An effective version of definability in metric model theory

03/04/2020 ∙ by Nazanin Roshandel Tavana, et al. ∙ AUT 0

In this paper, a computably definable predicate is defined and characterized. Then, it is proved that every separable infinite-dimensional Hilbert structure in an effectively presented language is computable. Moreover, every definable predicate in these structures is computable.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Definability is a basic but important notion in classical model theory. In classical model theory, it is important to describe which set is defined by a first-order formula. But this concept is studied in metric model theory which is introduced in the rest of this paper.
The usual first-order logic is not a suitable framework for mathematical structures such as Banach spaces, Banach lattices, -algebras, Hilbert spaces, etc. Logic for metric structures was first studied in the 1960s; then stopped [4]. After that, some efforts in recent years are done and the following approaches appeared;

  1. the logic of the positive bounded formulas with an approximate semantics [8] and [9], then

  2. compact abstract theories (CAT) [1].

These attempts end to a new continuous version of the first-order logic; it is equivalent to the both past approaches [2]. In section 2.1, this logic is briefly introduced. In this new framework, for a metric structure and , a definable predicate in over is one which is approximated by a sequence of formulas in the language. Likewise, a closed set is definable in over , if the distance predicate is a definable predicate in over .

In section 2.2, one of the approaches of computable analysis, TTE, is explained. TTE is used to study the effectiveness of definability in the metric structures in this paper. Computable analysis is a branch of computability theory studying the functions defined on real numbers. Type-two theory of effectivity, TTE, is based on the definitions of computable real numbers and functions by A. Turing [14], A. Grzegorczyk [7], and D.Lacombe [11]. In this framework first, computability on finite and infinite sequences of symbols are defined. Then, the computability on these sequences can be transferred to other sets by using them as names [15]. This way can be used to study computable versions of problems and theorems in analysis in mathematical style. Also, since metric model theory is the logic of metric structures, and the relations and functions in this logic are uniformly continuous, TTE is a suitable way to study effective versions of problems in metric model theory.

In section 2.3, an implementation of TTE to study effectiveness of metric model theory is expressed. These definitions first appeared in [12].

In section 3, a computably definable predicate is defined . Then, an effective version of a basic theorem in definability in the metric structures is presented. This theorem says that a predicate is computably definable iff there are a -computable function and computable -formulas such that for all ,

. So, with the mathematical approach, it will be shown that in which situation, there is an algorithm to estimate a definable predicate.


In section 4, an example is studied. Issac Goldbring [5] proved that a definable operator in a Hilbert space is of the form , where is a compact operator, is the identity operator, and . In this example, first, it is proved that a separable infinite-dimensional Hilbert structure in an effectively presented language is decidable. Then, every definable operator in this structure is computable.

2 Preliminaries

2.1 Metric model theory (Continuous logic)

In the following, a logic which is suitable to study metric structures is explained. Note that continuous logic is an extension of the first-order logic with discrete metric.

Assume is a complete metric space. A predicate on is a uniformly continuous function from (for some ) into some bounded interval in . Just uniformly continuous functions from into (for some ) are observed as functions on . For both of them, is called the arity of the predicate or the function.
A metric structure based on is denoted by

where is a predicate on , is a function on , and is a distingushed element in , for . Note that can be a family of complete subspaces of a metric space; in this case, is called many-sorted.

For each metric structure , , and are the interpretations of the predicate symbol , the function symbol and the constant symbol , respectively. Moreover, with each predicate symbol , a modulus of uniform continuity and a closed bounded interval are associated. It means takes its values in and uniformly continuous with modulus . Also, for every function symbol , there is a modulus of uniform continuity which means is uniformly continuous with modulus . Also, consists of a real number which is the diameter of . Note that the metric can be a binary predicate symbol and interpreted as the metric of .

The terms are defined as in first-order logic. An atomic formula is of the form , for terms and a predicate symbol . Also, is an atomic formula for every two terms and . Every atomic formula is a formula. Moreover, for every formula and continuous function , is a formula. And, for every formula and variable , and are formulas. Note that continuous functions are connectives. The interpretation of each formula without free variables, a sentence, is as usual and defined by induction. A structure is a model of a sentence if .

The key concept studied in this paper is definability which is defined as follows.

Definition 2.1.

Assume is a metric structure and .

  1. A predicate is definable in over , if there is a sequence of -formulas such that

  2. A function is definable in over if and only if the function on is a definable predicate in over .

  3. A set is definable in over if the distance predicate is definable in over .

The following lemmas are Theorems 2.13 and 2.15 of [10]. The Lemmes will be used in the proof of Lemma 3.6.

Lemma 2.2.

