When Every Total Program Is a Finite Tree-Program: A Study of Program-Saturated Classes of Structures

Abstract

This paper investigates classes of structures and individual structures where programs implementing functions defined everywhere (total programs) are equivalent to finite tree-programs. The programs considered may include cycles and contain at most countably many nodes. The analysis begins with programs where arbitrary terms of a given signature are used in function nodes, and arbitrary formulas of this signature are used in predicate nodes. The results are then extended to programs that closely resemble computation trees: if such a program is a finite tree-program, it can be classified as an ordinary computation tree.

1. Introduction

Finite tree-programs, including decision trees [1,2,3,4] and computation trees [5,6,7,8], are explored in various fields of computer science. Decision trees are employed as classifiers and predictors in data analysis, while both decision and computation trees serve as algorithms for solving problems in areas such as combinatorial optimization and computational geometry.

Finite tree-programs represent a simple class of programs that implement functions defined everywhere. However, in certain cases, all programs from sufficiently broad classes that implement functions defined everywhere are equivalent to finite tree-programs. This paper focuses on exploring these situations.

In this paper, a program is a pair $(S,U)$, where S denotes the scheme of the program of a signature $\sigma$ and U denotes a structure of the signature $\sigma$. The signature $\sigma$ consists of a finite or countable set of predicate and function symbols, along with their arity and constant symbols. The structure U is a pair $(A,I)$, where A is a nonempty set called the universe of U, and I is an interpretation function that maps the symbols of $\sigma$ to predicates, functions, and constants in A. The schemes of the programs under consideration may have cycles and finite or countable sets of nodes. Of particular interest are program schemes that are finite trees, with the corresponding programs referred to as finite tree-programs.

Let K be a nonempty class of structures of the signature $\sigma$. A scheme of program S is called total relative to K if, for any structure $U\in K$, the function implemented by the program $(S,U)$ is defined everywhere. The class K is called program-saturated if any scheme of the program of signature $\sigma$ that is total relative to K is equivalent to a scheme of the program that is a finite tree.

This study begins with the general case, where schemes of programs may use arbitrary terms of a given signature $\sigma$ in function nodes and arbitrary formulas of that signature in predicate nodes. A discussion of these schemes of programs can be found in Section 2.

Section 3 studies program-saturated classes of structures. A necessary and sufficient condition for a class of structures to be program-saturated is its compactness: any finitely satisfiable in K set of formulas with free variables from a finite set is satisfiable in K. In particular, any axiomatizable class of structures possesses the property of compactness and is, therefore, a program-saturated class.

Section 4 is devoted to the study of individual structures, each of which forms a class that is program-saturated. Such structures include, in particular, all models of cardinality $\alpha$ of $\alpha$-categorical theories. An example of such a model is the field of complex numbers.

In Section 5, the possibility of an elementary extension of a structure to a program-saturated structure is examined. It is shown that such an extension is always possible, and the minimum cardinality of such an extension is determined.

Section 6 is devoted to the transfer of the obtained results to programs that are essentially close to the computation trees: if the scheme of such a program is a finite tree, then the program is an ordinary computation tree. Some results in this direction were published earlier without proofs in [9].

Section 7 contains brief conclusions.

To the best of the author’s knowledge, studies similar to those presented in this paper have not been conducted previously. However, decision trees and computations trees that are special cases of finite tree-programs have been studied intensively. We mention only the literature related to the consideration of decision and computation trees as algorithms. Linear decision trees and algebraic decision and computation trees have been studied most intensively in this regard.

Lower bounds on the complexity were obtained in [10,11,12] for linear decision trees, in [4,13,14] for algebraic decision trees, in [5,6] for algebraic computation trees, and in [7] for Pfaffian computation trees. Upper bounds on the complexity were obtained in [2,15,16] for linear decision trees and in [3] for quasilinear decision trees, which include linear decision trees and some types of algebraic decision trees. Nondeterministic linear decision and computation trees were studied in [3,17,18].

The complexity of deterministic decision trees over arbitrary infinite sets of k-valued attributes, $k\ge 2$, was studied in [3,19]. The relations between deterministic and various kinds of nondeterministic decision trees over arbitrary infinite sets of k-valued attributes were investigated in [20,21,22]. The complexity of computation trees over arbitrary structures was studied in [8].

The proposed paper continues the line of research introduced by the works mentioned in the last paragraph: we do not concentrate on specific types of finite tree-programs, such as linear or algebraic, but study them over arbitrary structures.

2. Schemes of Programs and Programs

The study begins with the general case, where schemes of programs may use arbitrary terms of a given signature in function nodes and arbitrary formulas of that signature in predicate nodes.

Let $\omega =\{0,1,2,\dots \}$ and $X=\{x_i:i\in \omega \}$ be the set of variables. Let $\sigma$ be a finite or countable signature: a set of predicate and function symbols with their arity and constant symbols. The concept of a formula of signature $\sigma$ is defined in a standard way. First, the concept of a term is defined, and then the concept of an atomic formula using, in particular, the equality symbol =, and finally the concept of a formula using additionally the logical symbols ∧, ¬, and ∀ (see definitions on pp. 22 and 23 [23]). For $n\in \omega \setminus \left\{0\right\}$, we denote $X_n=\{x_0,\dots ,x_{n-1}\}$ and $F_n\left(\sigma \right)$ as the set of formulas of the signature $\sigma$ with free variables from the set $X_n$. Since $\sigma$ is finite or countable, for any $n\in \omega \setminus \left\{0\right\}$, the set $F_n\left(\sigma \right)$ is countable.

