein et al.addressed universal quantum Turing machines [1]. However, the logical foundation for these machines isstill Boolean logic. The relation between the above Turing machines and the proposed Turing machinesis similar to the relation between quantum mechanics and quantum logic.In this paper, we mainly consider two algebraic models of unsharp quantum logic for Turing ma-chines, namely extended lattice-ordered-effect algebras and lattice-ordered QMV algebras. Here wecall them E -valued lattices. Although similar to finite-state automata and pushdown automata basedon unsharp quantum logic, some important properties of Turing machines based on unsharp quantumlogic depend heavily on the distributivity of the underlying logic. However, we find that E -valued non-deterministic Turing machines ( E NTMs) are not equivalent to E -valued deterministic Turing machines( E DTMs) even if the distributivity of the underlying logic holds. This is a characteristic difference fromclassical Turing machines.The remainder of the paper is organized as follows. Section 2 provides some algebraic results usedlater in the paper. In Section 3, we introduce the concepts of E NTMs and E DTMs. We also define twopatterns of recursively enumerable language recognition for unsharp quantum Turing machines: width-first (namely, parallel) recognition and depth-first (namely, sequential) recognition, similar to the casein unsharp quantum automata. We prove that the width-first recognizability of a recursively enumerablequantum language is always equal to or less than its depth-first recognizability. We find that equivalencerequires the underlying E value lattice to degenerate to an MV algebra. In section 4, we discuss variantsof unsharp quantum Turing machines. E NTMs with a classical initial state and E NTMs with a classicalfinal state have the same power as E NTMs with quantum initial and final states. In particular, under acertain condition, the latter can be simulated by an E NTM with classical transitions. Using these results,we find that E NTMs are more powerful than E DTMs. This is different from the result in classicalcomputing theory. Section 5 presents our main conclusion.
First, we provide some notions and results in unsharp quantum logic.
Definition 2.1 [3] A supplement algebra (S-algebra for short) is an algebraic structure E = ( E , ⊞ , ′ , , ) consisting of set M with two constant elements , , a unary operation ′ and a binary operation ⊞ on M satisfying the following axioms:.Shang,X.Lu&R.Q.Lu 253(S1) a ⊞ b = b ⊞ a . (S2) a ⊞ ( b ⊞ c ) = ( a ⊞ b ) ⊞ c .(S3) a ⊞ a ′ = . (S4) a ⊞ = a .(S5) a ′′ = a . (S6) a ⊞ = .An MV algebra is an S-algebra that satisfies:(MV) ( a ′ ⊞ b ) ′ ⊞ b = ( a ⊞ b ′ ) ′ ⊞ a .For an S-algebra, we define the following three binary operations: a ⊙ b = ( a ′ ⊞ b ′ ) ′ , a ⊓ b = ( a ⊞ b ′ ) ⊙ b , and a ⊔ b = ( a ⊙ b ′ ) ⊞ b .A QMV algebra is an S-algebra that satisfies:(QMV1) a ⊔ ( b ⊓ a ) = a .(QMV2) ( a ⊓ b ) ⊓ c = ( a ⊓ b ) ⊓ ( b ⊓ c ) .(QMV3) a ⊞ [ b ⊓ ( a ⊞ c ) ′ ] = ( a ⊞ b ) ⊓ ( a ⊞ ( a ⊞ c ) ′ ] .(QMV4) a ⊞ ( a ′ ⊓ b ) = a ⊞ b .