Fuzzy control

By the expression Fuzzy logic one denotes several topics which are related with the notion of fuzzy subset defined in 1965 by Lotfi Zadeh at the University of California, Berkeley. Given a nonempty set S, a fuzzy subset of S is a map s from S into the interval [0,1]. Then elements in [0,1] are interpreted as truth values and, in accordance, for every x in S, the value s(x) is interpreted as a membership degree of x to s. In other words, a fuzzy subset is a characteristic function in which graded truth values are admitted. The notion of fuzzy subset enables us to represent the extension of predicates and relations as "big","slow", "near" "similar", which are vague in nature. Observe that there are two possible interpretations of the word "fuzzy logic". The first one is related with an informal utilization of the notion of fuzzy set and it is devoted to the applications. In such a case should be better expressions as "fuzzy set theory" or "fuzzy logic in board sense". Another interpretation is given in considering fuzzy logic as a chapter of formal logic. In such a case one uses the expression "fuzzy logic in narrow sense" or "formal fuzzy logic".

Formal fuzzy logic: a new chapter of multi-valued logic

Given a first order language, in fuzzy logic an interpretation is obtained by a domain D and by associating every constant with an element of D every n-ary operation name with an n-ary function in D and every n-ary predicate name by an n-ary fuzzy relation in D. In accordance with the fact that vagues properties and relations are admitted, these fuzzy relation are not necessarily crisp. Such a kind of semantics was proposed long time by people interested in multi-valued logic, obviously. As an example, since 1996 in the book "Continuous model theory" by Chang and Kisler all the main semantics notions of fuzzy logic where defined in a theoretical setting. Now, even if from a semantical point of view fuzzy logic is not different from first order multi-valued logic, in the deduction apparatus one manifests a basic difference. In fact, in multi-valued logic the deduction operator is a tool to associate every (classical) set of axioms with the related (classical) set of theorems. Also, several authors limite our attention to the generation of the set of valid formulas. From such a point of view the paradigm of the deduction in multi-valued logic is not different in nature from the one of classical logic. Instead in fuzzy logic the notion of approximate reasoning is crucial. This notion is based on the one of fuzzy set of logical axioms and graded inference rules and it enables us to associate any fuzzy set of proper axioms with the related fuzzy subset of consequences. Onother basic difference is in the origin and in the agenda. Indeed the origin of fuzzy logic is in control and its aim is to find applications, in general. Instead, multi-valued logic originates from philosophical and theoretical questions.

Fuzzy logic and probability

Many peoples compare fuzzy logic with probability theory since both refer to the interval [0,1]. However, they are conceptually distinct since we have not confuse a degree of truth with a probability measure. To illustrate the difference, consider the following example: Let $\alpha$ be the claim "the rose on the table is red" and imagine we can freely examine the rose (complete knowledge) but, as a matter of fact, the color looks not exactly red. Then $\alpha$ is neither fully true nor fully false and we can express that by assigning to $\alpha$ a truth value, as an example 0.8, different from 0 and 1 (fuzziness). This truth value does not depend on the information we have since this information is complete.

Now, imagine a world in which all the roses are either clearly red or clearly yellow. In such a world $\alpha$ is either true or false but, inasmuch as we cannot examine the rose on the table, we are not able to know what is the case. Nevertheless, we have an opinion about the possible color of that rose and we could assign to $\alpha$ a number, as an example 0.8, as a subjective measure of our degree of belief in $\alpha$ (probability). In such a case this number depends strongly from the information we have and, for example, it can vary if we have some new information on the taste of the possessor of the rose.

Fuzzy control

(to be completed)

Different formal fuzzy logics

There are several different fuzzy logics depending on the interpretation of the logical connectives. In Zadeh's original papers the logical operators $\vee ,\wedge ,\neg$ are usually interpreted by the operations $\oplus ,\otimes$, -defined by setting for every x and y in [0,1]:

$x\otimes y$ = minimum(x, y)

$x\oplus y$ = maximum(x,y)

- $(x)$ = 1 - x.

Further interpretations of the connectives

Zadeh's definitions of the connectives are not the only possible. We list the main definitions.

• Basic propositional fuzzy logic BL is an axiomatization of logic where conjunction is defined by a continuous t-norm, and implication is defined as the residuum of the t-norm. Its models correspond to BL-algebras.
• Łukasiewicz fuzzy logic is a special case of basic fuzzy logic where conjunction is Łukasiewicz t-norm. It has the axioms of basic logic plus an axiom of double negation (so it is not intuitionistic logic), and its models correspond to MV-algebras.
• Gödel fuzzy logic is a special case of basic fuzzy logic where conjunction is Gödel t-norm. It has the axioms of basic logic plus an axiom of idempotence of conjunction, and its models are called G-algebras.
• Product fuzzy logic is a special case of basic fuzzy logic where conjunction is product t-norm. It has the axioms of basic logic plus another axiom, and its models are called product algebras.
• Monoidal t-norm logic MTL is a generalization of basic fuzzy logic BL where conjunction is realized by a left-continuous t-norm. Its models (MTL-algebras) are prelinear commutative bounded integral residuated lattices.
• Rational Pavelka logic is a generalization of multi-valued logic. It is an extension of Łukasziewicz fuzzy logic with additional constants.

All these logics encompass the traditional propositional logic (whose models correspond to Boolean algebras).

Predicate fuzzy logics

These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic (to be completed).

Effectiveness for fuzzy set theory

The notions of a "decidable subset" and "recursively enumerable subset" are basic ones for classical mathematics and classical logic. Then, the question of a suitable extension of such concepts to fuzzy set theory arises. A first proposal in such a direction was made by E.S. Santos by the notions of fuzzy Turing machine, Markov normal fuzzy algorithm and fuzzy program. Successively, L. Biacino and G. Gerla proposed the following definition where Ü denotes the set of rational numbers in [0,1].

Definition A fuzzy subset μ : S $\rightarrow$[0,1] of a set S is recursively enumerable if a recursive map h : S×N $\rightarrow$Ü exists such that, for every x in S, the function h(x,n) is increasing with respect to n and μ(x) = lim h(x,n). We say that μ is decidable if both μ and its complement –μ are recursively enumerable.

An extension of such a theory to the general case of the L-subsets is proposed in a paper by G. Gerla. In such a paper one refer to the theory of effective domains. It is an open question to give supports for a Church thesis for fuzzy logic claiming that the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. To this aim, further investigations on the notions of fuzzy grammar and fuzzy Turing machine should be necessary (see for example Wiedermann's paper).

Effectiveness for fuzzy logic

Define the set Val of valid formulas as the set of formulas assuming constantly value equal to 1. Then it is possible to prove that among the usual first order logics only Goedel logic has a recursively enumerable set of valid formulas. In the case of Lukasiewicz and product logic, for example, Val is not recursively enumerable (see B. Scarpellini, and Belluce, also such a fact was extensively examined in the book of Hajek). Neverthless, from these results we cannot conclude that these logics are not effective and therefore that an axiomatization is not possible. Indeed, if we refer to the just exposed notion of effectiveness for fuzzy sets, then the following theorem holds true (provided that the deduction apparatus of the fuzzy logic satisfies some obvious effectiveness property).

Theorem. Any axiomatizable fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to utilize the notion of recursively enumerable fuzzy subset to find an extension of[Gödel]]’s theorems to fuzzy logic.

See also

• Neuro-fuzzy
• Fuzzy subalgebra
• Fuzzy associative matrix
• FuzzyCLIPS expert system
• Fuzzy control system
• Fuzzy set
• Paradox of the heap
• Pattern recognition
• Rough set

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