Weak essentially undecidable theories of concatenation

In the language {0,1,∘,⪯}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lbrace 0, 1, \circ , \preceq \rbrace $$\end{document}, where 0 and 1 are constant symbols, ∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\circ $$\end{document} is a binary function symbol and ⪯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\preceq $$\end{document} is a binary relation symbol, we formulate two theories, WD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textsf {WD} $$\end{document} and D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}$$\end{document}, that are mutually interpretable with the theory of arithmetic R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {R}} $$\end{document} and Robinson arithmetic Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf {Q}} $$\end{document}, respectively. The intended model of WD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textsf {WD} $$\end{document} and D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}$$\end{document} is the free semigroup generated by {0,1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lbrace {\varvec{0}}, {\varvec{1}} \rbrace $$\end{document} under string concatenation extended with the prefix relation. The theories WD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textsf {WD} $$\end{document} and D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {D}}$$\end{document} are purely universally axiomatised, in contrast to Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textsf {Q}} $$\end{document} which has the Π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _2$$\end{document}-axiom ∀x[x=0∨∃y[x=Sy]]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall x \; [ \ x = 0 \vee \exists y \; [ \ x = Sy \ ] \ ] $$\end{document}.


Introduction
This paper follows the line of work that focuses on determining whether there is a weakest theory that is essentially undecidable. We formulate two natural essentially undecidable theories in the language of concatenation that are purely universally axiomatised.
A countable first-order theory is called essentially undecidable if any consistent extension, in the same language, is undecidable (there is no algorithm for deciding whether an arbitrary sentence is a theorem). A countable first-order theory is called essentially incomplete if any recursively axiomatizable consistent extension is incomplete. It is known that a theory is essentially undecidable if and only if it is essentially incomplete. Indeed, if a theory is not essentially undecidable, then by definition it has a consistent decidable extension which can be extended to a decidable complete consistent theory (see Chapter 1 of Tarski et al. [18]). On the other hand, if a theory is not essentially incomplete, then by definition it has a complete consistent recursively B Juvenal Murwanashyaka juvenalm@math.uio.no 1 Department of Mathematics, University of Oslo, Oslo, Norway axiomatizable extension which clearly is decidable. Two theories that are known to be essentially undecidable are Robinson arithmetic Q and the related theory R. Rosser's generalization of Gödel's first incompleteness theorem is usually taken as the statement that Q is essentially undecidable.
The Axioms of R The Axioms of Q The main objective of this paper is to show that two theories, WD and D, are mutually interpretable with R and Q, respectively. The axioms of WD and D are given below using juxtaposition instead of the binary function symbol of the formal language. Later we regard D as the theory without the axiom D 5 . We do this because the two versions of D are mutually interpretable. The theories WD and D are theories in the language of concatenation extended with a binary relation symbol. That is, the language {0, 1, •, } where 0 and 1 are constant symbols, • is a binary function symbol and is a binary relation symbol. The intended structure D is the free semigroup with two generators extended with the prefix relation which we denote D . As in number theory, each element α in the universe is associated with a canonical term α. The inclusion of a binary relation in the languages makes it easier to define Σ 1 -formulas and to give a purely universal Σ 1 -complete axiomatization of D. We observe that any theory that proves all true Σ 1 -sentences is an extension of WD. Indeed, instances of WD 1 and WD 2 are true Σ 1 -sentences, and each instance of WD 3 follows from the true Σ 1 -sentences ∀x α [ γ D α x = γ ] and γ D α γ α. The theory WD is thus the weakest Σ 1 -complete axiomatization of D (modulo closure under logical implication) and the theory D is a natural finitely axiomatizable extension of WD. A variant of D where we have an identity element was introduced in Kristiansen and Murwanashyaka [9] as a Σ 1 -complete axiomatization of the structure D extended with the empty string. In [9,10], we identify a number of decidable and undecidable fragments of D and related structures.

The Axioms of WD
The Axioms of D The theory of concatenation TC was introduced by Grzegorczyk in [4] where he also showed that it is undecidable. The language of TC consists only of the two constant symbols 0 and 1 and the binary function symbol •. The intended model of TC is a free semigroup with at least two generators. That is, a structure of the form (Γ + , a 1 , . . . , a n , ) where Γ = {a 1 , . . . , a n } is a finite alphabet with at least two symbols, Γ + is the set of all finite non-empty strings over Γ and is the binary operator that concatenates elements of Γ + . Grzegorczyk's motivation for introducing the theory TC was that, as computation involves manipulation of text, the notion of computation can be formulated on the basis of discernibility of text without reference to natural numbers. Then, undecidability of first-order logic and essential undecidability can be explained using a theory of strings thereby avoiding complicated coding of syntax based on natural numbers. In [5], Grzegorczyk and Zdanowski showed that TC is essentially undecidable. This was further improved in Ganea [3], Visser [19] and Švejdar [16] where it was shown that TC is mutually interpretable with Robinson arithmetic. In Higuchi and Horihata [6], it was shown that TC is minimal essentially undecidable. That is, removing any one of the axioms of TC gives a theory that is not essentially undecidable. We will refer to TC as TC −ε and we will let TC refer to the theory where we have an identity element. The two theories are known to be mutually interpretable (see Grzegorczyk and Zdanowski [5] and Visser [19]). In the article [6], it was also shown that a weak theory of concatenation WTC −ε is minimal essentially undecidable and mutually interpretable with R. That WTC −ε is minimal essentially undecidable means that removal of any one of the axiom schemas of WTC −ε gives a theory that is not essentially undecidable.

