Subtypes in fuzzy type theory☆
Introduction
Higher-order logic, usually called type theory (TT), has irreplaceable role in mathematical logic and has also many kinds of applications. It was introduced by Bertrand Russell in [31]. An important and often cited contribution was given in 1940 by Church [5] who introduced a special formal language now called lambda calculus. Since then, TT was developed in various directions by many authors (see [19], [32] and citations therein).
In 1980s and 1990s of the previous century, propositional and first-order fuzzy logics were successfully established (see, e.g., [6], [7], [15], [20], [28], [29] and elsewhere) as constituents of mathematical fuzzy logic. The picture was completed by developing the higher-order fuzzy logic called fuzzy type theory (FTT) in [21] which naturally generalizes the classical type theory by replacing the two-valued boolean algebra of truth values for classical TT by a more general algebraic structure. In the beginning, the IMTL-algebra of truth values was considered but later on, also other versions of FTT have been established (cf. [23], [24], [25]). Běhounek in [4] constructed a minimalistic many-valued type theory that can be extended to a more sophisticated one by adding proper axioms. It follows that each fuzzy logic can have also a higher-order version.
The crucial notion in TT as well as in FTT is that of type (usually denoted by greek letters ) that can be seen as an index used to denote a set of elements of a certain kind, namely truth values, individuals, or functions. The fuzzy type theory is developed as a special logic which follows especially the way developed by Henkin and Andrews in [1], [16], [17]. From this point of view, it is a generalization of the simple type theory in the sense of Church [5] with the semantics based on interpretation of formulas in a frame where the latter is a set of sets with fuzzy equality constructed iteratively starting with the basic sets of truth values and individuals . Sets of higher-order types consist of functions between two sets of lower order type (and so, already constructed). If holds for sets of all higher-order types then the model is standard, otherwise it is general. Type theory is not complete with respect to standard models but, due to Henkin [16], [17], it is complete with respect to general models.
Remark 1 Let us emphasize that our approach to the type theory is mathematical and it differs from the other approaches to TT where the concept of function is taken as a procedure without explicit specification of its domain (cf., e.g., [18]). In [19], the construction of a function is given as follows: if K is an expression and x a variable then we can construct a new expression which constructs a function (note that this procedure is called abstraction). A possible semantics of it is execution of programs while types correspond to various data structures. Such type theory is applied in software engineering (cf. [3]). But not only, for example in [8], functions understood as special constructions are applied to logical analysis of the semantics of concepts. In abstract mathematics we may consider also the concept of mapping that is a special case of a relation. In set theory (cf., e.g., [14]), however, the concepts of function and mapping are taken as equal. The same is considered also in this paper, i.e., we speak about a function f from a set M to a set N, in symbols , that is just an abstract mapping between M and N satisfying the uniqueness condition.
It should be noted that any formula of type βα is interpreted by a total function (cf. Remark 1). In practice, however, one often needs to work with functions defined on a subset of a given set. For example, we often work with functions defined on a set of rational numbers , or natural numbers , where . This situation can be in FTT (and also in TT) expressed only by means of representation of a given function, say , as a special binary (fuzzy) relation . Such a solution, however, makes the notation fairly complex and is not very neat. Therefore, it is desirable to extend FTT by special objects called subtypes. These are types representing subsets of given sets forming a frame. Recall that classical type theory with subtypes has been developed by Farmer in [12]. Following his ideas and also those from the other papers on subtypes [13], [30], we adopt formalization of subtypes by introducing a special relation between types, namely saying that the type α is a subtype of β. In the semantics, this relation corresponds to inclusion of the corresponding sets .
When introducing the frame for such a type theory, however, it turns out that we cannot avoid the assumption that sets forming the frame may contain also partial functions. On the other hand, it should be emphasized that our theory is not the theory of FTT with partial functions.1 To make it more clear, let be a set of total functions and let . If we add a total function to the set then f becomes partial in it. Hence, the frame in our theory will consist of couples where contains both total functions as well as some partial functions that are total on selected subsets of , namely those corresponding to subtypes. One of the problems that must be solved is to extend the concept of fuzzy equality to all kinds of elements contained in .
The fuzzy type theory with subtypes will be denoted by sFTT. The assumed algebra of truth values is EQ-algebra (cf. [25], [27]) which is (probably) the most general structure of truth values suitable for fuzzy type theory. As each residuated lattice is an EQ-algebra, our theory includes also all fuzzy logics based on the former.
The structure of the paper is as follows. In Section 2, we briefly overview the concept of EQ-algebra, introduce types and subtypes and the modified concept of frame. In Section 3 we introduce syntax of sFTT that is the modified syntax of the original FTT. Of course, the concepts introduced in the latter are valid also for the special cases of sFTT, in which specific kinds of structures of truth values are taken into account. In Section 4 we introduce fuzzy equality as a syntactic connective over various types and demonstrate its basic properties. In Section 5 we introduce semantics of sFTT, the concept of frame and interpretation of formulas. In Section 6 we show how a canonical frame can be constructed and prove completeness of sFTT (w.r.t. general models).
Section snippets
Algebra of truth values
Truth values for FTT form a good, non-commutative, bounded, linearly ordered EQ-algebra with Δ operation, namely the algebra of type (2, 2, 2, 0, 0, 1) fulfilling the following axioms for all :
- (E1)
is a commutative idempotent monoid (∧-semilattice with the top element 1). We put iff , as usual.
- (E2)
is a monoid such that ⊗ is isotone w.r.t. ≤.
- (E3)
, (reflexivity)
- (E4)
, (substitution)
- (E5)
, (congruence)
- (E6)
,
Language and formulas
The language J of FTT consists of variables for types, variables for objects of certain types, special constants where , auxiliary symbol λ, and brackets.
Each formula A is assigned a type and we write .5 The set of formulas6
Fuzzy equality in syntax
We have already noted that the fundamental concept in FTT is that of a fuzzy equality. While in semantics it is a special binary fuzzy relation, in syntax, it is represented by a formula of type for which the standard properties of reflexivity, symmetry and transitivity are provable (cf. Definition 2). In syntax, they are expressed as follows.
Definition 7 Let T be a theory and for some type . Then the formula is a fuzzy equality if the following is
Interpretation of formulas
Let be a frame due to Definition 5. To define an interpretation of formulas, we must first consider an assignment p of elements from to variables. Namely, p is a function from the set of all variables of the language J to elements from in keeping with the corresponding types. Given an assignment p, we define a new assignment which equals to p for all variables except for . The set of all assignments over is denoted by .
Given a language J of FTT. An interpretation of
Initial concepts
The basic concepts and properties of our theory are the same as in [25], namely the deduction theorem, rules of modus ponens and generalization, properties of Δ and the other ones. Recall from [25] the following definition.
Definition 8 Let T be a theory. We say that: T is contradictory if . Otherwise it is consistent. T is maximal consistent if each its extension , is inconsistent. T is linear if for every two formulas . T is extensionally complete if for every closed formula
Conclusion
In this paper, we introduced a new version of fuzzy type theory extended by the subtypes, i.e., types that represent subsets of the other types (e.g., types of rational numbers being subsets of the real ones). If α is a subtype of β (i.e., ) and γ a type then we must accept that the sets of the frame contain both total functions representing formulas as well as partial functions representing formulas .
Further research will be focused on introduction of the possibility to consider
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