POTENCIAL DE MIGRACIÓN CARACTERISTICAS DE LOS CONTAMINANTES

The Weak Type Theory (WTT) was originally designed by R. Nederpelt as a re- finement of MV [Ned02]. Concurrently, F. Kamareddine gave WTT’s meta-theory which was eventually published in [KN04], see Section 2.2.4. WTT is an abstract language and a type system which could be used to express mathematical reason- ing in a formal way, see Sections 2.2.1, 2.2.3 and 2.2.5. To each element of the language, the system attributes a weak type, see Section 2.2.2.

2.2.1

Linguistic categories

WTT defines an abstract syntax for a formal language of mathematics. The el- ements of this syntax are classified according to four language levels. Variables, constants and binders belong to the atomic level. Terms, sets, nouns and adjec- tives belong to the phrase level. Statements and definitions belong to the sentence level. Finally, contexts, lines and books belong to the discourse level. Statements are also sometimes listed in both phrase and sentence levels.

WTT makes explicit the grammatical role of each linguistic piece of text. There- fore, the linguistic categories of the atomic level are split into disjoint subsets de- pending on the grammatical category of the identifier. This grammatical category is indicated by upper indices adjoint to the symbol representing the linguistic cate- gory (see Section 4.4). Variables (respectively constants and binders) are split into term and set variables (respectively term, set, noun, adjective and statement con- stant, and term, set, noun, adjective and statement binders). Statement constants are sometimes divided into relational constants, such as ≥, and logical constants, such as ∧. The sets of variables, constants and binders are given beforehand and are infinite. Variables have to be declared in a context prior to being instantiated

to form a phrase. Constants have to be defined by a definition prior to forming a phrase or a sentence.

Similarly to MV, a context is a sequence of assumptions and declarations. A line is the context plus sentence tuple. And a book is an ordered sequence of lines. WTT inherits MV flags described in Section 2.1.2.2.

2.2.2

Weak types

The use of the adjective weak as an attribute for a type system would indicate that WTT is a small type system. But weak should be seen as an attribute to the word type. A weak type is by definition not prevalent nor potent. By extension, weak typing is a light or generic judgment. WTT defines eight weak types. These types – book, cont, term, set, noun, adj, stat and def – correspond directly to the grammatical categories of the the abstract syntax. This list of weak types is highly related to the way the WTT typing rules are written. Some weak types could have been added to this list for completeness but were not essential in the type system. For instance, line, dec and phrase could well be weak types as in MWTT Section 4.4 (see Section 4.2.1.1 where a homogeneous set of weak types is defined for MathLang-CGa).

2.2.3

Type system

WTT defines a set of well formation criteria to validate WTT mathematical texts. WTT implements N.G. de Bruijn’s idea of a line-by-line language where validity is defined according to the relation between a line and the book formed by the previous lines. A typing judgment for a book B is stated in the empty environment.

⊢ B • • • • book

A typing judgment for a context C is stated in the environment formed by a well- typed book B.

B ⊢ C • • • • cont

A typing judgment for a term t (respectively a set s, a noun n, an adjective a, a statement p and a definition d) is stated in the environment formed by a well-typed

book B and a well-typed context C (C being well-typed in B). B; C ⊢ t • • • • term B; C _{⊢ s} • • • • set B; C ⊢ n • • • • noun B; C _{⊢ a} • • • • adj B; C _{⊢ p}• • • • stat B; C ⊢ d • • • • def

WTT rules are equivalent to MV’s basic rules (BR1–BR9 of [dB87, _{§9]). See} Section 4.4.2 for the set of typing rules of MWTT which is an extension of WTT.

2.2.4

Meta-theory

In [KN04], one finds the followings:

Corollary 1 (Decidability of weak type checking) Weak type checking is de- cidable: there is a decision procedure for the question B; C⊢ E • •

• • t?.

[KN04, Corollary 4.21] Corollary 2 (Weak typability) Weak typability is computable: there is a proce- dure deciding whether an answer exists for B; C _{⊢ E} • •

• • ? and if so, delivering the

answer. [KN04, Corollary 4.21]

They show that WTT has subject reduction with respect to the unfolding of definitions in a book. The unfolding of definitions is equivalent to δ-reduction (see 2.1.1.2.2), we note it_→δ.

Theorem 1 (Subject reduction) If B; C _{⊢ E} • •

• • t and B ⊢ E →δ E′, then

B; C _{⊢ E}′ • •

• • t. [KN04, Theorem 4.28]

And finally, they prove that WTT’s unfolding of definitions is strongly normalising. Theorem 2 (Strong normalisation) Let ⊢ B • •

• • book. For all subformulas E

occurring in B, relation→δis strongly normalising (i.e., definition unfolding inside a well-typed book is a well-founded procedure). [KN04, Theorem 4.40]

2.2.5

Expressiveness

Make grammatical roles explicit. From a computer scientist point of view, WTT’s abstract syntax may sound highly redundant. The identifiers are to be given and their grammatical category fixed before any WTT text can be written. This makes WTT somehow more explicit than strongly typed programming language. From a mathematician point of view this set-based definition of WTT uses familiar jargon. This follows N.G. de Bruijn’s idea of a language for mathematics with typed sets.

Weak-validation. WTT does not make any analysis on the mathematical mean- ing of a text but because each identifier is part of a grammatical category, the grammatical correctness is validated. For example, it is possible to write in WTT that a variable x is equal to 1 and to 0:

x : N

x = 1 and x = 0

This is grammatically correct even if semantically incorrect. But WTT derivation rules will lead to an error with the following example:

x : N

x = 1 _{⇒ 1}

This example would correspond to the sentence “let x be a natural number, we have that x implies 1” which could not mean anything if 1 is a number and not a particular statement.

2.2.6

Perspectives

WTT inspired some researches at Technische Universiteit Eindhoven which are concerned with moving from WTT to Type Theory. G. Jojgov and R. Nederpelt describe in [JN04, Joj06] ways to extend the analysis of a WTT document by recording assumptions in a context as future proof obligation for a full formalisation. Ultimately, the goal is to reach a robust formalisation similar to those performed by theorem provers (see Section 2.4.3).

of WTT. We do not give the set of typing rules of WTT since the typing rules of MWTT (see Section 4.4.2) are based on those of WTT. Hence the reader can get the feel of the WTT rules from those of MWTT.