(QMV5) ( a ′ ⊞ b ) ⊔ ( b ′ ⊞ a ) = .A partial relation ≤ in QMV algebra can be defined as a ≤ b iff a = a ⊓ b .It is clear that a QMV algebra is not necessarily a lattice under the operations ⊓ and ⊔ . If E formsa lattice with ≤ , it is called a lattice-ordered QMV algebra, where ∧ denotes the infimum operation and ∨ denotes the supremum operation in the lattice. A QMV algebra M is quasilinear if a b implies a ⊓ b = b . A QMV algebra (or an MV algebra) M is linear if ∀ a , b ∈ M , either a ≤ b or b ≤ a . Thereexists a QMV algebra that is not quasilinear (Example 1, [6]). Every MV algebra is a QMV algebra;however, there exists a QMV algebra that is not an MV algebra (Example 2.7, [13]).An effect algebra is a set P with two particular elements 0 , ( = ) and with a partial binaryoperation ⊕ : P × P −→ P such that, for all a , b , c ∈ P :(E1) If a ⊕ b ∈ P , then b ⊕ a ∈ P and a ⊕ b = b ⊕ a .(E2) If b ⊕ c ∈ P and a ⊕ ( b ⊕ c ) ∈ P , then a ⊕ b ∈ P and ( a ⊕ b ) ⊕ c ∈ P and a ⊕ ( b ⊕ c ) = ( a ⊕ b ) ⊕ c .(E3) For any a ∈ P there is a unique b ∈ P such that a ⊕ b is defined and a ⊕ b = ⊕ a is defined, then a = Example 2.1
Let j = ( E , ⊕ , , ) be an effect algebra. The operation ⊕ can be extended to a totaloperation ⊞ : E × E −→ E by defining a ⊞ b = (cid:26) a ⊕ b , if ( a ⊕ b ) is defined1 , otherwise.We denote the resulting structure by ¯ j = ( E , , , ⊞ ) and call it an extended-effect algebra. It is easyto see that an extended-effect algebra ¯ j preserves the order of the effect algebra and is equivalent to aquasilinear QMV algebra [6]. Theorem 2.1 [13] Let E = ( E , ⊞ , ′ , , ) be a lattice-ordered QMV algebra. The following conditionsare equivalent:(i) E is an MV algebra.(ii) ( a ⊞ b ) ∧ ( a ⊞ c ) = a ⊞ ( b ∧ c ) for any a , b , c ∈ E . Theorem 2.2 [13] Let E = ( E , ⊞ , ′ , , ) be an extended lattice-ordered-effect algebra. The followingconditions are equivalent:(i) E is a linear MV algebra.(ii) ( a ⊞ b ) ∧ ( a ⊞ c ) = a ⊞ ( b ∧ c ) for any a , b , c ∈ E .54 Turing machines based on unsharp quantum logic If we let unsharp quantum logic denote the truth value set of the propositions, we can set up Turingmachines based on unsharp quantum logic. In the following, E denotes a lattice-ordered QMV algebra.If E denotes an extended lattice-ordered-effect algebra, we can obtain Turing machines based on anextended lattice-ordered-effect algebra without changing anything. Definition 3.1 An E -valued non-deterministic Turing machine ( E NTM) is a septuple: M = ( Q , S , G , d , B , I , T ) , where1. Q is a finite nonempty-state set.2. S is the finite set of input symbols.3. G is the complete set of tape symbols; S ⊆ G / B .4. d : Q × G × Q × G × { L , S , R } −→ E is the transition function. The symbols L , R and S indicatethat the head of the E NTM moves left or right or remains stationary, respectively.5. B is the blank symbol. The blank symbol appears initially in all but the finite number of initialcells that hold input symbols.6. I : Q −→ E is the initial-state function.7. T : Q −→ E is the final- or accepting-state function.As defined for classical Turing machines, a configuration or instantaneous description (ID) of an E NTM M is a sequence C = a q a , where q ∈ Q and a a is the finite sequence between the leftmost andrightmost nonblank symbols. We denote the state of C by St ( C ) and denote ID ( M ) as the set of allinstantaneous descriptions of M . An E NTM in ID a q a means the current state is q and the readinghead is looking at the first symbol of a . The value of M transforming from C to C is described as d ⋆ ( C , C ) = d ( p , a , q , b , L ) , if C = a cpa