Definition 1. 

A scheme of program (scheme in short) of the signature σ is a pair $S=(n,G)$, where $n\in \omega \setminus \left\{0\right\}$ and G is a nonempty directed graph with a finite or countable set of nodes. The nodes of the graph G are divided into three types: function, predicate, and terminal. A function node is labeled with an expression of the form $x_i\Leftarrow t$, where t is a term of the signature σ. Only one edge leaves this node and this edge is not labeled. A predicate node is labeled with a formula of the signature σ. Two edges leave this node. One edge is labeled with the number 1 and another one is labeled with the number 0. A terminal node is labeled with a number from ω. This node has no leaving edges. In addition, some node of the graph is selected as the initial one and marked with the * sign. The set $X_n$ will be called the set of input variables of the scheme S. Usually, we will not distinguish between a scheme and its graph.

Definition 2. 

Let $S=(n,G)$ be a scheme of the signature σ. A complete path of the scheme S is a directed path that starts at the initial node of the scheme S and is either infinite or ends at a terminal node of the scheme S.

Let $\tau =w_1,d_1,w_2,d_2,\dots$ be a complete path of the scheme S. For $i=1,2,\dots$, we correspond to the node $w_i$ of the path $\tau$ a sequence $M_i=t_{i0},t_{i1},\dots$ of terms of the signature $\sigma$ with variables from $X_n$. Set $M_1=x_0,x_1,\dots ,x_{n-2},x_{n-1},x_{n-1},x_{n-1},\dots$. Let the sequences $M_1,\dots ,M_i$ already be defined. We now define the sequence $M_{i+1}$ associated with the node $w_{i+1}$. If $w_i$ is a predicate node, then $M_{i+1}=M_i$. Let $w_i$ be a functional node labeled with the expression $x_j\Leftarrow t(x_{l_1},\dots ,x_{l_h})$. Then, $M_{i+1}=t_{i0},\dots ,t_{ij-1},t(t_{i{l}_1},\dots ,t_{i{l}_h}),t_{ij+1},\dots$.

A formula from the set $F_n\left(\sigma \right)$ is associated with each predicate node of the path $\tau$. Let $w_i$ be a predicate node that is labeled with a formula $\phi (x_{l_1},\dots ,x_{l_q})$ and let the edge $d_i$ of the path $\tau$ leaving the node $w_i$ be labeled with the number c. Then, we correspond to $w_i$ the formula $\phi (t_{i{l}_1},\dots ,t_{i{l}_q})$ if $c=1$ and the formula $\neg \phi (t_{i{l}_1},\dots ,t_{i{l}_q})$ if $c=0$.

Denote by $\mathrm{\Pi }\left(\tau \right)$ the set of formulas corresponding to the predicate nodes of the complete path $\tau$. If $\tau$ is a finite path, then denote by $t_{\tau }$ the number attached to the terminal node of this path.

A structure of the signature $\sigma$ is a pair $U=(A,I)$, where A is a nonempty set called the universe of the structure and I is an interpretation function mapping the symbols of $\sigma$ to appropriate predicates, functions, and constants in A. The cardinal $\left|A\right|$ is called the cardinality of the structure U. Let $\phi \in F_n\left(\sigma \right)$ and $\overline{a}\in A^n$. The expression $U\vDash \phi \left(\overline{a}\right)$ means that the formula $\phi$ is true in the structure U for the tuple $\overline{a}$. We will say that the formula $\phi$ is satisfiable in the structure U if $U\vDash \phi \left(\overline{a}\right)$ for some $\overline{a}\in A^n$. A set of formulas $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$ is referred to as satisfiable in the structure U if there is a tuple $\overline{a}\in A^n$ such that $U\vDash \phi \left(\overline{a}\right)$ for any formula $\phi \in \mathrm{\Phi }$. Let $\phi$ be a sentence of the signature $\sigma$, i.e., a formula of the signature $\sigma$ without free variables. The expression $U\vDash \phi$ means that the sentence $\phi$ is true in the structure U.

Definition 3. 

Let $S=(n,G)$ be a scheme of the signature σ. The pair $\mathrm{\Gamma }=(S,U)$ will be called a program over the structure U with the set of input variables $X_n$.

The scheme S will be called the scheme of the program $\mathrm{\Gamma }$. A complete path $\tau$ of the scheme S will be called satisfiable in U on the tuple $\overline{a}\in A^n$ if, for any formula $\phi \in \mathrm{\Pi }\left(\tau \right)$, $U\vDash \phi \left(\overline{a}\right)$. One can show that there exists exactly one complete path of S that is satisfiable in U on the tuple $\overline{a}$.

The program $\mathrm{\Gamma }$ is associated with a possibly partial function ${\pi }_{\mathrm{\Gamma }}:A^n\to \omega$. Let $\overline{a}\in A^n$ and $\tau$ be a complete path of S that is satisfiable in U on the tuple $\overline{a}$. If $\tau$ is a finite path, then ${\pi }_{\mathrm{\Gamma }}\left(\overline{a}\right)=t_{\tau }$. If $\tau$ is an infinite path, then the value ${\pi }_{\mathrm{\Gamma }}\left(\overline{a}\right)$ is undefined. We will say that the program $\mathrm{\Gamma }$ implements the function ${\pi }_{\mathrm{\Gamma }}$.

