lattice-valued fuzzy turing machines: computing power, universality and efficiency

Fuzzy Sets and Systems 160 (2009) 3453–3474www.elsevier.com/locate/fss

Lattice-valued fuzzy Turing machines: Computing power,universality and efficiency�

Yongming Li∗

College of Computer Science, Shaanxi Normal University, Xi’an 710062, China

Received 16 March 2008; received in revised form 21 December 2008; accepted 9 August 2009Available online 19 August 2009

Abstract

In this paper we study fuzzy Turing machines with membership degrees in distributive lattices, which we called them lattice-valued fuzzy Turing machines. First we give several formulations of lattice-valued fuzzy Turing machines, including in particulardeterministic and non-deterministic lattice-valued fuzzy Turing machines (l-DTMcs and l-NTMs). We then show that l-DTMcs andl-NTMs are not equivalent as the acceptors of fuzzy languages. This contrasts sharply with classical Turing machines. Second, weshow that lattice-valued fuzzy Turing machines can recognize n-r.e. sets in the sense of Bedregal and Figueira, the super-computingpower of fuzzy Turing machines is established in the lattice-setting. Third, we show that the truth-valued lattice being finite is anecessary and sufficient condition for the existence of a universal lattice-valued fuzzy Turing machine. For an infinite distributivelattice with a compact metric, we also show that a universal fuzzy Turing machine exists in an approximate sense. This means, forany prescribed accuracy, there is a universal machine that can simulate any lattice-valued fuzzy Turing machine on it with the givenaccuracy. Finally, we introduce the notions of lattice-valued fuzzy polynomial time-bounded computation (lP) and lattice-valuednon-deterministic fuzzy polynomial time-bounded computation (lNP), and investigate their connections with P and NP. We claimthat lattice-valued fuzzy Turing machines are more efficient than classical Turing machines.© 2009 Elsevier B.V. All rights reserved.

Keywords: Fuzzy systems model; Fuzzy Turing machine; Recursively enumerable language; Recursive language; Universal machine;Computational complexity

1. Introduction

Recent years have witnessed increasing interests in formal models of computations that go beyond the boundariesof classical computations as characterized by the Church-Turing thesis. Particular attention has been paid to for-mal models of fuzzy computation. The history of formal models of fuzzy computations goes back to the late 1960swhen Zadeh came with his intuitive notion of fuzzy algorithm [37]. Santos and other researchers gave the formaldescription of fuzzy algorithm by fuzzy variants of Turing machines, Markov algorithms, and finite automata (cf.[30,13,6,21,23,5,22,7,19,24–26]). Furthermore, fundamental results of the fuzzy language theory and formal modelsof computingwithwords have been established in [13,36,32,27].Wiedermann considered computability and complexity

� This work is supported by National Science Foundation of China (Grant nos. 10571112, 60873119), and the Higher School Doctoral SubjectFoundation of Ministry of Education of China (Grant No. 200807180005).

∗ Tel.: +862985310166; fax: +862985310161.E-mail address: [email protected].

0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2009.08.003

http://www.elsevier.com/locate/fss

mailto:[email protected]

3454 Y. Li / Fuzzy Sets and Systems 160 (2009) 3453–3474

of fuzzy computation in [35]. He claimed that non-deterministic fuzzy Turing machines were able to solve undecidableproblems (more precisely, fuzzy Turing machines could accept non-recursively enumerable (r.e.) sets). Afterwards,the author of this paper further discussed the variants of fuzzy Turing machines, the stratified characterizations offuzzy languages (it was denoted by fuzzy r.e. languages) accepted by fuzzy Turing machines, the universality of fuzzycomputation and fuzzy complexity theory [15,17].We particular showed that deterministic and non-deterministic fuzzyTuring machines are not equivalent as the acceptors of fuzzy languages. Some stratified characterization of fuzzy r.e.languages were given there [15,17]. In fact, these stratified characterizations revealed the reasons that fuzzy Turingmachines can accept non-r.e. languages as demonstrated by Wiedermann [35]. We shall further exploit this point inthe current paper. Using these characterizations, we further showed that universal fuzzy Turing machine did not exist[15–18]. That is to say, there is no fuzzy Turing machine which can simulate any fuzzy Turing machines on it exactly.This fact was also observed by Bedregal and Figueira in [3] more recently using Gödel number technique. However, ifone restricts the membership degrees of fuzzy sets to a fixed finite subset D of [0, 1], such a universal machine existsas shown in [3,15–18]. The most important thing is that we showed that universal fuzzy Turing machine existed in theapproximate sense [17,18]. This means, for any prescribed accuracy, there is a universal machine that could simulateany fuzzy Turing machine on it with the given accuracy. More recently, Bedregal and Figueira carefully analyzedWiedermann’s statement mentioned above about the computational power of fuzzy Turing machines. They stated it ina more rigorous manner and they claimed that Wiedermann’s statement is not completely correct since there are fuzzyTuring machines which could also recognize n-r.e. sets.All the work mentioned above considered fuzzy Turing machines under narrow fuzzy logic. Under narrow fuzzy

logic, the truth values or membership degrees of atomic propositions are taken in the unit interval [0, 1]. In the modernfuzzy logic systems [8], even in multi-valued logic systems, one can give to the atomic propositions rich values like“uninitialized”, “unknown”, “high impedance”, “don’t care”, “logic 1”, “logic 0”, and more (cf., the IEEE StandardMultivalue Logic Systems for VHDLModel Interoperability [11]). In fact, there are several different kinds (for variousapplications) of fuzzy logic systems in modern fuzzy logic (cf. [8]). The truth values of the existing fuzzy logic systemsusually took in some particular type of distributive lattices instead of the unit interval [0, 1]. So distributive lattice canbe considered as a general truth values set for fuzzy logic. To get some more uniform frame of fuzzy Turing machinesin modern fuzzy logic systems, it is necessary to study fuzzy Turing machines with membership degrees taking ina general distributive lattices instead of the unit interval [0, 1]. We call the corresponding machines as lattice-valuedTuring machines in this paper. In the lattice-setting frame, we shall introduce some variants of Turing machines,discuss the relationships between them, characterize their recognized languages, study their super-computing power,their universality and their efficiency as computing devices.Some related work for lattice-valued fuzzy finite automata was done in [12,20]. In fact, the idea of fuzzy automata

valued in some structured abstract set come back to Wechler [34], Qiu [24] further proposed the same structure forresiduated lattices and some fundamental work were done under this way [24–28]. Moreover, some scholars have dealtwith the relationship between the distributivity of truth-value lattices and some theorems in automata, e.g. pumpinglemma in [28]. Indeed, as far as automata theory based on fuzzy logic (residuated lattice-valued logic) is concerned in[24]. This issue is also related to the question of what is the weakest logic to hold a theorem or proposition. Qiu in [29]has early addressed this issue.The remainder of this paper proceeds as follows. In Section 2, we first recall the definition of classical Turing

machine (TM) and computations of TM, then we naturally introduce the concept of lattice-valued non-deterministicfuzzy TM (l-NTM). We also introduce the concepts of non-deterministic fuzzy TM with crisp transition function(l-NTMc), deterministic fuzzy TM (l-DTMc) and multi-tape fuzzy TM. Then we study the relationships among them.Furthermore, we define the notions of l-fuzzy recursively enumerable (l-r.e.) languages and l-fuzzy recursive languagesand then we characterize them by classical r.e. languages and recursive languages by their stratified and level sets. Bythese characterizations, we prove that l-NTM can be seen as an acceptor of n-r.e. sets, which shows the super-computingpower of l-NTMs in Section 3. In Section 4, we study the universality of l-NTMs. A sufficient and necessary conditionis given for the existence of universal l-NTM, that is, the lattice l is finite. Moreover, we also show that there is auniversal l-NTM that can simulate any l-NTM with a preassigned accuracy � for any infinite lattice with compactmetric. Section 5 introduces the l-fuzzy theory of computational complexity, and presents the fuzzy P and fuzzy NPclasses. Characterizations of lP and lNP in terms of classical P and NP are also obtained. It is claimed that l-NTMsare more efficient than classical Turing machines. Section 6 concludes the paper and points out several directions forfuture study.

Y. Li / Fuzzy Sets and Systems 160 (2009) 3453–3474 3455

2. Lattice-valued fuzzy Turing machines and their variants

In this section, we introduce some variants of lattice-valued fuzzy Turing machines, discuss the relationships amongthem and give some characterizations about their recognized languages.

2.1. Variants of lattice-valued fuzzy Turing machines

A (one-way infinite tape) Turing machine is a 7-tuple M = (Q, �, �, �, q0, B, F), where Q is a finite set of states;� is a finite set of allowable tape symbols; B is a special symbol of �, called blank; � is a finite set of input symbolswhich is a subset of � not including B; q0 in Q is the initial state; F, a subset of Q, is the set of final states; and � is amapping from Q × � to P(Q × � × {L , R}), where P(X ) represents the family of all subsets of a set X, and L andR decide the directions for the tape-head to move. Obviously, �(q, x) is a finite subset of Q × � × {L , R}, for any(q, x) ∈ Q × �. If �(q, x) contains at most one element of Q × � × {L , R} for any (q, x) ∈ Q × �, then we call M adeterministic TM (DTM), otherwise, we call M a non-deterministic TM or NTM.We can define the operation of a Turing machine formally by using the notion of an instantaneous description (ID).

An ID of a TM M is described by a string as �1q�2. Here q, the current state of M, is in Q; �1�2 is the string in �∗that is the contents of the tape up to the rightmost non-blank symbol or the symbol to the left of the head, whicheveris rightmost, where �∗ denotes all these finite strings over � including empty string �, and the tape-head is scanningthe leftmost symbol of �2 or the blank B in case �2 = �. Usually one assumes that the first symbol of an ID is in theleftmost cell (for each step of computation; it may be blank). The move of the tape-head depends on the transitionrelation �.For a TMM and its two IDs D1 and D2, we define a move from D1 to D2, write D1 ≺M D2, if one of the following

three conditions is satisfied.If D1 = qx1x2 · · · xk, xi ∈ �, i = 1, 2, . . . , k, x1 is in the leftmost cell, and �(q, x1) = (p, y, R), then

D2 = ypx2x3 · · · xk .If D1 = x1x2 · · · xi−1qxi · · · xk , 2� i � n, xi ∈ �, i = 1, 2, . . . , k, then D2 = x1x2 · · · xi−1ypxi+1 · · · xk provided

that �(q, xi ) = (p, y, R), or D2 = x1x2 · · · pxi−1yxi+1 · · · xk provided �(q, xi ) = (p, y, L).If D1 = x1x2 · · · xkq , xi ∈ �, i = 1, 2, . . . , k, then D2 = x1x2 · · · xk yp provided that �(q, B) = (p, y, R), or

D2 = x1x2 · · · xk−1 pxk y provided that �(q, B) = (p, y, L).Denote ≺∗

M the reflexive and transitive closure of ≺M . Then the language accepted by TMM is defined to be the set

{� : � ∈ �∗, q0� ≺∗M �1 p�2, p ∈ F, �1�2 ∈ �∗}.

In what follows, we always use l to denote a non-trivial distributive lattice with the least element 0 and the largestelement 1 such that 0 � 1. We use ∨ (or

∨) to represent the finite supremum operation, and ∧ (or

∧) to represent the

finite infimum operation. Then the following distributive laws hold for any a, a1, a2, . . . , an ∈ l, where n is a positiveinteger,

a ∧(

n∨i=1

ai

)=

n∨i=1

(a ∧ ai ), a ∨(

n∧i=1

ai

)=

n∧i=1

(a ∨ ai ).