The Axioms of TC
We use x s y as shorthand for The diagram in Fig. 1 summarizes the relationships between R, Q, WD, D, WTC −ε and TC −ε .
Kristiansen and Murwanashyaka [11] have also introduced two essentially undecidable theories, WT and T, that are purely universally axiomatised. While WD and D are theories of concatenation, WT and T are theories of terms. The intended model of WT and T is the extended term algebra given by the language L T = {⊥, ·, · , } where ⊥ is a constant symbol, ·, · is a binary function symbol and is a binary relation symbol. The universe is the set of all variable-free L T -terms and ·, · is interpreted as the function that maps a pair (s, t) to the term s, t . Every variable-free term is thus realized as itself. The relation symbol is interpreted as the subterm relation. The theory WT has a compact axiomatization that consists only of analogues of WD 2 and WD 3 . An analogue of WD 1 is not necessary since it holds by pure logic. The theory T has two axioms that describe ·, · ; and two axioms that describe . Kristiansen and Murwanashyaka give interpretations of R and Q in WT and T, respectively. A result of Visser [20] ensures that WT is interpretable in R since it is locally finitely satisfiable. They conjecture that T and Q are mutually interpretable.

Preliminaries
In this section, we clarify a number of notions that we only glossed over in the previous section. We also introduce a number of intermediate theories that will be useful in showing that WD and R are mutually interpretable and that D and Q are mutually interpretable. We consider the structures In first-order number theory, each natural number n is associated with a numeral n. Each non-empty bit string α ∈ {0, 1} + is associated by recursion with a unique L − BTterm α, called a biteral, as follows: 0 ≡ 0, 1 ≡ 1, α0 ≡ (α • 0) and α1 ≡ (α • 1). The biterals are important if we, for example, want to show that certain sets are definable since we then need to talk about elements of {0, 1} + in the formal theory.
In the language L BT , we can define the Σ 1 -formulas as follows: atomic formulas and their negations are Σ 1 -formulas. If α and β are Σ 1 -formulas, then α ∧ β, α ∨ β and ∃x α are Σ 1 -formulas. Furthermore, if the variable x does not occur in the term t, then ∃x [ x t ∧ α ] and ∀x [ x t → α ] are also Σ 1 -formulas. We use ∃x t α and ∀x t α as abbreviations for ∃x [ x t ∧ α ] and ∀x [ x t → α ], respectively. To define Σ 1 -formulas over the language L − BT , we first need to define a binary relation in order to have bounded quantifiers. Two natural choices are Over the structure D − , the defined relation i is realized as the prefix relation while s is realized as the substring relation. Given a bit string α, the set of those bit strings that are substrings of α is denoted ...α and consists of those bit strings β such that β = α or there exist bit strings u and v such that α = uβ or α = βv or α = uβv. We choose to work with s since the intended interpretation of is the prefix relation. We observe that the number of substrings of a string α is quadratic in the length of α, whereas the number of prefixes of α is linear in the length of α. This means that the choice between i and s could make a difference in the context of very weak theories.
Having introduced Σ 1 -formulas, it is natural to try to find Σ 1 -complete axiomatizations of the structures D − and D. That is, to find theories that prove all true Σ 1 -sentences (sentences are formulas without free variables) and that are such that the non-logical axioms are true over the intended structure. A natural first step is to introduce the theories WBT and WD defined below. It is not difficult to see that these two theories are Σ 1 -complete. The theories WBT and WD are not finitely axiomatizable but they are the weakest possible Σ 1 -complete axiomatizations of D − and D, respectively, modulo closure under logical implication. Once we have WBT and WD, the theories BTQ and D + ∀x [ x 1 ↔ x = 1 ] defined below are natural finitely axiomatizable extensions (T is an extension of S if the language of S is a subset of the language of T and every theorem of S is a theorem of T ). The reason for not having ∀x [ x 1 ↔ x = 1 ] as an axiom of D is that it is not necessary for essential undecidability (we could, of course, very well have worked with the theory where we have ∀x [ Although the theories WBT, BTQ, WD and D are Σ 1 -complete, it is not at all obvious that they are essentially undecidable. When proving that R is essentially undecidable, the axiom schema R 5 ≡ ∀x [ x ≤ n ∨ n ≤ x ] is essential. It is however not straightforward to define a binary relation that provably satisfies the analogue of R 5 . The method of relative interpretability then becomes important for establishing that these theories are T means S is interpretable in T but it is unknown whether S interprets T essentially undecidable.

The Axioms of WBT
The Axioms of WD The Axioms of BTQ The Axioms of D The Axioms of BT The Axioms of C The overall relationship among the various theories introduced so far is summarized in Fig. 2. The constants 0 and 1 are atoms in BTQ. This means that BTQ ∀x y [ x y = 0 ] ∧ ∀x y [ x y = 1 ] (see Lemma 8 in Ganea [3] for a proof). This observation is used in the proof of Theorem 7 where we show that D is interpretable in BTQ.
We recall the method of relative interpretability introduced by Tarski [18] for showing that first-order theories are essentially undecidable. Let L 1 and L 2 be computable first-order languages. A relative translation τ from L 1 to L 2 is a computable map given by: 1. An L 2 -formula δ(x) with exactly one free variable. The formula δ(x) is called a domain. 2. For each n-ary relation symbol R of L 1 , an L 2 -formula ψ R (x 1 , ..., x n ) with exactly n free variables. The equality symbol = is treated as a binary relation symbol.
3. For each n-ary function symbol f of L 1 , an L 2 -formula ψ f (x 1 , ..., x n , y) with exactly n + 1 free variables. 4. For each constant symbol c of L 1 , an L 2 -formula ψ c (y) with exactly one free variable.
We extend τ to a translation of atomic L 1 -formulas by mapping a L 1 -term t to a L 2 -formula (t) τ,w with a free variable w that denotes the value of t: where v 1 , . . . , v n are distinct variable symbols that do not occur in t 1 , . . . , t n and where w 1 , . . . , w n are distinct variable symbols that do not occur in n j=1 (t j ) τ,w . We extend τ to a translation of all L 1 -formulas as follows: Let S be an L 1 -theory and let T be an L 2 -theory. We say that S is (relatively) interpretable in T if there exists a relative translation τ such that -For each constant symbol c of L 1 T ∃!y [δ(y) ∧ ψ c (y)] .
-T proves φ τ for each non-logical axiom φ of S. If equality is not translated as equality, then T must prove the translation of each equality axiom.
If S is relatively interpretable in T and T is relatively interpretable in S, we say that S and T are mutually interpretable.
The following proposition summarizes important properties of relative interpretability (see Tarski et al. [18] for the details).
Proposition 1 Let S, T and U be computably enumerable first-order theories.
1. If S is interpretable in T and T is consistent, then S is consistent.