Definition 4. 

A scheme $(n,G)$ of the signature σ will be called a tree-scheme of the signature σ if G is a tree with the root that coincides with the initial node of G. A tree-scheme $(n,G)$ of the signature σ will be called finite if G is a finite tree.

Definition 5. 

Let S be a finite tree-scheme of the signature σ and U be a structure of the signature σ. Then, the program $(S,U)$ will be called a finite tree-program over the structure U.

3. Program-Saturated Classes of Structures

This section studies classes of structures that are program-saturated. It is proven that a necessary and sufficient condition for a class of structures to be program-saturated is its compactness. In particular, any axiomatizable class of structures has the property of compactness.

Definition 6. 

Let K be a nonempty class of structures of the signature σ. We will say that a scheme S of the signature σ is total relative to K if, for any structure U from K, the program $(S,U)$ implements a total (everywhere defined) function.

Let $S_1=(n_1,G_1)$ and $S_2=(n_2,G_2)$ be schemes of the signature $\sigma$. We will say that complete paths ${\tau }_1=w_1^1,d_1^1,w_2^1,d_2^1,\dots$ and ${\tau }^2=w_1^2,d_1^2,w_2^2,d_2^2,\dots$ of schemes $S_1$ and $S_2$ are isomorphic if they have the same length and, for $i=1,2,\dots$, the nodes $w_i^1$ and $w_i^2$ are labeled with the same expressions or numbers if they are terminal and either the edges $d_i^1$ and $d_i^2$ are not labeled or they are labeled with the same numbers. We will say that the schemes $S_1$ and $S_2$ are strongly equivalent relative to the class K if $n_1=n_2$ and, for any structure U from K and for any tuple $\overline{a}\in A^{n_1}$, where A is the universe of the structure U, the complete paths of the schemes $S_1$ and $S_2$ satisfiable in U on the tuple $\overline{a}$ are isomorphic.

Definition 7. 

The class K will be called program-saturated if any total relative to the K scheme of the signature σ is strongly equivalent relative to K to a finite tree-scheme of the signature σ.

Let $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$. We will say that the set of formulas $\mathrm{\Phi }$ is satisfiable in the class K if there exists a structure $U\in K$ in which this set of formulas is satisfiable. We will say that the set $\mathrm{\Phi }$ is finitely satisfiable in the class K if any finite subset of $\mathrm{\Phi }$ is satisfiable in the class K.

Definition 8. 

We will say that the class K has the property of compactness if, for any $n\in \omega \setminus \left\{0\right\}$, any set of formulas $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$ finitely satisfiable in K is satisfiable in K.

A complete path $\tau$ of the scheme S will be called satisfiable in the class K if the set of formulas $\mathrm{\Pi }\left(\tau \right)$ is satisfiable in the class K.

Let $S=(n,G_1)$ be a scheme of the signature $\sigma$. We denote by $R\left(S\right)=(n,G_2)$ a tree-scheme of the signature $\sigma$, which has the following property: there exists a one-to-one correspondence between the set of complete paths of S and the set of complete paths of $R\left(S\right)$ for which the corresponding paths are isomorphic. Denote by C a subgraph of the graph $G_2$ induced by the set of nodes that belong to the complete paths of the scheme $R\left(S\right)$, which are satisfiable in the class K. For each predicate node of the graph C with one leaving edge, we add to C a new terminal node labeled with the number 0 into which we draw the missing edge. We denote the obtained graph $G_3$. Denote $R(S,K)=(n,G_3)$. Evidently, $R(S,K)$ is a tree-scheme of the signature $\sigma$.

Lemma 1. 

Let K be a nonempty class of structures of the signature σ, which has the property of compactness, and S be a scheme of the signature σ that is total relative to K. Then, all complete paths of the scheme S that are satisfiable in the class K are finite and the set of complete paths of the scheme S that are satisfiable in the class K is finite.

Proof. 

Evidently, all complete paths of the scheme S that are satisfiable in the class K are finite. Let us assume that the set of complete paths of the scheme S that are satisfiable in the class K is infinite. Similar to the proof of Kënig’s lemma [24], one can show that there is an infinite complete path $\tau$ of the scheme $R\left(S\right)$ such that each node of the path $\tau$ belongs to a complete path of the scheme $R\left(S\right)$ that is satisfiable in the class K. Therefore, the set of formulas $\mathrm{\Pi }\left(\tau \right)$ is finitely satisfiable in K. Taking into account that the class K has the property of compactness, we obtain that the set of formulas $\mathrm{\Pi }\left(\tau \right)$ is satisfiable in K. Thus, there is an infinite complete path in S that is satisfiable in the set K, but this is impossible. □

Theorem 1. 

A nonempty class K of structures of the signature σ is program-saturated if and only if it has the property of compactness.

Proof. 

Let K have the property of compactness and S be a total relative to K scheme of the signature $\sigma$. Using Lemma 1, we obtain that $R(S,K)$ is a finite tree-scheme of the signature $\sigma$. Evidently, the scheme S is strongly equivalent relative to the class K to the scheme $R(S,K)$. Therefore K is a program-saturated class.