Let be a double sequence and . Then the iterated limits

exist and both are equal to if and only if

  • exist for each , and

  • exist for each .

Lemma 2.3.

If is a double sequence such that

  • the iterared limit , and

  • exists uniformly in ,

then the double limit exists and is equal to .

2.2 Type-two theory of the effectivity (TTE)

In this section, the approach used to study the effectivity is introduced briefly. The computability notions on natural numbers, are as usual. For a fixed finite set of alphabet including , assume is the set of words (finite sequences on ) and is the set of strings (infinite sequences on ). It is emphasized that it is a mathematical way to study the computability of problems in the mathematical analysis.

Definition 2.4.

A naming system on a set is a surjective function where . If , is called a notation and if , is called a representation.

In the following, there are some examples of naming systems.

Example 2.5.
  1. The binary notation of natural numbers is defined by where .

  2. A notation of integers, is and for .

  3. A notation of rational numbers, is where , and .

  4. The Cauchy representation is defined as follows: if and only if there are words such that , for and , which is called rapidly converges.

By the following definition, a new name can be obtained by the former ones.

Definition 2.6.
  1. The wrapping function is defined by

    for all and .

  2. For , and with , define tupling function as follows:

If there exists a naming system for a set , a new one can be obtained for and , for every .

Definition 2.7.

Let be a naming system for a set where . Then, and are representations of and , respectively, which are defined by

and

A prefix of is a finite word such that there is a with . Then, it is denoted by . To define a continuous and then a computable function, a topology should be set which is Cantor topology. Open sets in this topology are . So, the function is continuous if it is continuous with respect to this topology. Also, is continuous with respect to the discrete topology. Note that a computable function is continuous.

In the following, a computable function on and is defined, ([13], Definition 5.1 and Lemma 5.2).

Definition 2.8.
  1. A function is computable if is a computable function from into in the sense of computability theory.

  2. A function is monotone-constant iff

    For monotone-constant function , define by

    A function is Turing computable iff for some Turing computable monotone-constant function .

  3. A function is monotone iff

    For a monotone function define by

    A function is Turing computable iff for some Turing computable monotone function .

When the notion of a computable function on and is established, a general computable function can be defined.

Definition 2.9.
  1. Let and be two naming systems where . A function is a -realization of the function if , for all .

  2. The function is -computable if it has a computable -realization. (Figure1)

Figure 1: is a -realization of whenever , for all such that exists.

2.3 Effective metric model theory

In the following, the concepts of computable and decidable metric structures are explained. This approach to study the effectiveness of the metric structures is firstly introduced in [12].

Definition 2.10.

[15]

  1. An effective metric space is a tuple such that

    1. is a separable complete metric space.

    2. is a notation of a dense and countable subset .

  2. A computable metric space is an effective metric space such that

    1. is c.e.

    2. is an -computable function.

Similar to the definition in Example 2.5.3, a generalization of Cauchy representation can be defined for an effective metric space. This representation is defined to study the computability of functions and predicates in a metric structure.

Definition 2.11.

[15] For an effective metric space , the Cauchy representation is defined by , where , for , for , and , rapidly converges.

For instance, if we let to be Euclidean metric over , is a computable metric space. In this case, is exactly the Cauchy representation in the Example 2.5.

There exists a representation for , the set of all partial continuous functions with -domain. It means is a name for a continuous function with a -domain which on input returns the value . For more details of this representation, see [6] and [12].

By the above representation, a continuous function is computable if there is a computable such that .

Below, by the representation , a new one for the set of continuous total functions can be obtained, for every two sets and .

Definition 2.12.

[6] Let and be two representations. For the set of continuous total functions , define a representation as follows:

for every such that exists.

Next, the notion of an effectively presented language and then a computable and a decidable -structure will be established [12].

Definition 2.13.

A countable signature is effectively presented if

  1. The sets of variable, predicate, function and constant symbols are computable. It means if , , and are the naming systems for the sets of variables, predicate, function and constant symbols, respectively, then , , and are computable subsets of .

  2. Modulus of uniform continuity of predicate and function symbols are -computable functions.

Similar to computability theory, a notation for , the set of -formulas exists such that is a c.e. set. So, let be an effective list of the set of all -formulas.

Now, let be an effective metric space. Put the Cauchy representations on and on . Let be a metric -structure based on . Assume

To define a representation on , take the representation , where

for any . Since it follows that the function defined by for each , is a representation for . A similar representation can be defined for the set of all interpretations of atomic -formulas in , , instead of the set .

Therefore, a computable and a decidable metric structure can be defined.

Definition 2.14.
  1. With the preceding assumption, a metric structure is computable iff the sequence

    has a computable -name.