If S is interpretable in T and T is interpretable in U , then S is interpretable in U .
3. If S is interpretable in T and S is essentially undecidable, then T is essentially undecidable.

Intermediate theories
We show in our Master's thesis [12] that WBT and WTC −ε are equivalent, i.e., they prove the same formulas. We do not include a proof of this result, but the interested reader may find the fairly straightforward proof in [12] . We also show in [12] that BTQ and TC −ε are mutually interpretable. Although this result is not trivial, we omit a proof since there is another way of seeing that BTQ is mutually interpretable with Q. The theory BTQ is a fragment of the theory F, introduced first by Alfred Tarski at the end of Chapter 3 of [18]. The theory BTQ also resembles the theory Q bin in Visser [19]. One can view Q bin as the analogue of BTQ where we have an identity element and associativity has been weakened to ∀x y The theory F differs from BTQ in that the axiom BTQ 2 is replaced with the axioms In Ganea [3], it is shown that F − F 2 is mutually interpretable with Q. Clearly, BTQ is interpretable in F − F 2 . The other way, we can interpret F − F 2 in BTQ by simply relativizing quantification to the domain It follows from BTQ 2 that 0, 1 ∈ J . We now show that J is closed under •. Suppose z, w ∈ J . We need to show that zw ∈ J . We have Hence, zw ∈ J . Thus, J satisfies the domain condition. Clearly, the axioms of F−F 2 hold on J . We show that the axiom BTQ 4 holds on J . Suppose x ∈ J . By BTQ 4 , if Hence, x ∈ J implies y ∈ J . Thus, BTQ 4 holds on J . It is not difficult to verify that the other axioms also hold on J . We can thus state the following theorem which will be used implicitly to show that D and Q are mutually interpretable.
Theorem 1 BTQ and Q are mutually interpretable.
The theories BT and C are examples of theories that lie strictly between R and Q w.r.t. relative interpretability. It follows from ∀x , it would be interpretable in a finite sub-theory. Since any finite sub-theory of WBT or WD clearly has a finite model, we cannot interpret BT (C) in WBT (WD). We now show that BTQ is not interpretable in BT. Similar reasoning shows that D is not interpretable in C. We let BT 4 denote WBT 3 and for 1 ≤ i ≤ 3 we let BT i denote BTQ i .

Theorem 2 BTQ is not interpretable in BT.
Proof Suppose τ is an interpretation of BTQ in BT. Then, there is a finite subset Σ of the axioms of BT such that To see that τ cannot exist, it suffices to show that the theory given by Σ is interpretable in BT−BT 4 , which is not essentially undecidable by minimality of BTQ (see Lemma 11).
To see that Σ is interpretable in BT−BT 4 , it suffices to show that for each natural number n ≥ 1, the theory BT ≤n is interpretable in BT. The theory BT ≤n is like BT except that the axiom schema is limited to those α such that the length of α, denoted |α|, is bounded by n. To do this, we define by recursion a sequence of domains such that we obtain an interpretation of BT ≤n in BT by simply relativizing quantification to I n . So, we proceed to construct these domains. We will omit parentheses most of the time since we have the axiom

Construction of I 1 :
We let Suppose x00 = 0 ∨ x01 = 1. Then, 11x00 = 110 ∨ 11x01 = 111. By BT 2 , we then have 11x0 = 11, contradicting BT 3 . Hence, 0 ∈ A 1 . By similar reasoning, we have 1 ∈ A 1 . We now show that A 1 is closed under •. Suppose y, y ∈ A 1 . By BT 1 , we have We let Suppose 00 = 0 ∨ 01 = 1. Then, 1100 = 110 ∨ 1101 = 111. By BT 2 , we then have 110 = 11, contradicting BT 3 . Thus, 0 ∈ B 1 . By similar reasoning, we have 1 ∈ B 1 . We now show that B 1 is closed under •. Suppose y, y ∈ B 1 . We observe that We let We clearly cannot have x1 = 0 ∨ x0 = 1 since 1x1 = 10 ∨ 1x0 = 11 would follow, contradicting BT 3 . It follows from the definition of B 1 that we have Hence, 0, 1 ∈ I 1 . We now show that Hence, yy ∈ I 1 . Thus, I 1 is closed under •. This means that I 1 satisfies the domain conditions. We interpret BT ≤1 in BT−BT 4 by simply relativizing quantification to I 1 . Since BT 1 , BT 2 and BT 3 are universal sentences that are theorems of BT−BT 4 , they hold in I 1 . It remains to show that BT−BT 4 proves the translation of We only show (1). The case (2) is handled similarly. So, suppose x ∈ I 1 and the translation of x s 0 holds. We then have one of the following cases for some z, w ∈ I 1 : We notice that (ii)-(v) contradict the definition of I 1 . We thus see that the translation of (1) is a theorem of BT−BT 4 . Similarly, the translation of (2) is a theorem of BT−BT 4 .