We give here some examples of distributive lattices. The simplest non-trivial distributive lattice is {0, 1} which isused as the truth values set in classical logic. The unit interval [0, 1] is one such example which is often used in fuzzylogic systems. For any a, b ∈ [0, 1], a ∨ b = max{a, b} and a ∧ b = min{a, b}. Then [0, 1] is a linear order lattice.Let N∞ = N ∪ {∞} denote the set of non-negative integers with an additional largest element ∞. Then N∞ is a linearorder lattice. For any set X, the powerset P(X ) endowed with the set union and set intersection is a distributive lattice.In fact, the powerset P(X ) is also a Boolean algebra.For a distributive lattice l, an element a ∈ l is called join-irreducible if and only if a = b ∨ c implies a = b or

a = c. In a distributive lattice, if a is join-irreducible and a � b∨ c, then a � b or a � c always holds. Let J I (l) denotethe set of all join-irreducible elements of l. Then it is well-known that, for any finite distributive lattice l, J I (l) is ajoin-generating set for l [2]. That is, for any a ∈ l, a = ∨{r ∈ J I (l) : r � a}. We shall use this fact in the characterizingof l-fuzzy languages recognized by l-fuzzy Turing machines.


For a given set X, an l-fuzzy subset of X is just a function from X to l, which is also called fuzzy subset of X in thispaper.We assume all fuzzy Turing machines considered in this paper taking truth values in l, which are defined as follows.

Definition 1. An l-valued (fuzzy) non-deterministic Turingmachine (l-NTM, for short) is 7-tupleM = (Q, �, �, �, q0,B, F), where Q, �, �, q0 are the same as before, � is an l-fuzzy subset of Q × � × Q × � × {L , R}, i.e., a function� : Q × � × Q × � × {L , R} → l, which represents the fuzzy transition function, F is an l-fuzzy subset of Q, whichrepresents the fuzzy final state.

Intuitively, �(q, x, q ′, x ′, d) stands for the degree (in l) that the current control state q and the tape symbol x beingscanned are capable of turning to state q ′ and rewriting tape symbol x ′, together with moving left (when d = L) orright (when d = R).For an l-NTM M, if the fuzzy transition function � only takes values 0 and 1 in l and F is a fuzzy subset of Q, then

we also call M an l-fuzzy non-deterministic Turing machine with crisp transition function (l-NTMc, for short). In thecase of l-NTMc M, the transition function � can be seen as a partial function from Q × � to P(Q × � × {L , R}), i.e.(p, y, d) ∈ �(q, x) iff �(q, x, p, y, d) = 1, and �(q, x) = ∅ if �(q, x, p, y, d) = 0 for any (p, y, d) ∈ Q×�×{L , R}.Furthermore, if � is a partial function from Q×� to Q×�×{L , R} (notably, �may be undefined for some arguments)and F is a fuzzy subset of Q, then we call M an l-fuzzy deterministic Turing machine (l-DTMc, for short). In the caseof l-NTMc or l-DTMc, only the final states are fuzzy.Similarly, we can define a move from an ID D1 to another ID D2, also denoted by ≺M , as a binary fuzzy relation on

D×D whereD = �∗ ×Q×�∗ denotes the set of all IDs ofM. To make explicitly the machineM, we also use ID(M)to represent D in the following. The fuzzy relation ≺M is defined in terms of the preceding three cases as follows: for�, � ∈ �∗, x, y, z ∈ � and p, q ∈ Q

≺M (D1, D2) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

�(q, x, p, y, R) if D1 = �qx�, D2 = �yp�,

�(q, x, p, y, L) if D1 = �zqx�, D2 = �pzy�,

�(q, B, p, y, R) if D1 = �q, D2 = �yp,�(q, B, p, y, L) if D1 = �zq, D2 = �pzy,0 otherwise.

Furthermore, ≺∗M is the reflexive and transitive closure of ≺M , that is, ≺∗

M (D, D) = 1 for any D ∈ D, and≺∗

M (D, D′) = ∨{≺ (D, D1)∧ ≺ (D1, D2)∧ · · · ∧ ≺ (Dk, D′) : Di ∈ ID(M), i = 1, 2, . . . , k, k ∈ N} if D, D′ ∈ Dand D � D′. For simplicity, we often substitute ≺ and ≺∗ for ≺M and ≺∗

M , respectively, if no confusion results and Mis clear from the context.It is evident that on each cycle, the machine executes a program line (instruction), which is an element in �, and

makes the current ID D1 change to another ID D2 in one step, which can be written as

D1a� D2,

where≺ (D1, D2) = a > 0 is observed. In general, a chain of derivations from ID D1 to Dk via executing successivelysome instructions, is expressed as follows:

D1a1� D2

a2� D3 · · ·

ak−1� Dk

and the degree corresponding to such chain of derivations is as

a1 ∧ a2 ∧ · · · ∧ ak−1.

Then, ≺∗ (D0, D) is the supremum of all those degrees, each of which corresponds to a chain of derivations from IDD0 to ID D.We say an l-NTM halts if it enters a state q, scanning a symbol x, and there is no move in this situation; i.e.,

�(q, x, p, y, d) = 0 for any p ∈ Q, y ∈ � and d ∈ {L , R}. If this is the case, we also say �(q, x) is undefined.We can always assume that an l-NTM halts if it accepts. That is, without changing the accepted language defined

below, we can make �(q, x) undefined whenever q is an accepting state, i.e., F(q) > 0. We assume, if not otherwisestated, that an l-NTM always halts when it is in an accepting state.


The l-fuzzy language accepted or recognized by an l-NTM M is a function fM from �∗ to l defined as

fM (�) =∨

{≺∗ (q0�, �1 p�2) ∧ F(p) : p ∈ Q, �1�2 ∈ �∗}

for any � ∈ �∗.In particular, if the l-NTM M is an l-NTMc, then the fuzzy language accepted or recognized by M is a function fM

from �∗ to l, which can be simply defined as

fM (�) =∨

{F(p) : q0� ≺∗ �1 p�2, p ∈ Q, �1�2 ∈ �∗}

for any � ∈ �∗, where q0� ≺∗ �1 p�2 means ≺∗ (q0�, �1 p�2) = 1 and it is the same as that in classical case whenwe see � as a partial function from Q × � to P(Q × � × {L , R}).

For an l-DTMc, since we assume that an l-DTMcM halts if it accepts, if q0� ≺∗ �1 p�2 for some accepting state p,i.e., F(p) > 0, then fM (�) = F(p), and fM (�) = 0 in the other cases.

We say two l-NTMs, M1 and M2, are equivalent if they recognize the same fuzzy language, i.e., fM1 = fM2 .

Remark 1. For any l-NTM, we can require the fuzzy final set F to be a single subset of Q. Since for an l-NTM M =(Q, �, �, �, q0, B, F), we can define another equivalent l-NTM M ′ = (Q′, �, �, �′, q0, B, {q f }) by Q′ = Q ∪ {q f },�′(q, x, q f , x, R) = F(q) and �′(q, x, q ′, y, d) = �(q, x, q ′, y, d) for any q, q ′ ∈ Q, x, y ∈ � and d ∈ {L , R}, and 0in the remainder cases. Then M ′ has crisp final subset which consists of a single state. This shows that our definitionof fuzzy TM is equivalent to those defined in [32,35]. However, it is necessary to require final states to be fuzzy in thedefinitions of l-NTMc or l-DTMc if we use l-NTMc or l-DTMc to accept fuzzy languages.

One reason for accepting Turing machine as a general model of a computation is that this model is equivalent tomany of its modified versions that would seem off-hand to have increased computing power. What about fuzzy TM?We discuss these equivalent theorems in this section. First, we study multitape fuzzy TM, which is the basic model ofour notion of the time and space expended by fuzzy TM computations.

Definition 2. A k-tape l-NTM,where k � 1 is an integer, is a 7-tupleM = (Q, �, �, �, q0, B, F), where Q, �, �, q0, Fare exactly as in classical one-tape l-NTMs, and � is a fuzzy program that must reflect the complexities of multitapestrings. Intuitively, � decides the next state as before, but also decides for each string the symbol overwritten, and thedirection of tape-head motion by looking at the current state and the current symbol at each string. Formally, � is afunction from Q × �k × Q × (� × {L , R})k to l. Intuitively, a = �(q, x1, . . . , xk, p, y1, d1, . . . , yk, dk) means thatif M is in state q, the tape-head of the first string is scanning an x1, that of the second an x2, and so on, and then themembership degree of the next state to be p, the next tape-head to write y1 and move in the direction indicated by d1,and so on for the other tape-heads, is a.In particular, if � only takes values 0 and 1 in l, i.e., � can be seen as a partial function from Q × �k to P(Q × (� ×

{L , R})k) or Q × (� × {L , R})k in the above definition, then we call M a k-tape l-NTMc or k-tape l-DTMc.

Intuitively, a multitape lattice-valued fuzzy TM (l-NTM, l-NTMc or l-DTMc) consists of a finite control with k tapeheads and k tapes; each tape is infinite in one direction. On a single move, depending on the state of the finite controland symbol scanned by each of the tape heads, the machine can: (1) change state; (2) print a new symbol on each ofthe cells scanned by its tape heads; (3) move each of its tape heads, independently, one cell to the left or right.An ID or configuration of a k-tape l-NTM is defined analogously with ordinary l-NTM. It is a (2k + 1)-tuple

(q, w1, u1, . . . , wk, uk), where q is the current state, the ith string reads wi ui , and first symbol of ui is holding theith tape-head. We say that (q, w1, u1, . . . , wk, uk) yields in one step (q ′, w′

1, u′1, . . . , w

′k, u

′k) with degree a (denoted

≺ ((q, w1, u1, . . . , wk, uk), (q ′, w′1, u

′1, . . . , w

′k, u

′k)) = a) if the following is true. First, suppose that xi is the first

symbol of ui , for i = 1, 2, . . . , k, and suppose that �(q, x1, . . . , xk, p, y1, d1, . . . , yk, dk) = a. Then, for i = 1, 2, . . . , kwe have the following: If di = R, then w′

i is wi with yi attached; u′i is ui with the first symbol (which was xi ) removed

(or, if ui was xi , u′i is empty). If di = L , then w′

i is wi its last symbol (say zi ) (zi is B if wi is the empty string)omitted from its end, and u′

i is zi yi u′′i , where u

′′i is ui with the first symbol (which was xi ) removed. In other words, the

conditions for yielding in single-tape l-NTM must hold at each string. The relation “n moves” or the similar notation


“≺∗”, and plain “move” are defined analogously with ordinary l-NTM. Then, the language accepted or recognized bya k-tape l-NTM M is a function fM from �∗ to l defined as

fM (�) =∨

{≺∗ ((q0, �, B, . . . , B), (p, w1, u1, . . . , wk, uk)) ∧ F(p) : p ∈ Q, wi , ui ∈ �∗

for i = 1, 2, . . . , k}for any � ∈ �∗.In particular, if the k-tape l-NTM is a k-tape l-NTMc, then the language recognized or accepted by M is a function

from �∗ to l simply defined as

fM (�) =∨

{F(p) : (q0, �, B, . . . , B) ≺∗ (p, w1, u1, . . . , wk, uk), p ∈ Q, wi , ui ∈ �∗ for i = 1, 2, . . . , k}

for any � ∈ �∗.