Let K not have the property of compactness. Then there exists a number $n\in \omega \setminus \left\{0\right\}$ and a finitely satisfiable in the class K set of formulas $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$, which is not satisfiable in the class K. Let $\mathrm{\Phi }=\{{\phi }_1,{\phi }_2,\dots \}$. We denote by $S=(n,G)$ a scheme of the signature $\sigma$, where G is the graph depicted in Figure 1. Since the set $\mathrm{\Phi }$ is not satisfiable in the class K, the scheme S is total relative to K. Taking into account that the set $\mathrm{\Phi }$ is finitely satisfiable in the class K, one can show that the set of complete paths of S that are satisfiable in the class K is infinite. Therefore, there is no finite tree-scheme of the signature $\sigma$, which is strongly equivalent relative to the class K to the scheme S. Thus, the class K is not program-saturated. □

A theory of the signature $\sigma$ is a nonempty set T of sentences of the signature $\sigma$. A model of the theory T is a structure of the signature $\sigma$ for which all sentences from T are true. The theory T is called complete if, for any sentence $\phi$ of the signature $\sigma$, either $\phi \in T$ or $\neg \phi \in T.$

Definition 9. 

A class K of structures of the signature σ is called axiomatizable if there exists a theory T of the signature σ such that the set of models of T coincides with the class K.

The next lemma follows directly from Proposition 2.2.7 [23].

Lemma 2. 

Let T be a theory of the signature σ and $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$. Then, the following statements are equivalent:

(a) T has a model in which Φ is satisfiable.

(b) Every finite subset of Φ is satisfiable in some model of T.

Theorem 2. 

Any nonempty axiomatizable class K of structures of the signature σ is program-saturated.

Proof. 

Let K be a nonempty axiomatizable class of structures of the signature $\sigma$, $n\in \omega \setminus \left\{0\right\}$, $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$, and the set of formulas $\mathrm{\Phi }$ be finitely satisfiable in the class K. Using Lemma 2, we obtain that the set $\mathrm{\Phi }$ is satisfiable in the class K. Therefore, the class K has the property of compactness. By Theorem 1, the class K is program-saturated. □

We now consider examples of axiomatizable classes of structures (see (pp. 38–41, [23])):

• Classes of Boolean algebras, atomic Boolean algebras, and atomless Boolean algebras.
• Classes of groups; abelian groups; abelian groups with all elements of order p, where p is a prime; and torsion-free abelian groups.
• Classes of commutative rings with unit; fields; fields of characteristic p, where p is a prime; fields of characteristic zero; algebraically closed fields; and real closed fields.

4. Program-Saturated Structures

This section is devoted to the study of individual structures, each of which forms a class that has the property of compactness. Such structures include, in particular, all models of cardinality $\alpha$ of $\alpha$-categorical theories. An example of such a model is the field of complex numbers.

Definition 10. 

Let U be a structure of the signature σ. The structure U will be called program-saturated if the class $\left\{U\right\}$ is program-saturated.

We denote by $\mathrm{Th}\left(U\right)$ the theory of the structure U: the set of all sentences of the signature $\sigma$ that are true in U. It is clear that the theory $\mathrm{Th}\left(U\right)$ is complete. Let $n\in \omega \setminus \left\{0\right\}$ and $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$.

Definition 11. 

We will say that the set Φ is consistent with the theory $\mathrm{Th}\left(U\right)$ if there exists a model of the theory $\mathrm{Th}\left(U\right)$ in which the set Φ is satisfiable.

Lemma 3. 

A set $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$ is consistent with the theory $\mathrm{Th}\left(U\right)$ if and only if the set Φ is finitely satisfiable in the class $\left\{U\right\}$.

Proof. 

Let $\mathrm{\Phi }$ be finitely satisfiable in the class $\left\{U\right\}$. Using Lemma 2, we obtain that $\mathrm{\Phi }$ is consistent with the theory $\mathrm{Th}\left(U\right)$.

Let $\mathrm{\Phi }$ be consistent with the theory $\mathrm{Th}\left(U\right)$. Let ${\phi }_1,\dots ,{\phi }_m\in \mathrm{\Phi }$. Using the completeness of the theory $\mathrm{Th}\left(U\right)$, we obtain that the sentence $\exists x_0\cdots \exists x_{n-1}({\phi }_1\wedge \cdots \wedge {\phi }_m)$ belongs to the theory $\mathrm{Th}\left(U\right)$. Therefore, the set of formulas $\{{\phi }_1,\dots ,{\phi }_m\}$ is satisfiable in U. Thus, the set $\mathrm{\Phi }$ is finitely satisfiable in the class $\left\{U\right\}$. □

Let U be a structure of the signature $\sigma$ with the universe A and $Y\subseteq A$. We denote by ${\sigma }_Y$ the signature obtained from $\sigma$ by adding constant symbol $c_a$ for any element $a\in Y$. We denote by $U_Y$ the expansion of the structure U to the signature ${\sigma }_Y$ such that each new constant symbol $c_a$ is interpreted as the element a.

Definition 12. 