Construction of I n+1 :
Let Γ n denote the set of all nonempty biterals of length at most n. Let I n s denote the realization of s in I n , i.e.
We assume I n has been constructed and satisfies We let By BT 3 , we cannot have x1 = α0 ∨ x0 = α1. By BT 2 , we have Hence, 0, 1 ∈ I n+1 . We now show that I n+1 is closed under •. Suppose y 0 , y 1 ∈ I n+1 . Let x ∈ I n . We observe that x y 0 ∈ I n since x, y 0 ∈ I n and I n is closed under •. By BT 1 , we have x(y 0 y 1 ) = (x y 0 )y 1 . Then It is not difficult to see that BT−BT 4 proves each instance of the axiom schemas This implies Hence, y 0 y 1 ∈ I n+1 . Thus, I n+1 is closed under •. This proves that I n+1 satisfies the domain conditions. It is clear that we get an interpretation of BT ≤n+1 in BT−BT 4 by simply relativizing quantification to I n+1 .
By a similar argument, we see that D is not interpretable in C.

Theorem 3 D is not interpretable in C.
The theories C and BT are strictly between R and Q w.r.t. interpretability. A natural question is whether they are mutually interpretable. It is not difficult to see that our interpretation of WD in WBT given at the end of Sect. 4 is also an interpretation of C in BT. It is not clear however whether BT is interpretable in C.

The theory WD
We now show that the theory WD is mutually interpretable with R. We give an interpretation of R in WD. To show that WD is interpretable in R we invoke the result of Visser [20]: a recursively enumerable theory is interpretable in R if and only if it is locally finitely satisfiable (each finite sub-theory has a finite model). We also show how to interpret WD in WBT , from which follows that WD is interpretable in R since WBT and WTC −ε are equivalent, and WTC −ε and R are mutually interpretable (see Fig. 2). The theory WD is purely universally axiomatised, in contrast to WTC −ε which has Π 2 -axioms.

Definition 1
The first-order theory WD contains the following non-logical axioms: for each α, β ∈ {0, 1} + We start by showing that a theory of arithmetic we call R − is interpretable in WD. The theory R − has been shown to be mutually interpretable with R (see Jones and Shepherdson [8]).

Definition 2
The first-order theory R − contains the following non-logical axioms: for each n, m ∈ N The most difficult step in interpreting R − in WD is translating ×. We start by proving two lemmas which show how we intend to translate ×. In the structure D, we can associate each natural number k with the bit string 1 k 0. For n = 0 and m > 1, we can then translate n × m = nm by saying that the sequence 1 n , 1 n+n , 1 n+n+n , . . . , 1 nm exists. The formulas given in Lemmas 1 and 2 try to capture this way of viewing multiplication. But we have to be careful since we are reasoning in a weak theory. For example, we lack full associativity. For readability, we dispense with parentheses whenever possible and expect terms to be read from right to left. That is, x yz should be regarded as shorthand for (x y)z. Although we do not have full associativity, by WD 1 , we have WD (αβ)γ = α(βγ ) for all α, β, γ ∈ {0, 1} + . Then, by WD 3 , we have The bounded quantifiers in Lemmas 1 and 2 are there to make full use of the axiom scheme WD 3 .

Lemma 1 Let n = 0 and m ≥ 2. Then, WD proves
(1) 1 nm 1 nm and 1 n , 1 m and 1 nm are the unique elements such that Then, w provably (in WD) satisfies: Proof Since we have the axiom schema WD 1 , we can skip parentheses in w. Clause (1) holds due to the axiom schemas WD 3 and WD 2 . We verify clause (a). By the rightleft implication of WD 3 , we have 1 n 1 n 0. By how biterals are defined, we have 1 n 0 = 1 n 0. For uniqueness, suppose By WD 3 , we have Then, by WD 2 , in order for x0 = 1 n 0 to hold we must have x = 1 n . By similar reasoning, one verifies clauses (b) and (c).
By the right-left implication of WD 3 , we have u w. By WD 1 , we have u001 nm 01 m 00 = w .
Then, by the right-left implication of WD 3 , we have u001 nm 01 m 00 w .
To see that clause (3) holds, suppose holds. By the axiom schemas WD 3 and WD 2 , the third and fourth conjunct of (*) imply y = 1 k+1 where 2 ≤ k + 1 ≤ m. Given w w ∧ z 1 nm , the axiom schema WD 3 gives us a set Γ of pairs (a, b) where a w and b 1 nm such that (w , z ) ∈ Γ . For each pair (a, b) ∈ Γ , we use WD 1 to compute a00b0y 00 and then use the left-right implication of WD 3 and WD 2 to determine whether a00b0y 00 w. We are then led to the conclusion w = 0001 n 01001 n+n 011001 n+n+n 011100 . . . 001 nk 01 k z = 1 n(k+1) .

Lemma 2 Let
By WD 3 and WD 2 , we have Furthermore, by the definition of φ M (1 n 0, 1 m 0, Z ), there exist w and w 1 w such that w 1 00z 1 0y 1 00 w .
By the axiom schemas WD 3 and WD 2 , we must have If m − 2 = 1, then by Φ(y 3 , z 3 , x) we have z 3 = 1 n . By WD 1 , this implies to repeat this procedure. We notice that after a finite number of steps, we have that there exist w m w, y m y m−1 and z m z 1 such that By the axiom schemas WD 3 and WD 2 , we must have By backtracking, we observe that z 1 = 1 n 1 n . . . 1 n where 1 n occurs m times. It then follows from the axiom schema WD 1 that z 1 = 1 nm . Thus, by WD 1 , we have Z = 1 nm 0. Hence, (***) holds.