b and C = a qcb bd ( p , a , q , b , S ) , if C = a pa b and C = a qb bd ( p , a , q , b , R ) , if C = a pa b and C = a bq b , otherwise,where a , b , c ∈ G and a , b ∈ G ∗ such that the leftmost symbol of a and the rightmost symbol of b arenot B . ⊢ ( C , C ) = ( p , a , q , b , D ) denotes that the E NTM can transform C to C through the transition ( p , a , q , b , D ) .Similar to finite-state automata theory based on unsharp quantum logic, by interacting ∧ and ⊞ , wecan adapt depth-first and width-first methods for defining the degree of acceptance of languages recog-nized by Turing machines. In fact, these correspond to parallel recognition and sequential recognition.We prove that the methods coincide only when the truth lattice is an MV algebra. Definition 3.2
A path of an E NTM M is a finite sequence of IDs. Definition 3.3
The E -valued language accepted by an E NTM M in a depth-first manner is defined as: | M | d ( s ) = ^ n ≥ ^ C i ^ q ∈ Q I ( q ) ⊞ d ⋆ ( q s , C ) ⊞ d ⋆ ( C , C ) ⊞ · · · ⊞ T ( St ( C n )) (1)for any s ∈ S + ..Shang,X.Lu&R.Q.Lu 255 Definition 3.4
The E -valued language accepted by an E NTM M in a width-first way is defined as: | M | w ( s ) = ^ n ≥ (cid:20) ^ C n (cid:18) · · · (cid:18) ^ C ^ C (cid:18) ^ q I ( q ) ⊞ d ⋆ ( q s , C ) (cid:19) ⊞ d ⋆ ( C , C ) (cid:19) ⊞ d ⋆ ( C , C ) (cid:19) · · · (cid:19) ⊞ T ( St ( C n )) (cid:21) (2)for any s ∈ S + . Remark 3.1
Similar to classical Turing machines, an E NTM M halts when it reaches some state q with T ( q ) < C with T ( St ( C )) = d ⋆ ( C , C ′ ) = C ′ . Each path inEquations (1) and (2) is required to halt. If the machine does not halt for some input s in all paths, thenthe E -value of s accepted by M is not defined. Definition 3.5 An E -valued deterministic Turing machine ( E DTM) is an E NTM whose transition func-tion d satisfies the following: for any p ∈ Q and a ∈ G , there exists at most one set { q , b , D } such that d ( p , a , q , b , D ) = E NTMs and E DTMs over alphabet S are denoted by NTM ( E , S ) and DTM ( E , S ) ,respectively. We denote L Td ( E , S ) = {| M | d : M ∈ NTM ( E , S ) } and L Tw ( E , S ) = {| M | w : M ∈ NTM ( E , S ) } . Definition 3.6
A partial function L : S + → E is called an E -valued d-recursively enumerable (d-RE)language or an E -valued w-recursively enumerable (w-RE) language if L ∈ L Td ( E , S ) or L ∈ L Tw ( E , S ) ,respectively. Proposition 3.1 (i) | M | w ≤ | M | d for any E NTM M .(ii) | M | w = | M | d for any E NTM M iff E is an MV algebra. Proof :
Point (i) is obvious since a ⊞ ( b ∧ c ) ≤ ( a ⊞ b ) ∧ ( a ⊞ c ) for a , b , c ∈ E in general. (ii) If E is anMV algebra, then ⊞ distributes over ∧ , so | M | w = | M | d . Conversely, for any a , b , c ∈ E we construct an E NTM M = ( { q , q , q } , S , G , d , B , I , T ) as follows. For some s ∈ S , I ( q ) = b , I ( q ) = c , I ( q ) = , T ( q ) = , T ( q ) = , T ( q ) = a d ( q , s , q , s , R ) = d ( q , s , q , s , R ) = d = s = s , all the effective paths are q s ⊢ s q and q s ⊢ s q .Thus, | M | d ( s ) = ( I ( q ) ⊞ d ⋆ ( q s , s q ) ⊞ T ( q )) ∧ ( I ( q ) ⊞ d ⋆ ( q s , s q ) ⊞ T ( q )) = ( a ⊞ b ) ∧ ( a ⊞ c ) .From the definition it is easy to see that | M | w ( s ) = [( I ( q ) ⊞ d ⋆ ( q s , s q )) ∧ ( I ( q ) ⊞ d ⋆ ( q s , s q )) ⊞ T ( q )] = ( b ∧ c ) ⊞ a . Therefore, a ⊞ ( b ∧ c ) = ( a ⊞ b ) ∧ ( a ⊞ c ) . Q.E.D.