Let α be a cardinal. The structure U is called α-saturated if, for any set $Y\subseteq A$ with cardinality less than α, for any set of formulas $\mathrm{\Phi }\subseteq F_1\left({\sigma }_Y\right)$, which is consistent with the theory $\mathrm{Th}\left(U_Y\right)$, the set Φ is satisfiable in $U_Y$. The structure U is called saturated if it is $\left|A\right|$-saturated.

Definition 13. 

Let α be a cardinal. A theory T of the signature σ is called α-categorical if T has a model of cardinality α and every two models of T of the cardinality α are isomorphic.

The next lemma follows directly from Proposition 2.3.6 [23].

Lemma 4. 

Let U be an ω-saturated structure of the signature σ with the universe A and $n\in \omega \setminus \left\{0\right\}$. Then, for each finite $Y\subseteq A$, each set of formulas $\mathrm{\Phi }\subseteq F_n\left({\sigma }_Y\right)$ consistent with the theory $\mathrm{Th}\left(U_Y\right)$ is satisfiable in $U_Y$.

Using statement (b) from the proof of Theorem 2.3.13 [23], we obtain the following lemma.

Lemma 5. 

Let T be a complete ω-categorical theory of the signature σ. Then, T has a countable ω-saturated model.

The next lemma follows directly from Corollary 7.1.8 [23].

Lemma 6. 

Let α be an uncountable cardinal and T be a complete theory of the signature σ, which has infinite models. Then, T is α-categorical if and only if every model of T of cardinality α is saturated.

Theorem 3. 

Let U be a structure of the signature σ and α be a cardinal.

(a) If U is an ω-saturated structure, then U is a program-saturated structure.

(b) If U is a model of the cardinality α of an α-categorical theory T, then U is a program-saturated structure.

Proof. 

(a) Let U be an $\omega$-saturated structure. Using Lemma 4, we obtain that, for any $n\in \omega \setminus \left\{0\right\}$, any set of formulas $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$ consistent with the theory $\mathrm{Th}\left(U\right)$ is satisfiable in U. Using Lemma 3, we obtain that the class $\left\{U\right\}$ has the property of compactness. From here and from Theorem 1, it follows that U is a program-saturated structure.

(b) Let U be a model of the cardinality $\alpha$ of an $\alpha$-categorical theory T. If U is a finite structure, then, evidently, U is a program-saturated structure.

Let $\alpha$ be an infinite cardinal. Evidently, $\mathrm{Th}\left(U\right)$ is an $\alpha$-categorical complete theory. If U is a countable structure, then using Lemma 5 and the completeness of the theory $\mathrm{Th}\left(U\right)$, we obtain that U is an $\omega$-saturated structure. If $\alpha$ is an uncountable cardinal, then using Lemma 6 and the completeness of the theory $\mathrm{Th}\left(U\right)$, we obtain that U is a saturated and consequently $\omega$-saturated structure. Using statement (a) of the theorem, we obtain that U is a program-saturated structure. □

Corollary 1. 

The following structures (see definitions in [23]), each of which, for some cardinal α, is a model of the cardinality α of an α-categorical theory, are program-saturated:

• Countable atomless Boolean algebra (see Proposition 1.4.5 [23]).
• Abelian group with all elements of order p, where p is a prime number (see Proposition 1.4.7 [23]).
• Uncountable divisible torsion-free abelian group (see Proposition 1.4.8 [23]), in particular, the additive group of real numbers $(\mathbb{R};+,0)$.
• Uncountable algebraically closed field of the characteristic zero or p, where p is a prime number (see Proposition 1.4.10 [23]), in particular, the field of complex numbers $(\mathbb{C};+,·,0,1)$.

5. Elementary Extensions

In this section, we study the possibility of elementary extension of the structure to a structure that is program-saturated. We show that this is always possible and find the minimum cardinality of such an extension.

Definition 14. 

Let $n\in \omega \setminus \left\{0\right\}$ and $F_n\left(\sigma \right)=\{{\phi }_i:i\in \omega \}$. The n-type of the structure U of the signature σ is any finitely satisfiable in $\left\{U\right\}$ set of formulas of the form $\{{\phi }_i^{{\delta }_i}:i\in \omega \}$, where, for any $i\in \omega$, ${\delta }_i\in \{0,1\}$ and ${\phi }_i^1={\phi }_i$, ${\phi }_i^0=\neg {\phi }_i$.

Lemma 7. 

A structure U of the signature σ is program-saturated if and only if, for any $n\in \omega \setminus \left\{0\right\}$, any n-type of the structure U is satisfiable in U.

Proof. 

Let the structure U be program-saturated. Using Theorem 1, we obtain that, for any $n\in \omega \setminus \left\{0\right\}$, any n-type of the structure U is satisfiable in U.

Let, for any $n\in \omega \setminus \left\{0\right\}$, any n-type of the structure U be satisfiable in U. Let $\mathrm{\Phi }\subseteq F_n\left(\sigma \right)$ and the set $\mathrm{\Phi }$ be finitely satisfiable in $\left\{U\right\}$. Using Lemma 3, we obtain that $\mathrm{\Phi }$ is satisfiable in some model $U_1$ of the theory $\mathrm{Th}\left(U\right)$ on some tuple $\overline{a}\in A_1^n$, where $A_1$ is the universe of the structure $U_1$. Denote $H=\{{\phi }_i^{{\delta }_i}:i\in \omega \}$, where, for any $i\in \omega$, ${\delta }_i\in \{0,1\}$ and $U_1\vDash {\phi }_i^{{\delta }_i}\left(\overline{a}\right)$. Using Lemma 3, we obtain that H is an n-type of the structure U. Hence, the set H is satisfiable in U. Taking into account that $\mathrm{\Phi }\subseteq H$, we obtain that $\mathrm{\Phi }$ is satisfiable in U. Thus, the class $\left\{U\right\}$ has the property of compactness. Using Theorem 1, we obtain that U is a program-saturated structure. □

Let $U_1$ and $U_2$ be structures of the signature $\sigma$ with universes $A_1$ and $A_2$, respectively.