Theorem 4 R − is interpretable in WD.
Proof We choose the domain δ(x) ≡ x = x. We translate, 0, S, × and ≤ as follows: The formula φ M is defined in Lemma 2. By the axiom schema WD 1 , our translation of S implies that we translate each numeral n as the biteral 1 n 0. By the axiom schema WD 2 , the translation of each instance of the axiom schema R − 2 is a theorem of WD.
We now show that the translation of each instance of the axiom schema R − 1 is a theorem of WD. By Lemma 2, for each natural number n and m, we have (1 n 0, 1 m 0, 1 nm 0) . (1 n 0, 1 m 0, 1 nm 0) .
Thus, the translation of each instance of the axiom schema R − 1 is a theorem of WD. We observe that WD can have models where not all elements in the universe are realization of terms of the form 1 k 0. Hence, the condition in the definition of φ × (x, y, z) ensures that × is translated as a total function.
We now show that the translation of each instance of the axiom schema R − 3 is a theorem of WD. We have The last equivalence is due to how biterals are defined. We thus see that the translation of each instance of the axiom schema R − 3 is a theorem of WD.

Theorem 5 WD and R are mutually interpretable.
Proof We have shown that R − is interpretable in WD. Since R − and R are mutually interpretable, R is interpretable in WD. To see that WD is interpretable in R, we first observe that WD is locally finitely satisfiable, i.e., any finite subset of the axioms has a finite model. In [20], Albert Visser shows that a recursively enumerable theory is interpretable in R if and only if it is locally finitely satisfiable. Hence, WD is interpretable in R. Thus, WD and R are mutually interpretable.
We could have shown that WD is interpretable in R by showing that it is interpretable in WTC −ε , which we know is mutually interpretable with R. Since we in our Master's thesis [12] show that WBT and WTC −ε are equivalent, it suffices to show how to interpret WD in WBT. We choose the domain δ(x) ≡ x = x. We translate 0, 1, • and as follows: Clearly, the translation of each instance of the axiom schemas WD 1 and WD 2 is a theorem of WBT. We now show that the translation of each instance of the axiom schema WD 3 is a theorem of WBT. We have We thus see that the translation of each instance of the axiom schema WD 3 is a theorem of WBT. Hence, WD is interpretable in WBT.

The theory D
We now show that the theory D is mutually interpretable with Q. In contrast to Q and TC −ε , which have Π 2 -axioms, the theory D is purely universally axiomatised. When interpreting one theory into another, handling existential quantifiers can become cumbersome. This is clearly illustrated in the proof of Theorem 6. Therefore, having a theory with purely universal axiomatization and that is mutually interpretable with Q could be advantageous in some circumstances.

Definition 3
The first-order theory D is defined by the following non-logical axioms: For D to be an extension of WD, we need the axiom ∀x [ x 1 ↔ x = 1 ]. The theory D extended with this axiom is what we call D at page3. The next lemma shows why we have decided to not include this axiom. We could very well have replaced The proof of the next lemma also illustrates some of the advantages of not having to worry about existential quantifiers when defining a domain.
Proof We translate 0, 1, • and as We choose the domain Clearly, I contains 0100, 0110 and is closed under •. We use D 1 when showing that I is closed under •. We now proceed to show that the translation of each non-logical By D 1 and the definition of φ • (x, y, z), the translation of D 1 is a theorem of D. The translation of D 2 is a theorem of D since by D 2 we have The translation of D 3 is also a theorem of D since by D 2 we have x0100 = y0110 ⇒ x010 = y011 and x010 = y011 contradicts D 3 .
We now show that the translation of D 4 is a theorem of D. We have We need to show that we cannot have Since x ∈ I , by the definition I , we have that x = z0100 ∨ x = z0110 where z could possibly be empty (x = zu with z empty means x = u since we do not have an empty string in D). We have contradicting D 3 . We also have contradicting D 3 . Hence, x 0100 if and only if x = 0100. Thus, the translation of D 4 is a theorem of D. By similar reasoning, the translation of ∀x [ We now show that the translation of D 5 is a theorem of D. By D 5 , we have By D 6 , we have By D 5 , we have We need to show that we cannot have Since x, y ∈ I , by the definition I , we have x = z0100 ∨ x = z0110 and y = w0100 ∨ y = w0110 where z and w could possibly be empty. Reasoning as in the preceding paragraph shows that x = y010 ∨ x = y01 ∨ x = y0 leads to a contradiction. Hence, Thus, the translation of D 5 is a theorem of D. By similar reasoning, the translation of D 6 is a theorem of D.
Since D proves the translation of each axiom of D + ∀x [ We now proceed to show that Q and D are mutually interpretable. We do this indirectly by showing that D is mutually interpretable with the theory BTQ which we have seen is mutually interpretable with Q.