Definition 4.1
Let M = ( Q , S , G , d , B , I , T ) be an E NTM. We call d classical if d ( p , a , q , b , D ) = ∀ p , q ∈ Q , ∀ a , b ∈ G and ∀ D ∈ { L , S , R } . Similarly, we call I ( T ) classical if I ( p ) = T ( p ) = ∀ p ∈ Q . The subclass of all E NTMs with a classical initial-state (terminal-state) function is denotedas NTM I ( E , S ) (NTM T ( E , S ) ). We define NTM IT ( E , S ) = NTM I ( E , S ) ∩ NTM T ( E , S ) .The following results show that any E NTM can be simulated by an E NTM with a classical initial-state function. That is, E NTMs with classical initial states are as powerful as general E NTMs.56 Turing machines based on unsharp quantum logic
Lemma 4.1
For any M ∈ NTM ( E , S ) there exists M I ∈ NTM I ( E , S ) such that | M | d = | M I | d and | M | w = | M I | w . Proof:
Assuming M = ( Q , S , G , d , B , I , T ) , we construct M I = ( Q I , S , G , d I , B , I I , T I ) , where Q I = Q ∪ { p I } and p I / ∈ Q , I I ( p I ) = , and I I ( q ) = , ∀ q ∈ QT I ( p I ) = , and T I ( q ) = T ( q ) , ∀ q ∈ Q d I ( p , a , q , b , D ) = d ( p , a , q , b , D ) , ∀ p , q ∈ Q d I ( p I , a , q , a , S ) = I ( q ) , ∀ q ∈ Q and d I = M I the new state p I is the unique initial state. It is straightforward to see that | M | d = | M I | d . We can directly prove the width-first method. | M I | w ( s ) = ^ n ≥ (cid:20) ^ C n (cid:18) · · · (cid:18) ^ C (cid:18) ^ q ∈ Q I I ( q ) ⊞ d ⋆ I ( q s , C ) (cid:19) ⊞ d ⋆ I ( C , C ) (cid:19) · · · (cid:19) ⊞ T I ( q n ) (cid:21) = ^ n ≥ (cid:20) ^ C n (cid:18) · · · (cid:18) ^ C (cid:18) I I ( p I ) ⊞ d ⋆ I ( p I s , C ) (cid:19) ⊞ d ⋆ I ( C , C ) (cid:19) · · · (cid:19) ⊞ T I ( q n ) (cid:21) = . . . = ^ n ≥ (cid:20) ^ C n (cid:18) · · · (cid:18) ^ q ∈ Q I ( q ) ⊞ d ⋆ ( q s , C ) (cid:19) · · · (cid:19) ⊞ T ( q n ) (cid:21) = | M | w ( s ) Q.E.D.
Symmetrically, any E NTM can be simulated by an E NTM with a classical terminal-state function.
Lemma 4.2
For any M ∈ NTM ( E , S ) there exists M T ∈ NTM T ( E , S ) such that | M | d = | M T | d . Proof:
Let M = ( Q , S , G , d , B , I , T ) and M T = ( Q T , S , G , d T , B , I T , T T ) , where Q T = { ( p , T ( p )) : p ∈ Q } , I T (( p , T ( p )) = I ( p ) d T (( p , T ( p )) , a , ( q , T ( q )) , b , D ) = ( d ( p , a , q , b , D ) ⊞ T ( q ) , if T ( q ) < d ( p , a , q , b , D ) , if T ( q ) = T T (( p , T ( p ))) = ( T ( p ) <
11 if T ( p ) = d T = E NTM halts in two cases: (i) it reaches some state p such that T ( p ) < a pa b such that T ( p ) = d ( p , a , q , b , D ) = ∀ q , b , D .Let s ∈ S + be an arbitrary input and let M halt along the path P = ( C = p I s , C , · · · , C n ) . Suppose ⊢ ( C i − , C i ) = ( p i − , a i − , p i , a i , D i ) , i = , · · · , n . Then there is a path M T : P T = ( ˜ C = ( p I , T ( p I )) s , ˜ C , · · · , ˜ C n ) ,where ˜ C i = a ( p , T ( p )) b if C i = a p b and ⊢ ( ˜ C i − , ˜ C i ) = (( p i − , T ( p i − )) , a i − , ( p i , T ( p i )) , a i , D i ) , i = , · · · , n . If M halts in case (i), then T ( St ( C n )) <
1. Obviously M T halts along P T and the E -values of P and P T are the same. If M halts in case (ii), then T ( St ( C n )) = d ⋆ ( C n , C ′ ) = C ′ . By the def-inition, d T (( p , T ( p )) , a , ( q , T ( q )) , b , D ) ≥ d ( p , a , q , b , D ) , so d ⋆ T ( ˜ C n , ˜ C ′ ) = C ′ and T T ( St ( ˜ C )) = M T also halts along P T and the E -values of P and P T all equal 1..Shang,X.Lu&R.Q.Lu 257Conversely, assume that M T halts along the path P T = ( ˜ C = ( p , T ( p )) s , ˜ C , · · · , ˜ C n ) . Suppose that ⊢ ( ˜ C i − , ˜ C i ) = (( p i − , T ( p i − )) , a i − , ( p i , T ( p i )) , a i , D i ) , i = , · · · , n . Then there is a path P = ( C = ps , C , · · · , C n ) where C i = a p b if ˜ C i = a ( p , T ( p )) b . If M T halts along P T in case (i), i.e. T T ( p n , T ( p n )) =
0, then M halts along P since T ( p n ) < E -values of P and P T are the same.If M T halts along P T in case (ii), then d T (( p n , T ( p n )) , a n , ( q , T ( q )) , b , D ) = q , b , D and T T (( p n , T ( p n ))) =
1. First, if d T (( p n , T ( p n )) , a n , ( q , T ( q )) , b , D ) = d ( p n , a n , q , b , D ) ⊞ T ( q ) = q , b , D , we have T ( q ) < M halts along P ′ = ( C , · · · , C n , C n + ) , where ⊢ ( C n , C n + ) =( p n , a n , q , b , D ) and the E -value of P ′ is 1. Otherwise, if d T (( p n , T ( p n )) , a n , ( q , T ( q )) , b , D ) = d ( p n , a n , q , b , D ) = q , b , D , then M halts along P and the E -value of P is 1.Therefore, we conclude that if M halts along some path, then M T also halts along the “mirror” pathwith the same E -value and vice versa. Q.E.D.