Definition 15. 

The structure $U_2$ is an extension of the structure $U_1$ if $A_1\subseteq A_2$, each predicate of $U_1$ is the restriction of corresponding predicate of $U_2$ to $A_1$, each function of $U_1$ is the restriction of corresponding function of $U_2$ to $A_1$, and each constant of $U_1$ is the corresponding constant of $U_2$. The notation $U_1\subset U_2$ means that $U_2$ is an extension of $U_1$. If $U_1\subset U_2$, we will say that $U_1$ is a substructure of $U_2$.

Definition 16. 

We will say that $U_2$ is an elementary extension of $U_1$ if $U_2$ is an extension of $U_1$ and, for any formula $\phi \in F_n\left(\sigma \right)$ and any n-tuple $\overline{a}\in A_1^n$, $U_1\vDash \phi \left(\overline{a}\right)$ if and only if $U_2\vDash \phi \left(\overline{a}\right)$. The notation $U_1\prec U_2$ means that $U_2$ is an elementary extension of $U_1$. If $U_1\prec U_2$, we will say that $U_1$ is an elementary substructure of $U_2$.

Definition 17. 

The structures $U_1$ an $U_2$ are called elementary equivalent if, for any sentence φ of the signature σ, $U_1\vDash \phi$ if and only if $U_2\vDash \phi$. The notation $U_1\equiv U_2$ means that the structures $U_1$ an $U_2$ are elementary equivalent.

The next lemma follows directly from Proposition 3.1.1 [23].

Lemma 8. 

• (a) If $U_1\prec U_2$, then $U_1\equiv U_2$.
• (b) If $U_1\prec U_3$, $U_2\prec U_3$, and $U_1\subset U_2$, then $U_1\prec U_2$.

Let U be a structure of the signature $\sigma$ and $n\in \omega \setminus \left\{0\right\}$. We denote by $S_{n}U$ the set of all n-types of the structure U.

Lemma 9. 

If a structure $U_2$ is an elementary extension of a structure $U_1$, then $S_n{U}_1=S_n{U}_2$.

Proof. 

Let $U_1\prec U_2$. Using statement (a) of Lemma 8, we obtain that $\mathrm{Th}\left(U_1\right)=\mathrm{Th}\left(U_2\right)$. From here and from Lemma 3 it follows that, for any $n\in \omega \setminus \left\{0\right\}$, $S_n{U}_1=S_n{U}_2$. □

The next lemma follows directly from Lemma 3.4 [25].

Lemma 10. 

If Y is a set of cardinals, then $supY$ is a cardinal.

Let $\alpha$ be a cardinal. We denote by ${\alpha }^{+}$ the successor of the cardinal $\alpha$, i.e., the least cardinal that is greater than $\alpha$. Denote $\left|\right|\sigma \left|\right|=\omega \cup \left|\sigma \right|$.

The next lemma follows directly from Lemma 5.1.4 [23].

Lemma 11. 

Let U be a structure of the signature σ with the universe A and α be a cardinal such that $\left|\right|\sigma \left|\right|\le \alpha$ and $\omega \le \left|A\right|\le 2^{\alpha }$. Then, there is an ${\alpha }^{+}$-saturated elementary extension of U of the cardinality $2^{\alpha }$.

The next lemma follows directly from Theorem 3.1.6 [23].

Lemma 12. 

Let U be a structure of the signature σ with the universe A, $\left|A\right|=\alpha$ and $\left|\right|\sigma \left|\right|\le \beta \le \alpha$. Then, for any set $B\subseteq A$ with cardinality at most β, the structure U has an elementary substructure of cardinality β for which the universe contains B.

The next lemma follows directly from Lemma 5.2 [25].

Lemma 13. 

Let S be a family of sets. Then, $|{\bigcup }_{P\in S}P|\le |S|·sup\{\left|P\right|:P\in S\}$.

Let U be a structure of the signature $\sigma$ with the universe A. For $n\in \omega \setminus \left\{0\right\}$, we denote ${\alpha }_n\left(U\right)=\left|S_{n}U\right|$. If the set A is a finite set, we denote $\alpha \left(U\right)=\left|A\right|$. If A is an infinite set, then denote $\alpha \left(U\right)=sup\left\{\right|A|,{\alpha }_1\left(U\right),{\alpha }_2\left(U\right),\dots \}$. By Lemma 10, $\alpha \left(U\right)$ is a cardinal.

Theorem 4. 

Let $U_1$ be a structure of the signature σ.

(a) The cardinality of any program-saturated structure, which is an elementary extension of the structure $U_1$, is greater than or equal to $\alpha \left(U_1\right)$.