Proof By Lemma 3, it suffices to show that BTQ is interpretable in
We also observe that D proves the axioms BTQ 1 , BTQ 2 and BTQ 3 (the axioms are identical with D 1 , D 2 , D 3 ). So, to translate BTQ in D + D 4 , we simply define a domain K such that the axiom holds restricted to K . Before defining K , we define auxiliary classes A ⊇ B ⊇ C ⊇ I ⊇ J ⊇ K . We need to ensure that there is y ∈ K such that x = y0 or x = y1 if x ∈ K and x = 0, 1. We do this by relying on . The idea is to first let I be such that if x ∈ I and x = 0, 1; then we can find y x such that x = y0 or x = y1. It will not necessarily be the case that y ∈ I . What we then do is to restrict I so that we have a subclass K that is downward closed under , that is, x ∈ K and y x implies y ∈ K . Since K is a subclass of I , this immediately ensures that BTQ 4 holds in K . We realize that in order for I to be closed under •, it is useful if for all x 0 , x 1 ∈ I , we have that x 0 x 0 x 1 and that y x 1 implies x 0 y x 0 x 1 . We therefore let I be a subclass of a class C with this property.
We let By D 5 and D 6 , we have D ∀x [ x0 x0 ∧ x1 x1 ]. Hence, 0, 1 ∈ A. We now show that A is closed under •. Suppose y 0 , y 1 ∈ A. Since y 1 ∈ A, we have By D 1 , we then have ∀x [ x(y 0 y 1 ) x(y 0 y 1 ) ]. Hence, y 0 y 1 ∈ A. Thus, A is closed under •. We let By D 4 and D 4 , we have D 0 0 ∧ 1 1.
We let By D 5 and D 4 , we have Similarly, by D 6 and D 4 , we have Hence, 0, 1 ∈ C. Next we show that C is closed under •. Suppose z 0 , z 1 ∈ C. Then We justify the last equivalence as follows: (⇒) Suppose u 0 z 0 ∧ x = yu 0 . Since z 1 ∈ C and u 0 z 0 , the right-left implication in the definition of C tells us that u 0 z 0 z 1 . We can thus let u = u 0 . Suppose now u 1 z 1 ∧ x = y(z 0 u 1 ). Since u 1 z 1 and z 1 ∈ C, the right-left implication in the definition of C tells us that z 0 u 1 z 0 z 1 . We can thus let u = z 0 u 1 . (⇐) Suppose u z 0 z 1 ∧ x = yu. Since z 1 ∈ C, by the left-right implication in the definition of C, we have that u z 0 or there exists u z 1 such that u = z 0 u . If u z 0 , then u 0 z 0 ∧ x = yu 0 by setting u 0 = u. If u z 1 ∧ u = z 0 u , then u 1 z 1 ∧ x = y(z 0 u 1 ) by setting u 1 = u .
We let By definition of I , we have 0, 1 ∈ I . We now show that I is closed under •. Suppose x 0 , x 1 ∈ I . Since x 1 ∈ I , we have one of the following cases: We first consider (1).
We now consider (2). We have We thus see that both cases imply x 0 x 1 ∈ I . Hence, I is closed under •. We let Since v 1 ∈ C, we have one of the following: In case of (a), we have x ∈ I since v 0 ∈ J . In case of (b), we have y ∈ I since v 1 ∈ J . We also have v 0 ∈ I since v 0 ∈ J ⊆ I . Since I is closed under •, we then have x = v 0 y ∈ I . We thus see that both cases imply v 0 v 1 ∈ J . Hence, J is closed under •.
We let By D 4 , we have Hence, 0 ∈ K . By D 4 , we have Hence, 1 ∈ K . We now show that K is closed under •. Suppose v 0 , v 1 ∈ K and We need to show that x w. From w v 0 v 1 and v 1 ∈ K ⊆ C, we have one of the following: In case of (I), since v 0 ∈ K , we have In case of (II), we observe that y w = v 0 u and u v 1 . Furthermore, since v 1 ∈ J , we have that u v 1 implies u ∈ I ⊆ C. Then, y w = v 0 u and u ∈ C implies that we have one of the following: (IIa) y v 0 (IIb) y = v 0 u for some u u.
In case of (IIa), In case of (IIb), we observe that Since v 1 ∈ K ⊆ J , we have that u v 1 implies u ∈ I ⊆ C. Then, x y = v 0 u and u ∈ C implies that we have one of the following: In case of (IIbi), since u v 1 and v 1 ∈ K ⊆ J implies u ∈ I ⊆ C, we have In case of (IIbii), we first observe that We thus see that Hence, v 0 v 1 ∈ K . Thus, K is closed under •. We now show that the class K has the following important property: Indeed, suppose v ∈ K and w v. We need to show that w ∈ K . By definition of K , we need to prove
We are now ready to give an interpretation of BTQ in D. We choose the domain K . We translate 0, 1 and • as It is clear that the translations of BTQ 1 , BTQ 2 and BTQ 3 are theorems of D. We now show that the translation of BTQ 4 is a theorem of D. Let x ∈ K . Since K ⊆ I , we have If the third disjunct is the case, then we have y ∈ K by (*). Thus, the translation of BTQ 4 is a theorem of D. Hence, BTQ is interpretable in D.

Theorem 7 D is interpretable in BTQ.
Proof We choose the domain J (x) ≡ x = x. We translate 0, 1, • and and as follows: It is clear that the translation of D 1 , D 2 and D 3 are theorems of BTQ. We now show that the translation of . It is not difficult to see that we cannot have ∃z [ 0 = xz ]. Hence, the translation of D 4 is a theorem of BTQ.
We now show that the translation of is a theorem of BTQ. We need to show that It suffices to show that The right-left implication of (*) is trivial. The left-right implication holds since Thus, the translation of D 5 is a theorem of BTQ.
By similar reasoning, the translation of D 6 is a theorem of BTQ. Thus, D is interpretable in BTQ.