Combining Lemmas 4.1 and 4.2, we know that the non-classical parts of E NTMs can exist only inthe transition processes.
Corollary 4.3
For any M ∈ NTM ( E , S ) there exists M IT ∈ NTM IT ( E , S ) such that | M IT | d = | M | d .Therefore, from now on we can denote an E NTM by M = ( Q , S , G , d , B , p I , T ) if needed. Definition 4.2
A path ( C , · · · , C n ) is effective if d ⋆ ( C i − , C i ) = i = , · · · , n . On an effective path,each d ⋆ ( C i − , C i ) = d ( St ( C i − ) , a , St ( C i ) , b , D ) for some a , b ∈ G and D ∈ { L , S , R } . Definition 4.3
Let M = ( Q , S , G , d , B , p I , T ) be an E NTM. For any s ∈ S + , we define ID M ( s , ) = { C ∈ ID ( M ) : ( p I s , C ) as an effective path } and ID M ( s , n + ) = { C ∈ ID ( M ) : ( C ′ , C ) as an effective path forsome C ′ ∈ ID M ( s , n ) } , n = , , · · · . Let ID M ( s ) = S n ID M ( s , n ) comprise all the IDs achievable from p I s .We omit the subscript M if no confusion is possible.From the definition above, Equation (1) can be simplified to | M | d ( s ) = ^ n ≥ ^ C i ∈ ID ( s , n ) d ⋆ ( p I s , C ) ⊞ · · · ⊞ d ⋆ ( C n − , C n ) ⊞ T ( St ( C n )) (3)and Equation (2) can be simplified to | M | w ( s ) = ^ n ≥ " ^ C n ∈ ID ( s , n ) · · · ^ C ∈ ID ( s , ) ^ C ∈ ID ( s , ) d ⋆ ( p I s , C ) ⊞ d ⋆ ( C , C ) ! ⊞ d ⋆ ( C , C ) ! · · · ! ⊞ T ( St ( C n )) . (4)We denote the range of a map f by R ( f ) . For an E NTM M = ( Q , S , G , d , B , I , T ) , let R M = R ( I ) ∪ R ( d ) ∪ R ( T ) . We assume that R M = { x , x , · · · , x k } since it is finite. Thus, the value of path P is e ( P ) = v x ⊞ v x ⊞ · · · ⊞ v l x k , or simply represented by a k -vector v ( P ) = ( v , · · · , v k ) . Two k -vectors ( v , · · · , v k ) and ( v ′ , · · · , v ′ k ) are called compatible if v i ≤ v ′ i for all i , denoted by ( v , · · · , v k ) ≤ ( v ′ , · · · , v ′ k ) .Obviously if v ( P ) ≤ v ( P ) then e ( P ) ≤ e ( P ) . That is, in this case P can be omitted from the calculus.A set of k -vectors is called independent if and only if all elements are not compatible with each other.In fact, Proposition 2 in [14] showed that any independent set of such k -vectors is finite. Thus, there arefinite ∧ operations in Equations (3) and (4).Next we show that under some finiteness condition, each E NTM can be simulated by some E NTMwith classical transitions.58 Turing machines based on unsharp quantum logic
Theorem 4.4
Let M be an E NTM and let S M denote the subalgebra generated by R M . If S M is finite,there exists an E NTM ¯ M with classical transitions such that | M | w = | ¯ M | w . Proof.