Theorem 1. If an l-fuzzy language f is accepted by a k-tape l-NTM (l-NTMc or l-DTMc)M operating within time h(n)(see the related definition of time complexity in Section 5), then we can construct a single-tape l-NTM (l-NTMc orl-DTMc, respectively) N operating within time O(h(n)2) and such that fN = f .

The proof is placed in Appendix A.

Remark 2. In the definition of l-NTM, if we replace the direction set {L , R} by {L , S, R}, where S means that thetape-head has nomove, thenwe obtain another generalized form of l-NTMwhich is still equivalent to l-NTM. The proofis analogous to that in classical TM. The construction is given as follows. Assume that MS = (Q, �, �, �, q0, B, F)is an l-NTM with direction set {L , S, R}, where � : Q × � × Q × � × {L , S, R} → l. We construct an equivalentl-NTM M with direction set {L , R}, write M = (Q1, �, �, �1, q0, B, F1), where for any (q, x, q ′, x ′, d) ∈ Q × � ×Q×�×{L , S, R}, if d � S, then �1(q, x, q ′, x ′, d) = �(q, x, q ′, x ′, d), if d = S, then a new state p�(q,x,q ′,x ′,S) /∈ Q isintroduced, we use two transitions to replace the move “stop”, i.e., �1(q, x, p�(q,x,q ′,x ′,R), x

′, R) = �(q, x, q ′, x ′, S),∀y ∈ �, �1(p�(q,x,q ′,x ′,S), y, q

′, y, L) = 1, �1 takes truth value 0 in other cases. The fuzzy final state F1 is defined as,F1(q) = F(q) for any q ∈ Q, and F1(q) = 0 whenever q ∈ Q1 − Q. Then it can be checked that fM = fM1 .As a comparison, this is not true for quantum Turing machines [4].

Remark 3. In the definition of l-NTM, the tape has a leftmost cell and is infinite to the right. We also call this machinea one-way infinite tape l-NTM. If we allow the tape to be infinite to the two directions, we obtain a two-way infinitetape l-NTM. Modifying the movement of one-way infinite tape l-NTM, we can define the fuzzy languages recognizedby a two-way infinite tape l-NTM. Then, we can show that a fuzzy language is recognized by an l-NTM with two-wayinfinite tape if and only if it is recognized by an l-NTM with a one-way infinite tape. The proof is analogous to that forclassic TM.

2.2. The stratified characterization of l-fuzzy recursively enumerable languages and l-fuzzy recursive languages

For an l-fuzzy language f : �∗ → l, if there is an l-NTMcM such that f = fM , then we call f an l-fuzzy recursivelyenumerable language (l-r.e, for short). Furthermore, if for any input � ∈ �∗, the l-NTMcM always halts, i.e.,M haltson all branches of its computation on input �, then we call f an l-fuzzy recursive language and say f is decided by M.In classical Turingmachines, it is well-known that DTM andNTM recognize the same class of languages: recursively

enumerable languages. On the contrary, we show that the l-DTMc and l-NTMc are not equivalent in the power ofrecognizing fuzzy languages, while l-NTMc and l-NTM are equivalent. To this end, we first give some characterizationsfor the fuzzy languages recognized by l-NTMc and l-DTMc, respectively. These results are also important in thediscussion of computing power of l-NTMs, l-universal fuzzy Turing machines and l-fuzzy complexity theory.

Lemma 1. The union of two l-r.e.(l-fuzzy recursive) languages is l-r.e. (l-fuzzy recursive).

Proof. Suppose that f and g are two fuzzy languages which can be recognized by two l-NTMcs M1 = (Q1, �, �1,

�1, q01, B, F1) and M2 = (Q2, �, �2, �2, q02, B, F2), respectively. It is no loose of generality to assume that


Q1 ∩ Q2 = ∅. We construct a new l-NTMc M = (Q1 ∪ Q2 ∪ {q0}, �, �1 ∪ �2, �, q0, B, F), where q0 /∈ Q1 ∪ Q2,�(q0, x) = �1(q01, x) ∪ �2(q02, x), F(q) = Fi (q) if q ∈ Qi for i = 1, 2 and F(q0) = 0, �(q, x) = �i (q, x) if q ∈ Qi .Then it can be readily verified that fM = f ∨ g. Furthermore, if f and g can be decided by M1 and M2, respectively,then it can be easily seen that fM can be decided by M. Hence, the union of two l-fuzzy recursive languages is alsol-fuzzy recursive. �

We introduce some notations here. Let X be a set. For a fuzzy subset f : X → l, let R( f ) = { f (x) : x ∈X, f (x) > 0} denote the non-zero range of f as a function. For any a ∈ l, two subsets fa and f[a] of X are definedas, fa = {x : x ∈ X and f (x)� a} and f[a] = {x : x ∈ X and f (x) = a}. The support of fuzzy set f is defined assupp( f ) = {x : f (x) > 0}.

Theorem 2. For any l-fuzzy language f : �∗ → l, the following three statements are equivalent.

(i) f is recognized by an l-NTM.(ii) R( f ) is finite, and fa is recursively enumerable (r.e.) for any a ∈ R( f ).(iii) f is an l-r.e. language.

Proof. (i) �⇒ (ii) Let f = fM , where M = (Q, �, �, �, q0, B, F) is an l-NTM. That is, for any � ∈ �∗, f (�) =∨{≺∗ (q0�, �1q�2) ∧ F(q) : q ∈ Q, �1�2 ∈ �∗}. Let X = R(�) ∪ R(F), then X is a finite subset of l. Let〈X〉 denote the sublattice of l generated by X, then 〈X〉 is a finite distributive lattice. The inclusion R( f ) ⊆ 〈X〉is obvious. Hence R( f ) is finite as a set. Moreover, for any r ∈ J I (〈X〉), and � ∈ �∗, � ∈ fr ⇔ f (�)� r⇔ ∃q ∈ Q, ≺∗ (q0�, �1q�2) ∧ F(q)� r ⇔ ∃q ∈ Q, ≺∗ (q0�, �1q�2)� r and F(q)� r ⇔ � ∈ L(Mr ), whereMr = (Q, �, �, �r , q0, B, Fr ) is an NTM, with �r to being a mapping from Q × � to P(Q × � × {L , R}) is definedas, �r (q, x) = {(p, y, d) : �(q, x, p, y, d)� r} for any (q, x) ∈ Q × �, and Fr = {q ∈ Q : F(q)� r}. That is tosay, fr is recognized by Mr , and thus fr is r.e. for any r ∈ J I (〈X〉). Then for any a ∈ R( f ), a ∈ 〈X〉, there arefinite r1, . . . , rk ∈ J I (〈X〉) such that a = r1 ∨ · · · ∨ rk . Observe that � ∈ fa ⇔ f (�)� a ⇔ f (�)� ri for anyi = 1, . . . , k ⇔ � ∈ ⋂k

i=1 fri . Hence, fa = ⋂ki=1 fri . Since each fri is r.e. and r.e. sets are closed under finite

intersection, it follows that fa is r.e. for any a ∈ R( f ).(ii) �⇒ (iii) Since fa is r.e. for any a ∈ R( f ), then fa can be recognized by an NTM M = (Q, �, �, �, q0, B, F).

Further, a ∧ fa can be recognized by an l-NTMc M ′ = (Q, �, �, �, q0, B, a ∧ F) and thus l-r.e., where a ∧ fa is afuzzy subset of �∗ defined as a ∧ fa(�) = a if � ∈ fa and 0 otherwise, a ∧ F is a fuzzy subset of Q defined similarly.Notice that R( f ) is finite and the family of l-r.e. languages is closed under finite union operation (Lemma 1), thusf = ∨

a∈R( f )(a ∧ fa) is l-r.e., that is, f can be recognized by an l-NTMc.(iii) �⇒ (i) is obvious, since l-NTMc can be seen as a special instance of l-NTM with crisp transition function. �

Theorem 2 thus suggests that an l-fuzzy language is recognizable if and only if all its non-fuzzy “threshold language”fa are recognizable. This result is very important, because it connects l-r.e. languages with recursively enumerablecrisp languages. From this point of view, we can extend many properties of crisp languages to that of fuzzy languages.By the proof of Theorem 2, we have the following observation: For a given l-NTM M = (Q, �, �, �, q0, B, F), if

we let X = R(�) ∪ R(F), then X is a finite subset of l. Let 〈X〉 denote the sublattice of l generated by X, then 〈X〉is a finite distributive lattice. The lattice l may be infinite as a set, but only a finite subset 〈X〉 of l is employed inthe computation of M. This observation is very important in the discussion of fuzzy-computability using l-NTM. It isreasonable to define a notion of fuzzy-computability on a computable framework. When we use l-NTM as a model offuzzy computation, it is necessary to use an effective given domain for distributive lattice as truth values set. To make〈X〉 an effective given domain, it is necessary that the elements of X are computable. For that reason, l must be aneffective given domain and X a subset of computable function.To show the similar characterization about the languages recognized by l-DTMcs, we need several lemmas.

Lemma 2. Let M = (Q, �, �, �, q0, B, F) be an l-DTMc and let f = fM . Then there is an l-DTMc M1 such thatfM1 = f and for any input � ∈ �∗, if f (�) = 0, i.e., � /∈ supp( f ), then M1 does not halt on the input �.

Proof. We construct such an l-DTMc M1 as follows, M1 = (Q ∪ {r}, �, �, �1, q0, B, F), where r /∈ Q and, for anyq ∈ Q − supp(F) and x ∈ �, �1(q, x) = �(q, x) if �(q, x) is defined, �1(q, x) = (r, x, R) if �(q, x) is undefined, and


�1(r, x) = (r, x, R) for any x ∈ �. By the definition of �1, M1 will stay in the state r once M1 enters the state r, andM1 may enter the state r only when M halts on the current non-final state.For any input � ∈ �∗, it is clear that q0� ≺∗ �1q�2 for some final state q ∈ supp(F) in M1 iff q0� ≺∗ �1q�2 for

the same final state q inM. Hence, fM1 = fM = f .Furthermore, if f (�) = 0, then there are two possibilities of the computation of M on the input �. Case 1: M halts

on � in finite steps, thenM will enter some non-final state q such that �(q, x) is undefined for some x ∈ �. In this case,M1 will enter state r and does not halt. Case 2: M does not halt on input �, then M1 will simulate M on this input anddoes not halt.Therefore, M1 constructed above satisfies the property required in this lemma. �

In the following, we will use M+ to denote M1 constructed in Lemma 2.

Lemma 3. Let M = (Q, �, �, �, q0, B, F) be an l-DTMc satisfying the following condition: for each � /∈ supp( fM ),i.e., fM (�) = 0, on this input �, M will enter a state q and remain there permanently, i.e., q ∈ Q1, where Q1 is asubset of Q defined as, Q1 = {r ∈ Q − supp(F) : �(r, x) = (r, yx , dx ) for any x ∈ � and for some yx ∈ � anddx ∈ {L , R}}. Then fM can be decided by an l-DTMc.