  2. Respectively, a metric structure is decidable iff the sequence

    has a computable -name.

Actually, is a naming system for which is the set of all sequences on . Hence, for a decidable metric structure , there is an algorithm such that for a given -formula and , it returns a good approximation of in rational numbers. This means that, for each , is computably found such that and .

3 Computably definable predicates

In this section, a computably definable predicate is defined and characterized. Let be a metric structure based on an effective metric space and assume .

Definition 3.1.

(Modulus of convergence) A function is called a modulus of convergence of a sequence if for

The following proposition is Theorem 4.2.3 of [15]. It explains in which situation the limit of a sequence is computable.

Proposition 3.2.

Let be a -computable sequence of real numbers with computable modulus of convergence . Then, its limit is computable.

In the following, a computable formula is defined.

Definition 3.3.

An -formula with free variables is computable in when is a -computable function.

Now, the concept of a computably definable predicate can be established.

Definition 3.4.

A predicate (with -arity) is computably definable in (over ) iff there is a sequence of computable -formulas such that the sequence of predicates is a -computable sequence with computable modulus of convergence and , for every .

Obviously, if an -arity predicate is computably definable in then by Proposition 3.2, is computable for every .

Below, Corollary 20 of [16] is expressed which is a computable version of Tietze Extension Theorem. It will be used in the proof of Theorem 3.7 to characterize a computably definable predicate.

Proposition 3.5.

Every -computable function with co-r.e domain has a -computable total -computable extension with the same and .

Assume

Also, let be a metric space such that the metric is defined by

for every . Since is compact, it is separable. Therefore, let be a countable and dense subset of and be a notation for . So, is an effective metric space.

Thus, the Cauchy representation can be defined for as follows

(1)

Every sequence in is Cauchy and so its limit exists in . We can define a function by and .

Lemma 3.6.

The above function has a closed and co-r.e domain and is -computable.

Proof.

It is obvious that is a closed and co-r.e subset of . Now, let be a -name of . So, is of the form such that , and for ,

and

So, if then for every ,

Thus,

(2)

The last equality is proved by Lemmas 2.2 and 2.3. Since

  1. , and

  2. is uniformly in .

by lemma 2.3, . And, since

  1. ,

  2. , and

  3. exists,

by lemma 2.2, .

The proof of the third item is as follows:
Since , there exists such that for every ,

Also, , for every implies that

And, since , there exists such that for every

Let and . Then,

(3)

The result is that is exists and for , , for every except finitely many numbers.
For every , define

So, . If is a -name of , for every , then is a computable -name for .

The next theorem says that in which situation a predicate is computably definable in metric structures.

Theorem 3.7.

Let be an effective metric space. Assume is a predicate. Then, is computably definable iff there are a -computable function and computable -formulas such that for all , .

Proof.

Let have the specified form. Then, by Prop 9.3 of [2], is definable. Then, for every , there is an in such that

whenever for . For the simplicity, let . Since is -computable, the function defined by

is also -computable. Since only accepts finite sequences, it implies is -computable. So, is a -computable, for some . Notice that an algorithm is presented to construct this sequence. If we define for every , then for ,

(4)

according to (*). So, the modulus of uniform convergence of the sequence is computable. By proof of Prop 9.3 of [2],

Therefore, , for every . by Proposition 3.2, is computable and is a computably definable predicate.

Now, let be computably definable. Consider the set

Each sequence in is a Cauchy sequence in . So, it converges to a limit that is denoted by . Moreover, is a closed and co-r.e subset of and computable formulas converges to is in for every . According to Lemma 3.6, the function is -computable. By Proposition 3.5, there is a -computable function that agrees with on . Therefore, for every ,

If is -computable then is -computable. ∎

Corollary 3.8.

An operator on an effective metric space is computably definable if and only if there are a -computable function and computable -formulas such that for all , .

Corollary 3.9.

Let be a first-order structure and is a definable set. So, there is a first-order formula such that . The structure can be assumed to be a metric structure with discrete metric . So, is definable in metric structure if there is a sequence of formula such that for all

Since is a discrete metric, it means there is a formula which is equivalent to . Moreover, according to Theorem 3.7, is computably definable iff there are a sequence of computable formula and a -computable function such that . Thus, is computably definable in if and only if whenever one gives a computable -name of then a -name of can be computed. It means if or not can be specified. So, is a computable set in the sense of classical computability theory.

4 An example

In the following example, assume that the language is effectively presented.

A separable infinite-dimensional Hilbert space over is a many-sorted structure

where

  • , for where . These sets are called domain,

  • is zero vector in

    ,

  • is the inclusion map for