Minimality results
This section is devoted to show that the axiomatizations of WD, WBT and BTQ are minimal essentially undecidable, which is to say that removing any one of the axioms (axiom schemas) gives a theory that is not essentially undecidable. We are not able to show that our axiomatization of D is minimal essentially undecidable, but we reduce the problem to showing that D− D 5 and D− D 6 are not essentially undecidable. However, as D has a finite axiomatization, we can make it minimal essentially undecidable by replacing some of the axioms with their conjunction. We now proceed to show that WBT−WBT 1 , WD−WD 1 and D−D 1 are not essentially undecidable by interpreting them in S2S. S2S is a monadic second order theory whose language is {e, 0, 1, S 0 , S 1 }, where e, 0 and 1 are constant symbols and S 0 and S 1 are unary function symbols. The axioms of S2S are the true sentences in the standard second-order structure where the universe is {0, 1} * . The symbol e is interpreted as the empty string, 0 is interpreted as 0 and 1 is interpreted as 1. The function symbol S 0 is interpreted as the function that takes a bit string and concatenates it with the bit 0, and the function symbol S 1 is interpreted as the function that takes a bit string and concatenates it with the bit 1. We have quantifiers that range over {0, 1} * , and we have quantifiers that range over subsets of {0, 1} * . It was proved in Rabin [13] that S2S is decidable. Our interpretation of WBT−WBT 1 in S2S does not use the monadic second order part, and this makes the induced algorithm more efficient. It is known that extending S2S with the prefix relation does not change the expressive power of S2S (see Börger et al. [1] p. 317). We also show this when we interpret WD−WD 1 and D −D 1 in S2S.
We recall that biterals are associated to the left. So, ((0 • 0) • 0) is a biteral while (0 • (0 • 0)) is not. Although we have so far not needed to take this into account, it now becomes important. Proof We interpret the two theories in S2S as follows: We choose the domain δ(x) ≡ x = x. We map 0 and 1 to So, 0 and 1 are realized as 0 and 1, respectively. We map • to This means that • is realized as the function Recalling that biterals are associated to the left, it is clear that the translation of each instance of WBT 2 and WD 2 is a theorem of S2S. We now show that the translation of each instance of is a theorem of S2S. We recall that Let α = ε. Suppose β is a substring of α w.r.t. φ • (x, y, z). We show that β = ε and that β is a substring of α in the actual sense. We have one of the following cases: We thus see that β is a substring of α in the actual sense. We map to This formula forces to be realized as the prefix relation on {0, 1} + . Indeed, suppose α, β ∈ {0, 1} + and α is related to β as defined by φ (x, y). The first line in φ (x, y) tells us that there exists Y ⊆ {0, 1} + such that The second line in φ (x, y) tells us that if γ ∈ Y , then all the non-empty prefixes of γ are also in Y . Let denote all the prefixes of α and β. The notation γ ≺ δ then means that γ is the longest proper prefix of δ. We show by induction that for each 0 ≤ j ≤ k, there exists 0 ≤ i ≤ m such that α j = β i , which implies that α is a prefix of β. The third line in φ (x, y) tells us that exactly one of 0 and 1 belongs to Y since we get for free that Hence, α 0 = β 0 . Suppose now that for 0 ≤ j < k there exists 0 ≤ i ≤ m such that α j = β i . We observe that we must have that i < m since α j = β i = β and α j+1 ∈ Y would contradict β0, β1 / ∈ Y . It remains to show that α j+1 = β i+1 . The third line in φ (x, y) tells us that exactly one of α j 0 and α j 1 belongs to Y . Hence, α j+1 = β i+1 . Thus, by induction, each prefix of α is a prefix of β.
We thus see that this translation has the desired properties. This translation shows us how to define decidable models of the theories in question.

Lemma 5 D− D 1 is not essentially undecidable.
Proof We modify the translation in Lemma 4 as follows: We map • to This means that • is realized as the function We need this modification to ensure that the axiom D 3 ≡ ∀x y [ x0 = y1 ] holds.
We observe that the simple translation of • we give in the preceding lemma does not work in the case of Lemma 4 since the axiom scheme WBT 3 then fails. Indeed, for any non-empty bit string α and any bit string β = 0, 1, we have φ • (α, β, α). This means that all substring different from 0 and 1 are substrings of α w.r.t. φ • (x, y, z).
We now define a relative interpretation of BTQ −BTQ 1 in Σ. We relativize quantification to λ(x) :≡ x = x and use the following translation of symbols We first observe that the map R → (−1, 1) defined by r → 2r is a bijection. The sections h(−, 0) and h(−, 1) are injective and have images (−∞, 0) and (0, ∞). This is because we have the following sequences of bijections: The fact that h(−, 0) and h(−, 1) are injective and have disjoint images implies that the translation of and are theorems of Σ. The translation of is also a theorem of Σ since we have defined ψ 0 (x) ≡ x = 0 and since the union of the images of h(−, 0) and h(−, 1) is The translation shows how to define a decidable model of BTQ−BTQ 1 .
One of the referees has observed that we get a simpler proof of the preceding lemma by considering the translation given in Lemma 4 and restricting the domain to the set of all non-empty strings. We can thus for example translate • as follows otherwise. 2 and WD−WD 2 are not essentially undecidable.

Lemma 7 WBT−WBT
Proof We obtain a one element model A of WBT −WBT 2 and WD−WD 2 as follows: Since • A is associative, WBT 1 and WD 1 hold in A. We observe that A satisfies since there is only one element in the universe. Thus, WBT 3 and WD 3 hold in A.