(b) There exists a program-saturated structure $U_2$, which is an elementary extension of the structure $U_1$ and for which the cardinality is equal to $\alpha \left(U_1\right)$.

Proof. 

Let $A_1$ be the universe of the structure $U_1$.

(a) Let $U_2$ be a program-saturated elementary extension of the structure $U_1$ and $A_2$ be the universe of the structure $U_2$. Evidently, $|A_2|\ge |A_1|$. Therefore, if $A_1$ is a finite set, then $|A_2|\ge \alpha \left(U_1\right)$. Let $A_1$ be an infinite set. Let $n\in \omega \setminus \left\{0\right\}$. Since $U_2$ is a program-saturated structure, by Lemma 7, any n-type from $S_n{U}_2$ is satisfiable in $U_2$. Evidently, different n-types from $S_n{U}_2$ are satisfiable on different n-tuples from $A_2^n$. Therefore, $|A_2^n|\ge |S_n{U}_2|$. Since $A_1\subseteq A_2$ and $A_1$ is an infinite set, $|A_2^n|=|A_2|$. Using Lemma 9, we obtain $|S_n{U}_2|={\alpha }_n\left(U_1\right)$. Therefore, $|A_2|\ge {\alpha }_n\left(U_1\right)$. Thus, $|A_2|\ge \alpha \left(U_1\right)$.

(b) Let $A_1$ be a finite set. Evidently, $U_1$ is a program-saturated structure, $U_1\prec U_1$, and the cardinality of $U_1$ is equal to $\alpha \left(U_1\right)$. Therefore, in the capacity of $U_2$, we can take $U_1$.

Let $A_1$ be an infinite set. From Lemma 11 it follows that there exists an $\omega$-saturated elementary extension $U_3$ of the structure $U_1$. From Theorem 3 it follows that $U_3$ is a program-saturated structure. Using Lemma 7, we obtain that, for any $n\in \omega \setminus \left\{0\right\}$, any n-type $H\in S_n{U}_3$ is satisfiable on some n-tuple ${\overline{a}}_H\in A_3^n$, where $A_3$ is the universe of $U_3$.

For an arbitrary $n\in \omega \setminus \left\{0\right\}$, we denote $B_n=\{{\overline{a}}_H:H\in S_n{U}_3\}$ and $C_n$ the set of all elements from $A_3$ belonging to the n-tuples from $B_n$. If $C_n$ is a finite set, then $|C_n|<|A_1|\le \alpha \left(U_1\right)$. If $C_n$ is an infinite set, then, as it is not difficult to check, $|C_n|=|B_n|$. Evidently, $|B_n|\le |S_n{U}_3|$. Using Lemma 9, we obtain $|S_n{U}_3|={\alpha }_n\left(U_1\right)$. Hence, $|C_n|\le {\alpha }_n\left(U_1\right)\le \alpha \left(U_1\right)$. Denote $C={\bigcup }_{n=1}^{\infty }{C}_n$. Taking into account that $\alpha \left(U_1\right)$ is an infinite cardinal and, for any $n\in \omega \setminus \left\{0\right\}$, $|C_n|\le \alpha \left(U_1\right)$, and using Lemma 13, we obtain $\left|C\right|\le$$\alpha \left(U_1\right)$.

Denote $Y=A_1\cup C$, $\beta =\left|Y\right|$, and $\alpha =|A_3|$. Evidently, $Y\subseteq A_3$. Therefore, $\left|\right|\sigma \left|\right|=\omega \le \beta \le \alpha$. Using Lemma 12, we obtain that there exists an elementary substructure $U_2$ of the structure $U_3$ such that $Y\subseteq A_2$ and $|A_2|=\beta$, where $A_2$ is the universe of $U_2$. Using Lemma 9, we obtain that, for any $n\in \omega \setminus \left\{0\right\}$, $S_n{U}_2=S_n{U}_3$. Let $n\in \omega \setminus \left\{0\right\}$. Since $C\subseteq$$A_2$, $U_2\prec U_3$, and any n-type $H\in S_n{U}_3$ is satisfiable on some tuple from $C^n$, we obtain that any n-type $H\in S_n{U}_2$ is satisfiable on some tuple from $A_2^n$. Using Lemma 7, we obtain that $U_2$ is a program-saturated structure. Evidently, $U_1\prec U_3$ and $U_2\prec U_3$. Taking into account that $A_1\subseteq A_2\subseteq A_3$, it is not difficult to show that $U_1$ is a substructure of the structure $U_2$, i.e., $U_1\subset U_2$. Using statement (b) of Lemma 8, we obtain $U_1\prec U_2$. It is easy to show that $|A_2|=\beta \le \alpha \left(U_1\right)$. Using statement (a) of the theorem, we obtain that the cardinality of the structure $U_2$ is equal to $\alpha \left(U_1\right)$. □

6. Computation Programs and Computation Trees

This section is devoted to the transfer of the obtained results to programs that are essentially close to the computation trees: if the scheme of such a program is a finite tree-scheme, then the program is an ordinary computation tree.

Let $\sigma$ be a finite or countable signature. We now define the notions of primitive term and primitive formula of the signature $\sigma$.