Proof. We construct a new l-DTMc M1 as follows, M1 = (Q, �, �, �1, q0, B, F), where for any q ∈ Q and x ∈ �, if�(q, x) = (r, y, d) for some r ∈ Q1, then �1(q, x) is undefined, otherwise, �1(q, x) = �(q, x).Then it is clear that q0� ≺∗ �1q�2 for some final state q ∈ supp(F) in M1 iff q0� ≺∗ �1q�2 for the same final state

q inM. Hence, fM1 = fM = f , where we write f as fM for the simplicity.For any � ∈ supp( f ), M1 halts on � as a succeeding computation. For any � /∈ supp( f ), by the assumption, M

will stay in a state, assumed as r, on this input �, then it is evident that r ∈ Q1. By the definition of M1, M1 will halton the state r. So M1 halts for any input. Hence f = fM1 = fM can be decided by l-DTMc M1. �

For two l-DTMcs M1 = (Q1, �, �1, �1, q01, B, F1) and M2 = (Q2, �, �2, �2, q02, B, F2), the tensor prod-uct of M1 and M2, denoted M1

⊗M2, is an l-DTMc constructed as follows, write M = M1

⊗M2, then M =

(Q, �, �, �, q0, B, F), where Q = Q1×Q2,� = �1∪�2, q0 = (q01, q02), F : Q → l by F(q1, q2) = F1(q1)∨F2(q2)and � : Q × �2 → Q × (� × {L , R})2 is defined by �((q1, q2), x1, x2) = (p1, p2, y1, d1, y2, d2), iff �1(q1, x1) =(p1, y1, d1), �2(q2, x2) = (p2, y2, d2), otherwise, � is undefined.

Lemma 4. If the l-fuzzy languages f and g can be recognized by two l-DTMcs M1 = (Q1, �, �1, �1, q01, B, F1) andM2 = (Q2, �, �2, �2, q02, B, F2), respectively, such that supp( f ) ∩ supp(g) = ∅, then the union f ∨ g can also berecognized by an l-DTMc. Moreover, if f and g can be decided by some l-DTMcs, then the union f ∨ g can also bedecided by an l-DTMc.

Proof. By Lemma 2, we can assume that if fMi (�) = 0, then Mi will not halt on input � for i = 1, 2. WriteM = M1

⊗M2, the tensor product of M1 and M2, then M is a two-tape l-DTMc. For any input � ∈ �∗, we have

� /∈ supp( f )∪supp(g),� ∈ supp( f ) or� ∈ supp(g). If� /∈ supp( f )∪supp(g), then it is obvious that fM (�) = 0 bythe construction ofM. If� ∈ supp( f ), since supp( f )∩ supp(g) = ∅, it follows that� /∈ supp(g), by the constructionof M, there are q1 ∈ supp(F1) and q2 ∈ Q − supp(F2) such that q0� ≺∗

M �1(q1, q2)�2 for some �1, �2 ∈ �∗, thenq01 ≺∗

M1q1. In this case, fM (�) = F1(q1) ∨ F2(q2) = F1(q1) = f (�). Similarly, if � ∈ supp(g), then � /∈ supp( f ),

there are q1 ∈ Q1 − supp(F1) and q2 ∈ supp(F2) such chat q0� ≺∗M �1(q1, q2)�2 for some �1, �2 ∈ �∗, then

q02 ≺∗M1

q2, and thus fM (�) = F1(q1) ∨ F2(q2) = F2(q2) = g(�). Hence fM = f ∨ g.For the second part of this lemma. Suppose that f and g can be decided by l-DTMcs N1 and N2, respectively. By

Lemma 2, it is of no loss of generality to assume that (N1)+ = M1 and (N2)+ = M2. Then it is easy to check that thetensor product M1

⊗M2 satisfies the conditions in Lemma 3. In fact, in M1

⊗M2, Q1 constructed in Lemma 3 is a

singleton {(r, r )}. By Lemma 3, f can be decided by an l-DTMc. �

Theorem 3. Let f : �∗ → l be an l-fuzzy language. Then the following two statements are equivalent.

(i) f is recognized by an l-DTMc.(ii) R( f ) is finite, and f[a] is r.e. for any a ∈ R( f ).


Proof. (i) �⇒ (ii) Suppose that f = fM , where M = (Q, �, �, �, q0, B, F) is an l-DTMc. Moreover, f (�) = F(q)if q0� ≺∗ �1q�2 for some accepting state q, i.e., F(q) > 0, and fM (�) = 0 in the other cases. Since R( f ) ⊆ R(F),R( f ) is finite. For any a ∈ R( f ), � ∈ f[a] ⇔ f (�) = a ⇔ ∃q ∈ Q, q0� ≺∗ �1q�2 and F(q) = a ⇔ � ∈ L(M[a]),where M[a] = (Q, �, �, �, q0, B, F[a]) is a DTM with F[a] = {q ∈ Q : F(q) = a}. Hence, f[a] = L(M[a]) is r.e. forany a ∈ R( f ).(ii) �⇒ (i) The equality f = ∨

a∈R( f ) (a ∧ f[a]) is obvious. Since f[a] is r.e. for any a ∈ R( f ), then f[a] can berecognized by a DTM M = (Q, �, �, �, q0, B, F). Furthermore, a ∧ f[a] can be recognized by an l-DTMc M ′, whereM ′ = (Q, �, �, �, q0, B, a ∧ F). If a, b are two distinct elements in R( f ), then it is evident f[a] ∩ f[b] = ∅ andthus supp(a ∧ f[a]) ∩ supp(b ∧ f[b]) = f[a] ∩ f[b] = ∅. By Lemma 4, and noting that R( f ) is finite, it follows thatf = ∨

a∈R( f ) (a ∧ f[a]) can be recognized by an l-DTMc. �

Theorem 3 gives an elegant characterization of l-fuzzy languages recognized by l-DTMcs. However, this character-ization is not valid for l-fuzzy languages recognized by l-NTMs. Indeed, we shall give an l-fuzzy language f below, fcan be recognized by an l-NTM, but there is a level set of f which is not r.e., and thus f cannot be recognized by anyl-DTMc. Therefore, l-NTM and l-DTMc are not equivalent, l-NTM can recognize non-r.e languages from the point ofview of level sets of its fuzzy language, which forms an essential difference between classical Turing machines andfuzzy Turing machines. Theorems 2 and 3 are the key for the sequel discussion about the super-power of l-NTMs.

Example 1. Let l be a distributive lattice containing at least three elements, and a be an element of l distinct with theleast and the largest elements. Take L to be any crisp language which is recursively enumerable but not recursive, andlet M = (Q, �, �, �, q0, B, F) be a crisp Turing machine accepting L. For each instruction (q, x, p, y, d) of M, let�′(q, x, p, y, d) = 1 and for each (q, x) ∈ (Q− F)×� let �′(q, x, p f , x, R) = a where p f is a new final state. For allother arguments �′ has value 0. The fuzzy final state F ′ : Q ∪ {p f } → l is just the characteristic function of F ∪ {p f },that is, F ′(q) = 1 if q ∈ F ∪ {p f }, and F ′(q) = 0 if q /∈ F ∪ {p f }. We shall not distinguish a set and its characteristicfunction in the following section. This defines an l-NTM M ′ = (Q ∪ {p f }, �, �, �′, q0, B, F ′). Let f = fM ′ , thenf[1] = L and f[a] = �∗ − L (the complement of L), �∗ − L is not r.e. By Theorem 3, f cannot be recognized by anyl-DTMc. This counterexample also implies that the languages accepted by l-DTMcs are not closed under finite union.Since f1 = L can be recognized by a DTM and thus an l-DTMc, and f2 as defined by f2(�) = a for any� ∈ �∗ can berecognized by a deterministic fuzzy finite automaton [19,20] and thus an l-DTMc. Clearly, fM ′ = f1 ∨ f2. However,fM ′ cannot be recognized by any l-DTMc as we just said.

However, l-NTMs and l-DTMcs are equivalent when they accept fuzzy recursive languages, i.e., they are equivalentwhen they are used as deciding machines, shown by Theorem 4 below.

Theorem 4. Let f : �∗ → l be an l-fuzzy language. Then the following statements are equivalent.

(i) f can be decided by an l-NTM.(ii) R( f ) is finite, and fa is recursive for any a ∈ R( f ).(iii) R( f ) is finite, and f[a] is recursive for any a ∈ R( f ).(iv) f can be decided by an l-DTMc.

Proof. The proofs of Theorems 2 and 3 can be applied to the equivalence between (i) and (ii), and the equivalencebetween (iii) and (iv) of this theorem, respectively. The implication of (iv) �⇒ (i) is obvious since l-DTMc can be seenas a special instance of l-NTM. The remainder is to prove the implication of (ii) �⇒ (iii). This is simple because wehave the following equality

f[a] = fa −⋃

{ fb : b ∈ R( f ), b > a}for any a ∈ R( f ), and the family of recursive languages is closed under finite union and complement operation. �

3. The computing power of l-NTMs

In this section, we show l-NTMs can recognize non-r.e. languages in the sense used by Wiedermann [35], and thusthey have more powerful computing ability than that of ordinary Turing machines.


Let L be a language over �. For two distinct elements a, b in l, we can define the following l-fuzzy language over �,denoted L(a, b), which is related with the language L and a, b, for any � ∈ �∗,

L(a, b)(�) ={a, � ∈ L ,

b, � /∈ L .

In fact, L(a, b) defined here corresponds exactly to the fuzzy language FL (a, b) defined in [3] when l = [0, 1]. Assaid in [3], L(a, b) can be seen as a special way of transforming or embedding an ordinary set L into an l-fuzzy set,exploiting the degree of acceptance to somehow codify the membership of each element of the set. Then as stated in[3], an l-NTM M “recognize” a set L ⊆ �∗ in the sense of Wiedermann iff M recognizes the associate l-fuzzy setL(a, b). It is obvious that L(a, b) = L(b, a), where L denotes the complement set of L in �∗.Based on the above discussion, we can give some characterizations about the recursively enumerable languages,

recursive languages and n-r.e. languages via l-NTMs as follows, which generalizes the main results in [35,3]. Here,we give some direct machine proofs instead of recursively proofs used in [3], which can be seen as corollaries ofTheorems 2–4.

Theorem 5. Let L be a language over �. Then the following statements are equivalent.

(i) L is a recursively enumerable language.(ii) For any two elements a, b in l, if b < a, then L(a, b) can be accepted by an l-NTM.

Proof. (i) �⇒ (ii) Note that R(L(a, b)) = {a, b} is finite and L(a, b)a = L , L(a, b)b = �∗, by Theorem 2, L(a, b) canbe recognized by an l-NTM.(ii) �⇒ (i) Since L(a, b) can be recognized by an l-NTM, by Theorem 2, L(a, b)a is r.e. Note that (L(a, b))a = L ,

it follows that L is r.e. �

Theorem 6. Let L be a language over �. Then the following statements are equivalent.

(i) L is a recursive language.(ii) For any two elements a, b in l, both L(a, b) and L(b, a) can be accepted by some l-NTMs.

Proof. (i) �⇒ (ii) This is an easy corollary of Theorem 5. Since L is recursive, both L and its complement L are r.e. ByTheorem 5, both L(a, b) and L(b, a) can be accepted by some l-NTMs if a and b can be comparable. If a and b cannotbe comparable, then R(L(a, b)) = R(L(b, a)) = {a, b}, L(a, b)a = L , L(a, b)b = L , L(b, a)b = L and L(b, a)a = L .By Theorem 2, both L(a, b) and L(b, a) can be accepted by some l-NTMs.(ii)�⇒ (i) There are two cases to be considered here. The first one is that a and b can be comparable, that is, a < b or

b < a. It is no loose of generality to assume that b < a. Then L(a, b)a = L and L(b, a)a = L . By Theorem 2, both Land L are r.e., and thus, L is recursive. The second case is that a and b cannot be comparable. In this case, L(a, b)b = Land L(a, b)a = L are r.e., then L is recursive. �

Similarly, we can give some characterizations about n-r.e. languages as done in [3].First, let us recall the definitions of n-r.e. sets and d.r.e. sets, refer to [31] for more detail. For any set A ⊆ �∗, we

identify A with its characterization function 1A. A set A ⊆ �∗ is n-r.e. for a non-negative integer n when there is acomputable approximation g : �∗ × N → {0, 1} such that

limt→∞ g(�, t) = A(�),(∀� ∈ �∗)g(�, 0) = 0,#{t : g(�, t) � g(�, t + 1)} � n.A set D is the difference of r.e. sets (d.r.e.) if D = A − B where A and B are r.e. sets. Then it is well-known thatr.e. sets are 1-r.e. sets and d.r.e sets are 2-r.e. sets.