Assume that M = ( Q , S , G , d , B , p I , T ) . We construct ¯ M = ( S QM , S , G , ¯ d , B , ¯ p I , ¯ T ) as follows:¯ p I ( q ) = ( , if q = p I , otherwise¯ T ( X ) = ∧ p ∈ Q X ( p ) ⊞ T ( p ) for any a , b ∈ G , X ∈ S QM and D ∈ { L , S , R } , ¯ d ( X , a , Y , b , D ) =
0, where Y ( q ) = ∧ p ∈ Q X ( p ) ⊞ d ( p , a , q , b , D ) ∈ S QM and ¯ d = d can be treated as a classical transition function S QM × G −→ S QM × G ×{ L , S , R } .We only need to consider effective paths ( Is , ¯ C , · · · , ¯ C n ) . For each effective path there exists a uniqueset { a i , b i , D i } ni = satisfying ¯ d ⋆ ( Is , ¯ C ) = ¯ d ( I , a , St ( ¯ C ) , b , D ) and ¯ d ⋆ ( ¯ C i − , ¯ C i ) = ¯ d ( St ( ¯ C i − ) , a i , St ( ¯ C i ) , b i , D i ) for i = , · · · , n . Thus, for any s ∈ S + , | ¯ M | w ( s ) = ^ n ≥ (cid:20) ^ ¯ C n (cid:18) · · · (cid:18) ^ ¯ C (cid:18) ^ ¯ C ¯ d ⋆ ( ¯ p I s , ¯ C ) ⊞ ¯ d ⋆ ( ¯ C , ¯ C ) (cid:19) ⊞ ¯ d ⋆ ( ¯ C , ¯ C ) (cid:19) · · · (cid:19) ⊞ ¯ T ( St ( ¯ C n )) (cid:21) = ^ n ≥ ^ ¯ C n ∈ ID ¯ M ( s , n ) ¯ T ( St ( ¯ C n )) = ^ n ≥ ^ ¯ C n ^ p n ∈ Q St ( ¯ C n )( p n ) ⊞ T ( p n )= ^ n ≥ ^ ¯ C n , p n (cid:18) ^ ¯ C n − ∈ ID M ( s , n − ) ^ p n − St ( ¯ C n − )( p n − ) ⊞ d ( p n − , a n , p n , b n , D n ) (cid:19) ⊞ T ( p n )= ^ n ≥ (cid:20) ^ ¯ C n , p n (cid:18) ^ ¯ C n − , p n − (cid:18) · · · (cid:18) ^ ¯ C , p St ( ¯ C )( p ) ⊞ d ( p , a , p , b , D ) (cid:19) · · · (cid:19) ⊞ d ( p n − , a n , p n , b n , D n ) (cid:19) ⊞ T ( p n ) (cid:21) = ^ n ≥ (cid:20) ^ ¯ C n , p n (cid:18) ^ ¯ C n − , p n − (cid:18) · · · (cid:18) ^ ¯ C , p d ( p I , a , p , b , D ) ⊞ d ( p , a , p , b , D ) (cid:19) · · · (cid:19) ⊞ d ( p n − , a n , p n , b n , D n ) (cid:19) ⊞ T ( p n ) (cid:21) = ^ n ≥ (cid:20) ^ ¯ C n , p n (cid:18) ^ ¯ C n − , p n − (cid:18) · · · (cid:18) ^ p , b , D d ( p I , a , p , b , D ) ⊞ d ( p , a , p , b , D ) (cid:19) · · · (cid:19) ⊞ d ( p n − , a n , p n , b n , D n ) (cid:19) ⊞ T ( p n ) (cid:21) = ^ n ≥ (cid:20) ^ ¯ C n , p n (cid:18) ^ ¯ C n − , p n − (cid:18) · · · (cid:18) ^ C ∈ ID M ( s , ) d ( p I , a , St ( C ) , b , D ) ⊞ d ( St ( C ) , a , p , b , D ) (cid:19) · · · (cid:19) ⊞ d ( p n − , a n , p n , b n , D n ) (cid:19) ⊞ T ( p n ) (cid:21) = ^ n ≥ (cid:20) ^ C n ∈ ID M ( s , n ) (cid:18) · · · (cid:18) ^ C ∈ ID M ( s , ) d ⋆ ( p I s , C ) ⊞ d ⋆ ( C , C ) (cid:19) · · · (cid:19) ⊞ T ( St ( C n )) (cid:21) = | M | w ( s ) . Q . E . D . .Shang,X.Lu&R.Q.Lu 259 Definition 4.4 [3] A QMV algebra is said to be locally finite iff ∀ a ∈ E s.t. a = ∃ n ∈ N s.t. n · a = M be an E NTM. Let R ⊞ M = { a ⊞ a ⊞ · · · ⊞ a n : a i ∈ R M , n ∈ N } ∪ { } . It is straightforward toprove that if E is locally finite, then R ⊞ M is also finite. In the following we can simulate any E NTM withsome E NTM with classical transitions.After Corollary 4.3 the question arises as to whether the transitions of an E NTM can be classicalwithout losing power. The next lemma shows that this can be obtained under a certain finite condition.