A primitive term of the signature $\sigma$ is a term of the signature $\sigma$ of the following form:

• A variable from the set X.
• A constant symbol from the signature $\sigma$.
• A term $f(x_{i_1},\dots ,x_{i_m})$, where f is a function symbol of arity m from $\sigma$ and $x_{i_1},\dots ,x_{i_m}\in X$.

A primitive formula of the signature $\sigma$ is a formula of the signature $\sigma$ of the following form:

• The formula $x_i=x_j$, where = is the equality symbol and $x_i,x_j\in X$.
• The formula $p(x_{i_1},\dots ,x_{i_m})$, where p is a predicate symbol of arity m from $\sigma$ and $x_{i_1},\dots ,$$x_{i_m}\in X$.

Definition 18. 

Let $S=(n,G)$ be a scheme of the signature σ. This scheme will be called a computation scheme if each of its function nodes is labeled with an expression of the form $x_i\Leftarrow t$, where t is a primitive term of the signature σ and $x_i\in X$, and each its predicate nodes is labeled with a primitive formula of the signature σ.

Definition 19. 

Let U be a structure of the signature σ and $\mathrm{\Gamma }=(S,U)$ be a program over the structure U. The program Γ will be called a computation program over the structure U if S is a computation scheme.

Definition 20. 

A computation scheme $(n,G)$ of the signature σ will be called a computation tree-scheme of the signature σ if G is a tree with the root that coincides with the initial node of G. A computation tree-scheme $(n,G)$ of the signature σ will be called finite if G is a finite tree.

Definition 21. 

Let U be a structure of the signature σ and $\mathrm{\Gamma }=(S,U)$ be a program over the structure U. The program Γ will be called a computation tree over the structure U if S is a finite computation tree-scheme.

6.1. Computation-Program-Saturated Classes of Structures

In this section, some results related to computation-program-saturated classes of structures are considered.

Definition 22. 

A nonempty class K of structures of the signature σ will be called computation-program-saturated if any total relative to K computation scheme of the signature σ is strongly equivalent relative to K to a finite computation tree-scheme of the signature σ.

It is clear that any program-saturated class K is a computation-program-saturated class. Using Theorem 1, we obtain the following statement.

Proposition 1. 

If a nonempty class K of structures of the signature σ has the property of compactness, then it is computation-program-saturated.

The next statement follows from Theorem 2.

Proposition 2. 

Any nonempty axiomatizable class K of structures of the signature σ is computation-program-saturated.

The following examples of axiomatizable classes of structures have already been considered:

• Classes of Boolean algebras, atomic Boolean algebras, and atomless Boolean algebras.
• Classes of groups; abelian groups; abelian groups with all elements of order p, where p is a prime; and torsion-free abelian groups.
• Classes of commutative rings with unit; fields; fields of characteristic p, where p is a prime; fields of characteristic zero; algebraically closed fields; and real closed fields.

6.2. Computation-Program-Saturated Structures

In this section, some results related to computation-program-saturated structures are considered.

Definition 23. 

Let U be a structure of the signature σ. The structure U will be called computation-program-saturated if the class $\left\{U\right\}$ is computation-program-saturated.

The next statement follows from Theorem 3.

Proposition 3. 

Let U be a structure of the signature σ and α be a cardinal.

(a) If U is an ω-saturated structure, then U is a computation-program-saturated structure.

(b) If U is a model of the cardinality α of an α-categorical theory, then U is a computation-program-saturated structure.

It has already been mentioned that each of the following structures is, for some cardinal $\alpha$, a model of the cardinality $\alpha$ of an $\alpha$-categorical theory:

• Countable atomless Boolean algebra.
• Abelian group with all elements of order p, where p is a prime number.
• Uncountable divisible torsion-free abelian group, in particular, an additive group of real numbers $(\mathbb{R};+,0)$.
• Uncountable algebraically closed field of the characteristic zero or p, where p is a prime number, in particular, the field of complex numbers $(\mathbb{C};+,·,0,1)$.

Each of these structures is a computation-program-saturated structure.

6.3. Elementary Extensions

In this section, we consider the possibility of elementary extension of the structure to a structure that is computation-program-saturated. In Section 5, for any structure U, we defined a cardinal $\alpha \left(U\right)$. The next statement follows from Theorem 4.

Proposition 4. 

Let $U_1$ be a structure of the signature σ. Then, there exists a computation-program-saturated structure $U_2$, which is an elementary extension of the structure $U_1$ and of which cardinality is equal to $\alpha \left(U_1\right)$.

7. Conclusions

In this paper, program-saturated classes of structures were studied. It was proven that a necessary and sufficient condition for a class to be program-saturated is its compactness. It was also shown that any axiomatizable class of structures is program-saturated. Additionally, individual structures were examined, with each forming a class that is program-saturated. It was demonstrated that such structures include all models of the cardinality $\alpha$ of $\alpha$-categorical theories.

The possibility of an elementary extension of a structure to one for which the corresponding singleton class of structures is program-saturated was studied. It was shown that such an extension is always possible, and the minimum cardinality of such an extension was determined. Finally, the obtained results were applied to programs that are essentially close to computation trees.

These results are of interest for applications. For instance, the field of complex numbers $(\mathbb{C};+,·,0,1)$ is a computation-program-saturated structure. As a result, when constructing a total computation program over this structure to solve a task, it is sufficient to consider only computation trees over $(\mathbb{C};+,·,0,1)$. This significantly simplifies the problem.