Lemma 5 (Soare [31]). (i) A set A is 2n-r.e. iff A is the union of n d.r.e. sets.(ii) A set A is (2n + 1)-r.e. iff A is the union of n d.r.e. sets and an r.e. set.

We first give a characterization of d.r.e. sets using l-NTM.


Theorem 7. Let l be a distributive lattice with at least three elements. Then the following are equivalent.

(i) A is a d.r.e. set.(ii) For any a1, a2 in l such that 0 < a1 < a2 � 1, there is an l-NTM such that R( f ) = {a1, a2} and A = f[a1].

Proof. (i) �⇒ (ii) Since A is a d.r.e. set, there are r.e. sets A1 and A2 such that A = A1 − A2. Since A1 − A2 =(A1 ∪ A2)− A2 and r.e. sets are closed under finite union, we can assume A2 ⊆ A1 in this case. Define f : �∗ → l by

f (�) =⎧⎨⎩a2, � ∈ A2,

a1, � ∈ A,

0, otherwise.

Then R( f ) = {a1, a2} is finite, fa2 = A2, fa1 = A2 ∪ A = A1 and f[a1] = A. Since A1 and A2 are r.e. sets, byTheorem 2, it follows that f can be recognized by an l-NTM.(ii) �⇒ (i) Let A1 = fa1 , A2 = fa2 . By Theorem 2, A1 and A2 are r.e. sets, A2 ⊆ A1 is obvious, and A = f[a1] =

A2 − A1. Hence A is a d.r.e. set. �

We need the following lemma to further characterize n-r.e. sets.

Lemma 6 (Bedregal and Figueira [3]). The following are equivalent.

(i) A is an n-r.e. set.(ii) For any a1, . . . , an in [0, 1]∩ Q such that 0 < a1 < · · · < an � 1, there is some [0, 1]-NTM M such that � ∈ A iff

fM (�) ∈ ⋃1� i � n/2 [a2i−1, a2i ) if n is even, and fM (�) ∈ ⋃1� i<n/2 [a2i−1, a2i ) ∪ [an, 1] if n is odd.

Corollary 1. The following are equivalent.

(i) A is an n-r.e. set.(ii) There are r.e. sets A1, . . . , An such that An ⊆ An−1 ⊆ · · · ⊆ A2 ⊆ A1 and A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪

(An−1 − An) if n is even, and A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪ An if n is odd.

Proof. (ii) �⇒ (i) By Lemma 5, this implication holds trivially.(i) �⇒ (ii) By Lemma 6, there are [0, 1]-NTMM such that � ∈ A iff fM (�) ∈ ⋃1� i � n/2 [a2i−1, a2i ) if n is even,

and fM (�) ∈ ⋃1� i<n/2 [a2i−1, a2i ) ∪ [an, 1] if n is odd, where 0 < a1 < · · · < an � 1 are in [0, 1] ∩ Q. Take

Ak = fak , then Ak is r.e. and � ∈ Ak iff f (�)� ak . So An ⊆ An−1 ⊆ · · · ⊆ A2 ⊆ A1. If n is even, let n = 2m, then� ∈ A iff f (�) ∈ [a1, a2) ∪ [a3, a4) ∪ · · · ∪ [a2m−1, a2m). Hence, A = ⋃m

i=1{� : f (�) ∈ [a2i−1, a2i )}. Observingthat f (�) ∈ [a2i−1, a2i ) iff f (�)� a2i−1 and f (�) < a2i iff � ∈ fa2i−1 and � /∈ fa2i iff � ∈ fa2i−1 − fa2i . Hence,f (�) ∈ [a2i−1, a2i ) iff � ∈ A2i−1 − A2i . Therefore, A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪ (A2m−1 − A2m).If n is odd, then � ∈ A iff f (�) ∈ [a1, a2) ∪ [a3, a4) ∪ · · · ∪ [an, 1]. In this case, f (�) ∈ [an, 1] iff f (�)� an iff

� ∈ fan = An and An is an r.e. set. Therefore, A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪ An if n is odd. �

The following theorem is a generalization of Theorems 5 and 6. It characterizes n-r.e. sets in terms of the l-fuzzysets accepted by l-NTMs.

Theorem 8. The following are equivalent.

(i) A is n-r.e.(ii) For any a1, . . . , an ∈ l such that 0 < a1 < · · · < an � 1, there is some l-NTM M such that R( f ) ⊆ {a1, . . . , an},

and

A =⋃

1� i<n/2+1

f[a2i−1]

where f = fM .


Proof. (i) �⇒ (ii) By Corollary 1, there are r.e. sets A1, . . . , An such that An ⊆ An−1 ⊆ · · · ⊆ A1 and A =⋃1� i<n/2+1 (A2i−1 − A2i ), where we assume An+1 = ∅. Define an l-fuzzy language f over � as follows, for any

� ∈ �∗, f (�) = ai iff� ∈ Ai −Ai+1, for i = 1, . . . , n, and f (�) = 0 if� /∈ A1. In this case, R( f ) ⊆ {a1, . . . , an}, andfai = Ai is r.e. for any i = 1, . . . , n. By Theorem 2, f can be recognized by an l-NTMM. Obviously, f[ai ] = Ai − Ai+1.Therefore, A = ⋃

1� i<n/2+1(A2i−1 − A2i ) = ⋃1� i<n/2+1 f[a2i−1].

(ii)�⇒ (i) Assume thatM is an l-NTM satisfying the stated conditions (ii), then A = ⋃1� i<n/2+1 f[a2i−1]. We prove

that A is an n-r.e. set. It is obvious that R( f ) ⊆ {a1, . . . , an} and fai is r.e. for i = 1, . . . , n. Hence, f[ai ] = fai − fai+1 isd.r.e. for any i = 1, . . . , n, wherewe assume fan+1 = ∅. Hence A = ⋃

1� i<n/2+1 f[a2i−1] = ⋃1� i<n/2+1 ( fai − fai+1 ).

By Lemma 5, A is an n-r.e. set. �

Remark 4. Theorem 8 shows that, when l is in linear order and is infinite as a set, in particular, l = [0, 1] or l = N∞as a linear order lattice, l-NTM is a good device to recognize n-r.e. sets in the sense of Definition 3. That is to say, forany non-negative integer n and for any n-r.e. set A, we can design an l-NTM to accept A, and for any l-NTM M, Mcan be used as an acceptor of n-r.e. sets if #R( f )� n. When l is not in linear order, the situation is more complex, thecorresponding l-NTMs may be used as acceptors for more generalized languages than n-r.e. sets.

Definition 3. For any l-NTM M, assume R( fM ) ⊆ D = {a1, . . . , an}, where 0 < a1 < · · · < an � 1, we say Mrecognizes a language over � with respect to D, denoted LD(M), provided, � ∈ LD(M) iff there is an integer i suchthat 1� i < n/2 + 1 and fM (�) = a2i−1, i.e., fM (�) only takes value ak with index k as an odd integer.

We summarize one of the main results of this section as follows.

Corollary 2. Let l be an infinite linear order lattice. For any language A over�, A is an n-r.e. set for some non-negativeinteger n iff A can be recognized by an l-NTM as defined in Definition 3.

Proof. Note that A can be recognized by an l-NTM as defined in Definition 3 iff A satisfies the condition (ii) in Theorem8, then this theorem follows by Theorem 8. �

4. Universal l-fuzzy Turing machine

We have proved that universal fuzzy Turing machine exists in the approximate sense [17]. That is to say, for anyprescribed accuracy, we can construct a universal machine that simulates any fuzzy Turing machine on it with thegiven accuracy. The existence of universal fuzzy Turing machine is very important to the realization of fuzzy Turingmachines, just as universal Turing machine laid the foundation for the Church-Turing thesis. We shall extend this tolattice-valued fuzzy Turing machines.

4.1. The existence of universal fuzzy Turing machine

A universal (or reprogrammable) l-fuzzy TM Mu is an automaton that, given as input the description or code of anyl-fuzzy Turing machine M and a string �, can simulate the computation of M on �. That is to say, if the (appropriate)code ofM is 〈M〉 (we shall give an appropriate code of l-DTMc or l-NTM in the sequel), and input toM is �, then wehave fMu (〈〈M〉, �〉) = fM (�). For convenience, we encode l-fuzzy TM with input alphabet as strings over � = {0, 1}as done in ordinary case.Let fu = fMu . We call fu universal since the question of whether any particular string � in �∗ is accepted by any

particular l-fuzzy TMM with degree a ∈ l is equivalent to the question of whether fu(〈〈M〉, �〉) = a. Of course, oncethe codes of all l-NTMs are given, fu is well defined on �∗. This means that fu exists once the codes of all l-NTMsare given, regardless whether Mu exists or not. We give the explicit expression of fu as follows:

fu() ={a, = 〈〈M〉, �〉 and fM (�) = a,

0 otherwise.

Then, we have the following obvious theorem by the above discussion.


Theorem 9. R( fu) = l − {0}.

Followed by Theorem 9, if l is infinite, then R( fu) is also infinite, but for any fuzzy TMM, R( fM ) is finite as statedin Theorem 2. This means that fu is not recognized by any fuzzy TM. Therefore, the universal fuzzy TM Mu does notexist in this case. We present it in the following corollary.

Corollary 3. Universal fuzzy Turing machine does not exist if l is infinite.

By Corollary 3, the lattice l being finite is a necessary condition for the existence of universal l-NTM or universall-DTMc. We shall further show that this necessary condition is also sufficient for the existence of universal l-NTM oruniversal l-DTMc.For the convenience of the notations, in the remainder of this subsection, we assume that l is a finite distributive

lattice and write l = {0, a1, . . . , ar = 1}. By Theorem 2, for any l-NTM there always exists an equivalent l-NTMc tosimulate the given l-NTM, so we only consider l-NTMc when we use l-NTM to recognize fuzzy language. We needthe following lemma in order to give an appropriate code of an l-NTM.

Lemma 7. Let M = (Q, �, �, �, q0, B, F) be an l-NTMc. Then there exists an l-NTMc M ′ = (Q′, �, �, �′, p1, B, F ′)satisfying the following conditions: fM ′ = fM and Q′ contains at least r + 1 elements such that p1 is the initial state,p2, . . . , pr+1 are all accepting states such that F ′(pi+1) = ai for i = 1, . . . , r .