Lemma 4.5
Let M be an E NTM. If E is locally finite, there exists some E NTM M c with classicaltransitions that accepts the same E -valued language. Proof.
Let M = ( Q , S , G , d , B , I , T ) and M c = ( Q c , S c , G c , d c , B , I c , T c ) . We assume that || Q × G × Q × G × { L , S , R }|| = N and we number all possible transitions ( p , a , q , b , D ) from 1 to N .The state set Q c = Q ∪ { q ( i , j ) x : q ∈ Q , x ∈ R ⊞ M , i = , · · · , N , j = , · · · , } ∪ { q ( f ) x : x ∈ R ⊞ M } is finitesince R ⊞ M is finite. The input alphabet is S c = S × { } , where 0 is the least element of E . The tapealphabet G c = S × R ⊞ M ∪ { B } is finite for finite R ⊞ M . The initial function is I c | Q = I and I c | Q c − Q = d ( p , a , q , b , D ) = y , suppose the index of ( p , a , q , b , D ) is i . We define the following classicaltransitions: d c ( p , ( a , x ) , q ( i , ) x ⊞ y , ( b , x ⊞ y ) , S ) = d c ( q ( i , ) x ⊞ y , ( b , x ⊞ y ) , q ( i , ) x ⊞ y , ( b , x ⊞ y ) , L ) = d c ( q ( i , ) x ⊞ y , ( c , z ) , q ( i , ) x ⊞ y , ( c , x ⊞ y ) , R ) = , ∀ c ∈ G , z ∈ R ⊞ M (7) d c ( q ( i , ) x ⊞ y , ( b , x ⊞ y ) , q ( i , ) x ⊞ y , ( b , x ⊞ y ) , R ) = d c ( q ( i , ) x ⊞ y , ( c , z ) , q ( i , ) x ⊞ y , ( c , x ⊞ y ) , L ) = , ∀ c ∈ G , z ∈ R ⊞ M (9) d c ( q ( i , ) x ⊞ y , ( b , x ⊞ y ) , q , ( b , x ⊞ y ) , D ) = d c ( q , ( c , z ) , q ( f ) z , ( c , z ) , S ) = , ∀ c ∈ G , z ∈ R ⊞ M (11)and d c = T c ( q ( f ) x ) = x ⊞ T ( q ) and T c ( p ) = M cantransform from ID a pa b to a qb b through the transition d ( p , a , q , b , D ) = y . Let ¯ a = ¯ a ′ ( c , z ) and¯ b = ( c , z ) ¯ b ′ ; then M c must run as follows:¯ a p ( a , x ) ¯ b ( ) −→ ¯ a q ( i , ) x ⊞ y ( b , x ⊞ y ) ¯ a ( ) −→ ¯ a ′ q ( i , ) x ⊞ y ( c , z )( b , x ⊞ y ) ¯ b ( ) −→ ¯ a ′ ( c , x ⊞ y ) q ( i , ) x ⊞ y ( b , x ⊞ y ) ¯ b ( ) −→ ¯ a ′ ( c , x ⊞ y )( b , x ⊞ y ) q ( i , ) x ⊞ y ( c , z ) ¯ b ′ ( ) −→ ¯ a ′ ( c , x ⊞ y ) q ( i , ) x ⊞ y ( b , x ⊞ y )( c , x ⊞ y ) ¯ b ′ ( ) −→ ¯ a ′ q ( c , x ⊞ y )( b , x ⊞ y )( c , x ⊞ y ) ¯ b ′ , if D = L ¯ a ′ ( c , x ⊞ y ) q ( b , x ⊞ y )( c , x ⊞ y ) ¯ b ′ , if D = S ¯ a ′ ( c , x ⊞ y )( b , x ⊞ y ) q ( c , x ⊞ y ) ¯ b ′ , if D = R . Since M c is non-deterministic, transition (11) would take the machine into state q ( f ) x ⊞ y and then it musthalt. To see this, if T c ( q ( f ) x ⊞ y ) = x ⊞ y ⊞ T ( q ) <
1, then M c halts. Otherwise the machine is in state q ( f ) x ⊞ y and then the E -values of all the next possible transitions are 1, so M c must halt. Therefore, we can seethat M c turns into ˜ a q ( b , x ⊞ y ) ˜ b from ¯ a p ( a , x ) ¯ b through transitions (5)–(10).60 Turing machines based on unsharp quantum logicNow suppose the input is s and there is an effective path for M : I ( p ) ⊞ d ⋆ ( p s , C ) ⊞ · · · ⊞ d ⋆ ( C n − , C n ) ⊞ T ( p n )= I ( p ) ⊞ d ( p , a , p , b , D ) ⊞ · · · d ( p n − , a n , p n , b n , D n ) ⊞ T ( p n ) . According to the above discussion, there is an effective path for M c : I ( p ) ⊞ d c ⋆ ( p s × { } , ¯ C ) ⊞ · · · ⊞ d c ⋆ ( ¯ C n − , ¯ C n ) ⊞ T c ( St ( ¯ C n ))= I ( p ) ⊞ ⊞ · · · ⊞ ⊞ T c ( p ( f ) x )= I ( p ) ⊞ d ( p , a , p , b , D ) ⊞ · · · d ( p n − , a n , p n , b n , D n ) ⊞ T ( p n ) , where x = d ( p , a , p , b , D ) ⊞ · · · ⊞ d ( p n − , a n , p n , b n , D n ) ⊞ T ( p n ) . The E -values of these two pathsare the same.Conversely, any M c input must be in the form s × { } , where s ∈ S + , so each effective path for M c can be simulated by some path of M . Q.E.D