Proof. Assume that p2, . . . , pr+1 are r elements not belonging to Q, and let Q′ = Q ∪ {p2, . . . , pr+1}. We use p1to denote q0 in Q. Define F ′ : Q′ → D by F ′(pi+1) = ai for i = 1, . . . , r and F ′(q) = 0 for any other state q. Atransition function �′ : Q′ × � → Q′ × � × {L , R} is defined as �′(q, x) = �(q, x) if F(q) = 0 and �(q, x) is definedfor x ∈ �, �′(q, x) = (pi , x, R) if F(q) = ai ∈ D and x ∈ �, �′(q, x) is undefined for other cases. Then we get anl-NTMc M ′ = (Q′, �, �, �′, p1, B, F ′) satisfying the conditions required. �

To beginwith, we encode l-NTMcswith restricted alphabet as strings over {0, 1}. LetM = (Q, {0, 1}, �, �, q1, B, F)being an l-NTMc with input alphabet {0, 1}. We further assume that Q = {q1, q2, . . . , qn} with q1 the initial state,q2, . . . , qr+1 are all accepting states satisfying F(qi+1) = ai for i = 1, . . . , r (this can be done because of Lemma 7).We shall assume � = {X1, X2, . . . , Xm} for some m. X1 always will be the symbol 0, X2 will be 1, and X3 will be B,the blank. However, other tape symbols can be assigned to the remaining integers arbitrarily. We also give directions Land R the symbols d1 and d2, respectively. Then a generic move (qk, Xl , ds) ∈ �(qi , X j ) is encoded by a binary string

0i10 j10k10l10s . (1)

A binary code for the entire l-NTMcM consists of all the codes for the transitions, in some order, separated by pairsof 1’s:

111code111code211 · · · 11codet111, (2)

where each codei is a string of the form (1), and each move ofM is encoded by one of the codei ’s. The move need notbe in any particular order, so each l-NTMc actually has many codes. Any such code forM will be denoted 〈M〉.Every binary string can be interpreted as the code for at most oneM; many strings are not the code of any l-NTMc.

To see that decoding is unique, note that no string of the form (1) has two 1’s in a row, so the codei ’s can be founddirectly. If a string fails to begin and ends with exactly with three 1’s, has three 1’s other than at the end, or has twopair of 1’s with other than five blocks of 0’s in between, then the string represents no l-NTMc.The pairM and � is represented by a string of form (2) followed by �. Any such string will be denoted by 〈M, �〉.

Example 2. Let D = {0, a1, a2, 1} be a distributive lattice such that a1 and a2 cannot be comparable. An l-NTMcM = ({q1, q2, q3, q4, q5}, {0, 1}, {0, 1, B}, �, q1, B, {(q2, a1), (q3, a2), (q4, 1)}) is given by

�(q1, 1) = {(q5, 0, R), (q1, 1, R)}, �(q5, 0) = {(q1, 1, R)}, �(q5, 1) = {(q2, 0, R)}, �(q5, B) = {(q5, 1, L)}.Thus one string denoted by 〈M, 1111000001〉 is 11101001000001010011 010010010010011 00000101010010011000001001001010011 000001000 10000010010111 1111000001.


Define fu : �∗ → l, the “universal fuzzy language of l-NTMs”, to be

fu() ={a, = 〈M, �〉 for an l-NTMc M satisfying Lemma 7 and fM (�) = a,

0 otherwise.

Theorem 10. If l is a finite distributive lattice, then fu is an l-r.e. language that can be recognized by anl-NTMc Mu .

Proof. We show this theorem by giving some modifications of that of ordinary case. We shall exhibit a three-tapel-NTMc Mu accepting fu . The first tape of Mu is the input tape, and the input head on that tape is used to look upmoves of the l-NTMc M when given code 〈M, �〉 as input. Note that the moves of M are found between the first twoblocks of three 1’s. The second tape of Mu will be used to hold the simulated tape ofM, using that same format as forthe code ofM. That is, tape symbol Xi ofM will be represented by 0i , and the tape symbols will be separated by single1’s. The third tape holds the state of M, with qi represented by 0i . There are at least r + 1 states of M. The operationof Mu can be summarized as follows:

(1) Check the format of tape 1 to see that it has a prefix of the form (2) and that there are no two codes that beginwith 0i10 j1 for the same i and j. Also check that if 0i10 j10k10l0s is a code, then i = 1 or i > r + 1, and 1� s � 2.If not, Mu halts without accepting. Since invalid codes are assumed to represent the l-NTMc with no moves, andsuch an l-NTMc accepts no inputs, this action is correct. Tape 3 can be used as scratch to fasciate the computation ofcodes.(2) Initialize the second tape to contain the input �, the portion of the input beyond the second block of three 1’s, in

its encoded form.(3) Set 0, the start state ofM, on the third tape, and move the head of second tape to the first simulated cell. All three

tape heads of Mu are positioned on the leftmost symbols. These symbols may be marked so the heads can find theirway back.(4) If tape 3 holds 0i for 2� i � r + 1, the code for the final state, Mu halts and accepts with membership degree ai .(5) Let X j be the symbol currently scanned by tape head 2 and let 0i be the current contents of tape 3. Scan tape

1 from the left end to the second 111, looking for the substring beginning 110i10 j1. There may be many substringsbeginning 110i10 j1. Using the non-deterministic ability of Mu , Mu will choose any substring as M does at this step.Any choice in this step will correspond to one possible path accepting by Mu . If no such string is found,M has no nextmove and has not accepted, thenMu halts and rejects. If such codes are found, let one of these codes be 0i10 j10k10l10s .Then Mu puts 0k on tape 3, prints Xl on the tape cell scanned by head 2 and moves that head in direction ds in thischoice. Once Mu completes all possible choices, go to step (4).

In this manner, Mu simulates M on �. It is straightforward to check that Mu accepts 〈M, �〉 with degree a ∈ lin one possible path if and only if M accepts � with degree a in the same path. Maximizing all those degrees, eachcorresponding to a path of derivation from initial ID to an accepting ID, then fMu (〈M, �)〉) = fM (�). Evidently, exceptstep (4) describing fuzzy final state, all the description ofMu is crisp as done in the classical case [10]. Hence,Mu is an l-NTMcwith structure similar to classical universal Turingmachine butwith fuzzyfinal state F = a1/q2+· · ·+ar/(qr+1),i.e., F(qi+1) = ai for i = 1, . . . , r and F(q) = 0 for any other state q ∈ Q, we refer to [10] for the detail constructionof classical universal Turing machine. This shows that fMu = fu . Therefore, fu is an l-r.e. language. �

Since the above theorem holds, we call fu the universal l-r.e. language and the l-NTM Mu recognizing fu a universall-NTM (since it dose the work of any l-NTM with input alphabet {0, 1}).Similarly, universal l-DTMc exists if l is a finite distributive lattice.

Theorem 11. Let l be a finite distributive lattice. Then there is an l-DTMc Mud which can simulate any l-DTMc on itexactly.

Proof. The proof is very similar to that of Theorem 10, the only difference lies in the operation (5) in that proof. Weuse the following operation (5′) instead of (5) in the proof of Theorem 10 for the construction of Mud :(5′) Let X j be the symbol currently scanned by tape head 2 and let 0i be the current contents of tape 3. Scan tape

1 from the left end to the second 111, looking for the substring beginning 110i10 j1. There is at most one substringsbeginning 110i10 j1. If no such string is found, M has no next move and has not accepted, then Mu halts and rejects.


If such codes are found, let it be 0i10 j10k10l10s . Then Mud puts 0k on tape 3, prints Xl on the tape cell scanned byhead 2 and moves that head in direction ds , go to step (4). �

We call Mud the universal l-DTMc.

4.2. Approximation of universal l-NTM

In this subsection, we assume that l admits a compact metric d. For example, if l is a closed sublattice of [0, 1]N, thenl admits a compact metric, where the natural metric on [0, 1]N is defined as, d(x, y) = ∑∞

i=112i

|xi − yi | for x = (xi )

and y = (yi ) in [0, 1]N and l is a compact metric space as a closed sub-space of [0, 1]N. In fact, if l is a completelydistributive lattice with countable generating base, then l is a compact metric space endowed with interval topology asa closed sublattice of [0, 1]N ([2]). In this case, for any small positive real number �, there is a finite subset D of l suchthat, for any x ∈ l, there is an element a ∈ D such that the distance of a and x, d(a, x), is smaller than �. That is to say,D is the �-dense subset of l. In particular, we can require that D is a finite sublattice of l.Although we have shown in Corollary 4 that there is no universal fuzzy Turing machine if l is infinite, the following

theorem shows that we can always approximate an l-NTM by a restricted universal fuzzy Turing machine with anygiven accuracy.

Theorem 12. Let l be an infinite distributive lattice with a compact metric. Then for any l-NTM, M, and any positiveinteger n, there always exists an l-NTM N such that the membership values of fuzzy transition and fuzzy final state ofN take values in a finite subset D of l and the following inequality holds for any input � ∈ �∗:

d( fN (�), fM (�))�1

n.

Proof. Since l admits a compact metric d, for any positive integer n, there is a finite 1/n-denseD in l. For any l-NTMMwith input alphabet �, by Theorem 2, there is an l-NTMc M1 such that fM1 = fM . Write M1 = (Q, �, �, �, q0, B, F).We construct the l-NTM N = (Q, �, �, �, q0, B,G) from M1 as follows. The only difference between N and M1 liesin the definitions of fuzzy final state F and G, where G is defined in the following manner: for any state q in Q, sinceD is 1/n-dense in l, there is aq ∈ D such that d(F(q), aq )� 1/n. Let G(q) = aq . It is clear that G is well defined,and d(G(q), F(q))� 1/n holds for any state q ∈ Q. Evidently, the membership values of fuzzy transition and fuzzyfinal state take values in D and N is an l-NTM. In fact, N is an l-NTMc. For any input string � ∈ �∗, suppose thatM1 halts on accepting ID �1q�2, i.e., q0� ≺ �1q�2 (halting) such that FM1 (�) = F(q), then fN (�) = G(q). By thedefinition of G, it follows that d( fM (�), fN (�)) = d( fM1 (�), fN (�)) = d(F(q),G(q))� 1/n. Therefore, N is anl-NTM satisfying the conditions required by this theorem. �

Remark 5. For an infinite distributive lattice lwith compact metric, the unit interval [0, 1] is such an example, althoughuniversal fuzzy Turing machine does not exist, universal l-fuzzy TM exists in the approximate sense. That is to say, byan appropriate choice of finite subset D = 〈D〉 of the lattice l, we can obtain a universal D-fuzzy Turing machine asdone in Theorem 10, which can simulate any l-NTM on it with the given accuracy �, where � � 1/n. As we know, fuzzysystems are the approximate description of practical systems [14,33], it is not surprising that an approximation erroroccurs when we use fuzzy system to model practical system. Accordingly, as a formal model of fuzzy computation,l-fuzzy Turing machine may not exactly describe practical system, but it can approximately describe the practicalsystem with any accuracy. So, just as for the case of universal quantum Turing machine studied in [4], universal l-fuzzyTM exists in the approximate sense, this justifies the significance of Theorem 12.

5. Lattice-valued fuzzy computational complexity theory

Consider a multi-tape l-DTMcM. Assume thatM has k-tapes, one of which contains the input. All tapes, includingthe input tape, may be written upon. If for every input word of length n, M makes at most T (n) moves before halting(i.e. M has no next move), then M is said to be T (n) time-bounded, or of time complexity T (n), and the languagerecognized by M is said to be of time complexity T (n).


The concepts of time-bounded l-DTMcs apply equally well to l-NTMs. An l-NTM is of time complexity T (n) if nosequence of choices of move causes the machine to make more than T (n) moves.

The family of fuzzy languages of time complexity T (n) is denoted lDTIME(T (n)) and that of non-deterministiccomplexity lNTIME(T (n)). All these families of languages are called complexity classes. It is clear that fuzzy languagesin the complexity classes are fuzzy recursive.The fuzzy languages recognizable in deterministic polynomial time form a natural and important class, the class⋃i � 1 lDTIME(ni ), which we denote by lP. It is an intuitively appealing notion that lP is the class of (fuzzy) problems

that can be solved efficiently. Similarly,we define⋃

i � 1 lNTIME(ni ), denoted lNP, lNP consists of the (fuzzy) problemswhich have efficient non-deterministic algorithms.It is worth noting that the definition of fuzzy time complexity class in this paper is not the same as that in [35].

In [35], Wiedermann defined fuzzy complexity classes just for those input strings accepted by fuzzy Turing machine,while our fuzzy TM halts on any input.From the proof of Lemmas 1 and 4, the time taken of the new constructed fuzzy Turing machine is no more than the

sum of the times taken to recognize each language for each given fuzzy Turing machine. If two fuzzy languages aretaken in lP or lNP, then each one can be recognized in polynomial time, and the sum of polynomials is a polynomial.Thus, their union is in lP or lNP. This shows that lP and lNP are closed under disjoint finite union and finite union,respectively. Therefore we have the following stratified characterization for lP and lNP.

Theorem 13. For a fuzzy language f : �∗ → [0, 1], the following statements are equivalent:

(i) f ∈ lN P .(ii) R( f ) is finite, and fa ∈ N P for any a ∈ R( f ).

Proof. (i) �⇒ (ii) Suppose that f can be recognized by an l-NTM M = (Q, �, �, �, q0, B, F) with polynomial time.Then R( f ) is finite and for any r ∈ J I (〈R( f )〉), Mr = (Q, �, �, �r , q0, B, Fr ) is an NTM halting with a polynomialtime no more than M for any input �, and fr = fMr . For any a ∈ R( f ), there are finite r1, . . . , rk ∈ J I (〈R( f )〉) suchthat a = r1 ∨ · · · ∨ rk . Then fa = fr1 ∩ · · · ∩ frk and fa ∈ N P because NP is closed under finite intersection.

(ii) �⇒ (i) Notice f = ∨a∈R( f ) (a ∧ fa), a ∧ fa is in lNP (since fa ∈ NP) and lNP is closed under finite union, it

follows that f ∈ lNP. �

Theorem 14. Let f : �∗ → [0, 1] be a fuzzy language. Then the following statements are equivalent:

(i) f ∈ l P .(ii) R( f ) is finite, and f[a] ∈ P for any a ∈ R( f ).(iii) R( f ) is finite, and fa ∈ P for any a ∈ R( f ).

Proof. (i) �⇒ (iii) Suppose that f can be recognized by an l-DTMc M = (Q, �, �, �, q0, B, F) with polynomial time.Then R( f ) is finite and for any a ∈ R( f ), Ma = (Q, �, �, �, q0, B, Fa) is a DTM halting with a polynomial time nomore than M for any input �, and fa = fMa .

(iii) �⇒ (ii) By the simple calculation, we have

f[a] = fa −⋃

{ fb : b ∈ R( f ) and b > a}

for any a ∈ R( f ), and observed that the class P is closed under finite union and complement operations, it follows thatf[a] ∈ P for any a ∈ R( f ).(ii)�⇒ (i) Notice f = ∨

a∈R( f ) (a∧ f[a]), a∧ f[a] is in lP (since f[a] ∈ P) and lP is closed under finite disjoint union(using Lemma 4, and noting all the constructions involved in the proof of Lemma 4 can be processed in polynomialtime), it follows that f ∈ l P . �

The above theorems show that the problemwhether l P = lNP is equivalent to the famous problemwhether P = NP.However, for an f ∈ lN P , and any a ∈ R( f ), we do not know whether f[a] is in NP. In fact, the mentioned

problem is equivalent to the famous problem whether N P = Co− N P , where Co− N P denotes all languages whosecomplements are in NP, as stated by the following theorem.


Theorem 15. Let l be a distributive lattice with at least three elements. Then the following are equivalent.

(i) NP = Co − NP(ii) For any f ∈ lNP, and for any a ∈ R( f ), f[a] ∈ NP.

Proof. (i) �⇒ (ii) Suppose that NP = Co − NP, then NP is closed under complement operation. Then the proof of(iii) �⇒ (ii) of Theorem 14 can be applied to lNP. Thus, the statement holds.(ii) �⇒ (i) Take a ∈ l − {0, 1}. First, we prove the inclusion NP ⊆ Co − NP. Let L ∈ NP. For this L, consider

fuzzy language L(1, a) defined in Section 3. Then R(L(1, a)) = {a, 1} is finite, L(1, a)1 = L and L(1, a)a = �∗ arein NP. By Theorem 13, L(1, a) ∈ lNP. Hence, by the assumed condition (ii), L(1, a)[a] ∈ NP. By a simple calculation,we know that L(1, a)[a] = L = �∗ − L , it follows that L ∈ NP and L ∈ Co − NP. Hence NP ⊆ Co − NP. Forthe converse inclusion Co − NP ⊆ NP, let L ∈ Co − NP, i.e., L ∈ NP. Consider fuzzy language L(1, a), we haveR(L(1, a)) = {a, 1}, L(1, a)1 = L and L(1, a)a = �∗ are in NP. By Theorem 13, L(1, a) ∈ lNP. Hence, by theassumed condition (ii), L(1, a)[a] ∈ NP. Note L = L(1, a)[a], it follows that L ∈ NP. Hence, Co − N P ⊆ NP.Therefore, NP = Co − NP. �

Similarly to the computing power of l-NTM compared to that of the ordinary Turing machines discussed in Section3, we can show that l-NTM are more efficient than that of the ordinary Turing machines by the following theorems.The proofs are similar to those of Theorems 5–2, we omit them here.


(i) L ∈ NP.(ii) For any a, b ∈ l such that a < b� 1, L(a, b) ∈ lNP.


(i) L ∈ Co − NP.(ii) For any a, b ∈ l such that a < b� 1, L(b, a) ∈ lNP.

Theorem 18. The following are equivalent for a language A ⊆ �∗.

(i) A = A1 − A2 for some A1, A2 ∈ NP.(ii) For any a1, a2 ∈ l such that 0 < a1 < a2 � 1, there exists f ∈ lNP such that R( f ) = {a1, a2} and f[a1] = A.

Theorem 19. The following are equivalent for a language A ⊆ �∗.

(i) There are NP-sets A1, . . . , An such that An ⊆ An−1 ⊆ · · · ⊆ A2 ⊆ A1 and A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪(An−1 − An) if n is even, and A = (A1 − A2) ∪ (A3 − A4) ∪ · · · ∪ An if n is odd.

(ii) For any a1, . . . , an ∈ l such that 0 < a1 < · · · < an � 1, there is f ∈ lNP such that R( f ) ⊆ {a1, . . . , an} andA =

⋃1� i<n/2+1

f[a2i−1].

6. Conclusion

In this work, we established the theory of Turing machines in the lattice-setting, which is the uniform form under theexisting fuzzy logic systems andmulti-valued logic systems. For a distributive lattice l, we proposed the notion of l-fuzzydeterministic Turing machines (l-DTMcs), and showed that l-DTMc was not equivalent to l-fuzzy non-deterministicTM (l-NTM). The notions of l-fuzzy recursively enumerable languages and l-fuzzy recursive languageswere introducedand their stratified characterizations were studied. Then we proved that l-NTMs can recognize n-r.e. sets when l is aninfinite linear order lattice, which reflected the super-computing power of l-NTMs. In this work, we proposed the notionof universal lattice-valued fuzzy Turing machine, a sufficient and necessary condition of the existence of a universall-NTM is given. In general, universal l-NTM exists in the approximate sense if the lattice endowed with a compactmetric. We could construct a universal l-NTM U that could simulate any l-NTM on U with any given accuracy, even


if there was no universal l-NTM that could simulate any l-NTM on it exactly. Fuzzy computational complexity theorywas also considered in this paper, in particular, the notions of lP and lNP were proposed and their connection to P andNP was studied. It was shown that l-NTMs are more efficient than the classical Turing machines.As said in the Introduction section, there are several authors doing some work on the fuzzy Turing machines. Zadeh

gave the intuitive concept of fuzzy algorithms in [37], then Santos presented the formal definition of fuzzy Turingmachines in [30] and gave some useful description of fuzzy algorithms there. Related work is the computing with wordusing the formal model of fuzzy Turing machines by Wang and Qiu in [32]. Using modern fuzzy logic, Wiedermannstudied the computability and complexity of fuzzy computation using fuzzy Turing machines in [35] and he claimedthat fuzzy Turing machines were able to solve undecidable problems. Afterwards, the authors of this paper furtherdiscussed the variants of fuzzy Turing machines, the approximation of fuzzy Turing machines and the universality offuzzy Turing machines in [15–18]. At the same time, Bedregal and Figueira in [3] studied the universality and supercomputing power of fuzzy Turing machines. All the work mentioned above considered fuzzy Turing machines undernarrow fuzzy logic with the truth values of fuzzy propositions are taken in the unit interval [0, 1]. We extend all theabove-mentioned work to lattice-valued fuzzy Turing machines in this paper. We use distributive lattices instead of theunit interval [0, 1] as the truth values set of our fuzzy Turing machines, some general results about fuzzy computationwere obtained. For example, in discussing the universality of fuzzy computation, the necessary and sufficient conditionof the existence of universal fuzzy Turing machines in lattice-setting is precisely presented, i.e., the truth values setis finite; we also can study the computing power of fuzzy Turing machines in a general frame. On the other hand, inWidermann work, he extended Zadeh fuzzy Turing machines, by considering arbitrary t-norms instead of the minimumoperation. In case we use t-norm on lattices to determine the fuzzy language accepted by an l-NTM, some of theresults would stop being valid, some further work should be done under this direction, some related work on finitefuzzy automata had been done in [19]. However, as shown in [17], fuzzy Turing machine using arbitrary t-norm canbe universally approximated by a fuzzy Turing machine using the minimum operation, lattice-valued fuzzy Turingmachines can be considered as the extension of fuzzy Turing machines studied byWidermann in [35] in approximatelysense.Of course, many work should be done on lattice-valued TMs and lattice-valued recursively enumerable languages,

especially on the subject of fuzzy computational complexity theory. For example, Hájek [9] studied the arithmeticalcomplexity of fuzzy logic and fuzzy predicate logic, cf. [1]. Future work will study fuzzy (arithmetical) complexity offuzzy logic and fuzzy predicate logic in the lattice-setting using the notions of the present paper. Another generalizationof this work is permitted the truth values set to be a general lattice instead of distributive lattice. As seen from thediscussion of this paper, many results are dependent on the distributive laws of the truth values set. How about thebehaviors of Turing machines with truth values in a general lattice is another fascinating problem worthy to study. Withrespect to the study of the behaviors of Turing machines with truth values in a general lattice, the further question iswhat is the weakest logic to hold a property in a lattice-valued Turing machine (cf. [29]).

Appendix A

To prove the equivalence between one tape l-NTM and multitape l-NTM, it is necessary to introduce another variantof l-NTM, multitrack l-NTM, a lattice-valued counterpart of classical multitrack Turing machine.A multitrack tape is one in which the tape is divided into tracks. A tape position in an k-track contains k symbols

from the tape alphabet. The diagram in Fig. 1 depicts a two-track tape with the tape head scanning the third position.The machine reads an entire tape position. Multiple tracks increase the amount of information that can be considered

when determining the appropriate transition.A tape position in a k-trackmachine is represented by a k-tuple (x1, . . . , xk),where xi is the symbol in the track i.