Using the same construction as in Lemma 4.5, we can show that if M is deterministic, then M c canalso be deterministic. Corollary 4.6
Let M be an E DTM. When E is locally finite, there exists some E DTM M c with classicaltransitions that accepts the same E -valued language.In fact we can assume that M in Lemma 4.5 has a single initial state by Lemma 4.1, and therefore M c has a single initial state and a classical transition function.In classical computation theory, deterministic Turing machines are equivalent to non-deterministicTuring machines, that is, they can recognize the same languages. However, this property does not holdfor fuzzy non-deterministic Turing machines [14, 7]. Fuzzy non-deterministic Turing machines are morepowerful than fuzzy deterministic Turing machines. Similarly, we show that E NTMs are also morepowerful than E DTMs.Let E be locally finite. By Lemmas 4.1 and 4.5, we can assume that M is an E DTM with classicaltransitions and a single initial state. Then we can construct a classical Turing machine with two tapes tocompute the E -valued language | M | d . For any input s , in the first tape, M ′ simulates M according to thetransition function of M . Since M is deterministic, the E value of each step can be recorded in the secondtape. Obviously, M ′ halts iff M halts. When M ′ halts, the final result for the second tape is just | M | d ( s ) .From the above discussion, we can conclude that there exists E DTM that can be simulated by aclassical Turing machine. However, in the following example we find that for some E NTM, there is noclassical Turing machine that can simulate it.
Example 4.1
Let L u be the standard universal language in classical computation theory and let M u =( Q u , S , G , d u , B , p I , Q T ) be the universal Turing machine accepting L u . We construct an E NTM M =( Q , S , G , d , B , q I , T ) such that, for any given 0 < x < • Q = Q u ∪ { q I , q T } , where q I , q T / ∈ Q u . • d ( q I , a , p I , a , S ) = d ( q I , a , q T , a , S ) = ∀ a ∈ S . • d ( p , a , q , b , D ) = ( q , b , D ) ∈ d u ( p , a ) , and d = • T ( p ) = p ∈ Q T , and T = M is an E NTM and its language is | M | d ( s ) = ∀ s ∈ L u and | M | d ( s ) = x ∀ s / ∈ L u . If thereexists some E DTM M ′ simulating M , then the classic language { s ∈ S ∗ : | M ′ | d ( s ) = x } = S ∗ − L u mustbe recursively enumerable, which contradicts the fact that L u is undecidable..Shang,X.Lu&R.Q.Lu 261As a result, we obtain the following theorem. Theorem 4.7 E NTMs are not equivalent to E DTMs and E NTMs have more computational power than E DTMs.
To set up a quantum computation theory for characterizing open quantum systems, we continue to discussTuring machines based on unsharp quantum logic. By reexamining some properties of classical Turingmachines, we found that some important properties are different from those of classical Turing machines,such as the relation between E NTMs and E DTMs. We also found that some E NTMs with some classicalcharacters have the same power as general E NTMs. The phrase structure grammar, the universality ofthe Turing machines, the multitape case and the closure properties of unsharp Turing machines will bepresented elsewhere.
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