Track 1

Track 2

q

Fig. 1. Two-track machine with the tape head scanning the third position.


q

Tape 1

Tape 2 a b c b

c b a

c

c

Fig. 2. Two-tape machine with the tape heads independently positioned on the two tape.

The state, input alphabet, tape alphabet, initial state and final states of k-track machine are the same as in the standardTuring machine. A k-track transition reads and rewrites the entire tape position. A transition of a k-track machineis written �(q, (x1, . . . , xk), p, (y1, . . . , yk), d), where di ∈ {L , R}. The input to a k-track machine is placed in thestandard input position in track 1. All the positions in other tracks are initially blank. Acceptance in multitrack l-NTMis by final state. We need the following lemma to prove Theorem 1.

Lemma 8. A fuzzy language f is accepted by a k-track l-NTM if, and only if, it is accepted by a single track l-NTM.

Proof. Clearly, if f is accepted by a single l-NTM, it is accepted by a k-track l-NTM. The equivalent k-track l-NTMsimply ignores the presence of the other tracks.Let M = (Q, �, �, �, q0, F) be a k-track l-NTM. A one-track machine will be constructed in which a single tape

square contains the same information as a tape position in the k-track tape. The representation of a k-track tape positionindicates how this can be accomplished. The tape alphabet of the equivalent one-track machine N consists of k-tupleof tape elements of M. The input to the k-track machine consists of k-tuples whose components are blank except thefirst component. The input symbol x ofM is identified with the k-tuple (x, B, . . . , B) of N. The one-track machine

N = (Q, � × ({B})k−1, (�)k, , q0, F)

with transition function

(q, (x1, . . . , xk), p, (y1, . . . , yk), d) = �(q, (x1, . . . , xk), p, (y1, . . . , yk), d)

accepted f. �

Proof of Theorem 1. Let f be accepted by M = (Q, �, �, �, q0, B, F), an l-NTM with k tapes. We can construct N,N = (Q′, �, �′, �′, q ′

0, B, F ′), a one-tape l-NTMwith 2k+1 tracks, two tracks for each ofM’s tape. One track recordsthe contents of the corresponding tape of M and the other is blank, except for a marker ↑ in the cell that holds thesymbol scanned by the corresponding head ofM. In track 2k+1, the leftmost cell is always placed the symbol #whichis used to reposition the tape head of N, the other cells are blank. The finite control of N store the state ofM, along witha count of the number of head markers to the right of N’s tape head.Each move ofM is simulated by a sweep from left to right and then from right to left by the tape head of N. Initially,

N’s head is at the leftmost cell containing a head marker. To simulate a move ofM, N sweeps right, visiting each of thecells with head markers and recording the symbol scanned by each head ofM. When N crosses a head marker, it mustupdate the count of head markers to its right. When no more head markers are to the right, N has seen the symbolsscanned by each ofM’s heads, so N has enough information to determine the move ofM. Now Nmakes a pass left, untilit reaches the leftmost head marker. The count of markers to the right enables N to tell when it has gone far enough. AsN passes each head marker on the leftward pass, it updates the tape symbol ofM “scanned” by that head marker, and itmoves the head marker one symbol left or right to simulate the move of M. The membership degree is 1 in the abovemoves of N. Finally, N change the state ofM recorded in N’s control to complete the simulation of one move ofM withthe same membership degree as that of M. If the new state of M is accepting, then N accepts. Hence, fN = fM .For simplicity, we give the complete description of the construction of N whenM is a two-tape machine. Figs. 2 and

3 are the related graphic description.The five track machine N is constructed as follows to simulate the computations of M. Tracks 1 and 3 maintain the

information stored on tapes 1 and 2 of the two-tape machine. Tracks 2 and 4 have a single non-blank square indicatingthe position of the tape heads of the two-tape machine.


#

↑a b c b

c b a↑

Track 1

Track 2

Track 3

Track 4

Track 5

c

c

Fig. 3. Two-tape machine simulated by five-track machine.

The initial action of the simulation in the multitrack machine is to write # in the leftmost position of track 5 and ↑in the leftmost position of tracks 2 and 4. The remainder of the computation of the multitrack machine consists of asequence of actions that simulate the transitions of the two-tape machine.The state set of N is as follows:

Q′ = { f 1, p1, f 2, p2} × Q × ((� ∪ {U }) × (� ∪ {U }) × {L , R,U })2.The states ofN are 8-tuples of the form (s, q, x1, y1, d1, x2, y2, d2). The element s represents the status of the simulationof M, where fi means finding ith symbol, pi means printing ith symbol for i = 1, 2. The symbol U, added tothe tape alphabet and the set of directions, indicates that this item is unknown. The initial state of N is with q ′

0 =( f1, q0,U,U,U,U,U,U ). The fuzzy finial state F ′ of N is then defined as

F ′(s, q, x1, y1, d1, x2, y2, d2) = F(q).

The new tape symbol set is as follows:

�′ = (� × {B, ↑})2 × {B,#}.The transition function �′ of N is defined by the transition function � ofM as follows.Let �(q, x1, x2, p, y1, d1, y2, d2) = a(> 0) be the applicable two-tape transition of M. N begins the simulation of

the transition in the state ( f1, q, (U,U,U )2). The following four actions simulated the transition ofM in the multitrackl-NTM N, where t1, t3 ∈ �, t2, t4 ∈ {B, ↑} and t4 ∈ {B,#}.1. f1 (find first symbol): N moves right until it reads the ↑ on track 2. State ( f1, q, x1,U,U,U,U,U ) is entered,

where x1 is the symbol in track 1 under the symbol ↑. We use the following transitions of N:

�′(( f1, q, (U,U,U )2), (t1, B, t3, t4, t5), ( f1, q, (U,U,U )2), (t1, B, t3, t4, t5), R) = 1,

�′(( f1, q, (U,U,U )2), (x1, ↑, t3, t4, t5), ( f1, q, x1,U,U,U,U,U ), (x1, ↑, t3, t4, t5), R) = 1.

After recording the symbol on the track 1 in the state, N returns to the initial position. The # on track 5 is used toreposition the tape head.

�′(( f1, q, x1,U,U,U,U,U ), (t1, t2, t3, t4, B), ( f1, q, x1,U,U,U,U,U ), (t1, t2, t3, t4, B), L) = 1.

2. f2 (find second symbol): The same sequence of actions records the symbol beneath the ↑ on track 4. N entersstate ( f2, q, x1, x2,U,U,U,U ), where x2 is the symbol in track 3 under the symbol ↑. The tape head is then returnedto the initial position. The transition of N are as follows:

�′(( f1, q, x1,U,U,U,U,U ), (t1, t2, t3, B, t5), ( f2, q, x1,U,U,U,U,U ), (t1, t2, t3, B, t5), R) = 1,

�′(( f1, q, x1,U,U,U,U,U ), (t1, t2, x2, ↑,#), ( f2, q, x1, x2,U,U,U,U ), (t1, t2, x2, ↑,#), R) = 1,

�′(( f2, q, x1,U,U,U,U,U ), (t1, t2, x2, ↑, B), ( f2, q, x1, x2,U,U,U,U ), (t1, t2, x2, ↑, B), L) = 1,

�′(( f2, q, x1, x2,U,U,U,U ), (t1, t2, t3, t4, B), ( f2, q, x1, x2,U,U,U,U ), (t1, t2, t3, t4, B), L) = 1.

3. p1 (print first symbol): Nmoves to the right to the ↑ in track 2 and writes the symbol y1 on track 1. The ↑ on track2 is moved in the direction designated by d1. The machine N enters the state (p1, q, x1, x2, y1,U,U,U ). The machinethen returns to the initial position:

�′(( f2, q, x1, x2,U,U,U,U ), (t1, B, t3, t4,#), (p1, q, x1, x2,U,U,U,U ), (t1, B, t3, t4,#), R) = 1,


�′(( f2, q, x1, x2,U,U,U,U ), (x1, ↑, t3, t4,#), (p1, q, x1, x2, y1,U,U,U ), (y1, B, t3, t4,#), d1) = 1.

�′((p1, q, x1, x2,U,U,U,U ), (t1, B, t3, t4, t5), (p1, q, x1, x2,U,U,U,U ), (t1, B, t3, t4, t5), R) = 1,

�′((p1, q, x1, x2,U,U,U,U ), (x1, ↑, t3, t4, t5), (p1, q, x1, x2, y1,U,U,U ), (y1, B, t3, t4, t5), d1) = 1,

�′((p1, q, x1, x2, y1,U,U,U ), (t1, B, t3, t4, B), (p1, q, x1, x2, y1,U, d1,U ), (t1, ↑, t3, t4, B), L) = 1,

�′((p1, q, x1, x2, y1,U,U,U ), (t1, B, t3, t4,#), (p1, q, x1, x2, y1,U, d1,U ), (t1, ↑, t3, t4,#), R) = 1,

�′((p1, q, x1, x2, y1,U, d1,U ), (t1, t2, t3, t4, B), (p1, q, x1, x2, y1,U, d1,U ), (t1, t2, t3, t4, B), L) = 1.

4. p2 (print second symbol): N moves to the right to the ↑ in track 4 and writes the symbol y2 on track 3. The ↑ ontrack 4 is moved in the direction designated by d2.

�′((p1, q, x1, x2, y1,U, d1,U ), (t1, t2, t3, B,#), (p2, q, x1, x2, y1,U, d1,U ), (t1, t2, t3, B,#), R) = 1,

�′((p2, q, x1, x2, y1,U, d1,U ), (t1, t2, x2, ↑, t5), (p2, q, x1, x2, y1, y2, d1,U ), (t1, t2, y2, B, t5), d2) = 1,

�′((p2, q, x1, x2, y1, y2, d1,U ), (t1, t2, t3, B, B), (p2, p, x1, x2, y1, y2, d1, d2), (t1, t2, t3, ↑, B), L) = a,

�′((p2, q, x1, x2, y1, y2, d1,U ), (t1, t2, t3, B,#), (p2, p, x1, x2, y1, y2, d1, d2), (t1, t2, t3, ↑,#), R) = a,

�′((p2, p, x1, x2, y1, y2, d1, d2), (t1, t2, t3, t4, B), (p2, p, x1, x2, y1, y2, d1, d2), (t1, t2, t3, t4, B), L) = 1,

�′((p2, p, x1, x2, y1, y2, d1, d2), (t1, t2, t3, t4,#), (p2, p,U,U,U,U,U,U ), (t1, t2, t3, t4,#), R) = 1,

�′((p2, p,U,U,U,U,U,U ), (t1, t2, t3, t4, B), ( f1, p,U,U,U,U,U,U ), (t1, t2, t3, t4, B), L) = 1.

The simulation of the transition �(q, x1, x2, p, y1, d1, y2, d2) = a(> 0) terminates by returning the tape head to theinitial position to process the subsequent transition.How long does the operation of N on an input x take? Since M halts within time h(|x |), during its operation none

of its strings ever becomes longer than h(|x |) (this is a very basic facts about any reasonable model of computation: Itcannot waste more space than times!). Thus the total length of the string of N is never more than k(h(|x |) + 1) + 1.Simulating a move thus takes at most two traversals of this string from left to right and back (4k(h(|x |)+ 1)+ 4 steps),plus at most 3k(h(|x |)+ 1)+ 3 steps per each string ofM simulated. The total is O(k2h(|x |)2), or, since k is fixed andindependent of x, O(h(|x |)2). �

Acknowledgments

The authors would like to thank the anonymous reviewers for their careful reading of this paper and for a number ofvaluable comments which improved the quality of this paper.

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lattice-valued fuzzy turing machines: computing power, universality and efficiency

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