Lemma 8 BTQ−BTQ 2 is not essentially undecidable.
Proof We obtain a two element model A of BTQ−BTQ 2 as follows: The operator • A is clearly associative. The axiom ∀x y [ x0 = y1 ] holds since The following proof of the next Lemma was suggested by one of the referees . We show that D−D 2 is not essentially undecidable by interpreting it in Presburger Arithmetic. Presburger Arithmetic refers to all sentences in first-order logic with equality and the language {0, 1, +, <} true in the structure (N, 0, 1, +, <). It is shown that Presburger Arithmetic is decidable in Chapter 3.4 of Smoryński [15]. The idea is to consider the free monoid generated by 0 and 1 modulo the equations 00 = 0 and 11 = 1. The universe of this structure then consists of strings of the form (01) n , 1(01) n , (01) n 0, 1(01) n 0, with n being a natural number. Since we do not have the empty string, we have n > 0 when the string is of the form (01) n . The concatenation operator on this set is described as follows Observe that u and v denote the empty string in the cases x, y ∈ {0, 1}. It is clear that axiom D 3 ≡ ∀x y [ x0 = x1 ] holds in this model. It also follows easily from the definition of that (x y) z = x (y z) when y has length at least two. To see that is indeed associative, we observe that given distinct a, b ∈ {0, 1}, we have .
We interpret as the prefix relation, i.e., x y if and only if x = y or there exists holds by the following reasoning: -Suppose x y 0 and y 0 = x z for some z. If y has 0 as a suffix, then y = x z, which implies x y. Assume now y has 1 as a suffix. We have that z = 0 or z = u10 where u is possibly empty. We first assume z = 0. If x has 0 as a suffix, then y 0 = x. If y and x both have 1 as suffices, then y = x, which in turn implies x y. Suppose now z = u10. Then y = x u1, which in turn implies x y.
It is then not difficult to see that is definable in Presburger Arithmetic. This in turn implies that the prefix relation is definable. We thus have the following result. Proof Since Presburger Arithmetic is decidable, we obtain a decidable model of BTQ−BTQ 3 and D−D 3 as follows: -the universe is the set of natural numbers It is not difficult to see that axioms of BTQ−BTQ 3 hold. Axiom Proof We start by showing that BTQ−BTQ 4 has a decidable model. We consider the set M 2 (R ≥0 ) of 2 × 2 matrices with coefficients in R ≥0 and which are such that the first entry is strictly positive. So, the elements of M 2 (R ≥0 ) are of the form We consider the binary operation of matrix multiplication, denoted ×. It is not difficult to verify that M 2 (R ≥0 ) is closed under matrix multiplication. We can consider this structure as a first-order structure for the language {c 0 , c 1 , ×} where c 0 and c 1 are constant symbols for the matrices 1 0 1 1 and 1 1 0 1 respectively. Let Th(M 2 (R ≥0 )) denote the set of sentences true in this structure. We show that BTQ − BTQ 4 is interpretable in Th(M 2 (R ≥0 )) and that Th(M 2 (R ≥0 )) is decidable. Since relative interpretability preserves the property of being essentially undecidable, this means that BTQ−BTQ 4 is not essentially undecidable. We interpret BTQ−BTQ 4 in Th(M 2 (R ≥0 )) as follows: Since matrix multiplication is associative, the translation of is a theorem of Th(M 2 (R ≥0 )). To see that the translation of is a theorem of Th(M 2 (R ≥0 )), we observe that To see that translation of is a theorem of Th(M 2 (R ≥0 )), we observe that But we cannot have x = 0 by how the set M 2 (R ≥0 ) is defined. We then see that the translation of BTQ 3 is a theorem of Th(M 2 (R ≥0 )) To see that Th(M 2 (R ≥0 )) is decidable, we use Tarski's result that the set Th(R) of sentences which are true in the ordered field of real numbers (R, 0, 1, +, ×, ≤) is decidable (see Tarski [17]). To see that this is the case, we associate each sentence φ in the language of Th(M 2 (R ≥0 )) with a sentence φ * in the language of Th(R) such that φ ∈ Th(M 2 (R ≥0 )) ⇔ φ * ∈ Th(R).
We start by associating c 0 and c 1 with 1 0 1 1 and 1 1 0 1 respectively, and by associating each variable x with We recall that matrix multiplication is defined by x y z w × a b c d = xa + yc xb + yd za + wc zb + wd .
By following this definition, we associate each term s in the language of Th(M 2 (R ≥0 )) with a matrix p s,1 p s,2 p s,3 p s,4 where each p s,i is a term in the language of Th(R). So, for example, the term x × y is associated with the matrix x 1 y 1 + x 2 y 3 x 1 y 2 + x 2 y 4 x 3 y 1 + x 4 y 3 x 3 y 2 + x 4 y 4 .
The first big conjunct in clause (1) reflects the fact that matrices in M 2 (R ≥0 ) have entries in R ≥0 . The second big conjunct in (1) reflects the fact that matrices in M 2 (R ≥0 ) are such that the first entry is in R >0 . The last big conjunct in (1) states that the matrix associated with s equals the matrix associated with t. It is obvious that φ is a theorem of Th(M 2 (R ≥0 )) if and only if φ * is a theorem Th(R). Thus, Th(M 2 (R ≥0 )) is decidable.
It is not difficult to see that the model of BTQ − BTQ 4 we have defined is also a model of WD−WD 3 and WBT−WBT 3 . We extend it to a decidable model of D −D 4 by interpreting as M 2 (R ≥0 ) × M 2 (R ≥0 ).
Our proof of decidability of Th(M 2 (R ≥0 )) is actually a 4-dimensional interpretation of Th(M 2 (R ≥0 )) in Th(R). This means that each object in the language of Th(M 2 (R ≥0 )) is associated with a quadruple of objects in the language of Th(R). For more on this more general notion of interpretability, see Visser [20].
We can now state the main theorem of this section.

Theorem 8 WD, WBT and BTQ are minimal essentially undecidable.
The only thing that is lacking to show that D is minimal essentially undecidable is to show that D−D 5 and D−D 6 are not essentially undecidable. A negative solution of the following problem would thus show that D is minimal essentially undecidable.
Open Problem 2 Show that neither D −D 5 nor D −D 6 is essentially undecidable.
We have not put much focus on the theories BT and C. The proofs of Lemma 5 and Lemma 11 show that C − C 1 , C − C 4 and BT − BT 4 are not essentially undecidable. Beyond that, the minimality of BT and C is an open problem.

Open Problem 3 Are BT and C minimal essentially undecidable?
In this section, we investigated whether our axiom sets are minimal w.r.t set inclusion and the property of being essentially undecidable. A different notion of minimality that we have been implicitly investigating is minimality w.r.t. interpretability. We have seen that WD is interpretable in all the essentially undecidable theories we have studied. It is however not minimal in the interpretability pre-order. In [7], Jeřábek shows that there is an essentially undecidable theory that is interpretable in R but that does not interpret R. The theory Jeřábek gives is such that all partially recursive functions are representable. In Section 3 of [2], Yong Cheng uses results in [7] to give many examples of essentially undecidable theories that are interpretable in R but that do not interpret R. One of the referees observed that the existence of theories strictly below R w.r.t. interpretability also follows from the work of Shoenfield [14].
Although there are many essentially undecidable theories below R w.r.t interpretability, to the best of our knowledge, it is not known whether there exists a minimal computably enumerable essentially undecidable theory, w.r.t. interpretability. For a more detailed discussion of this problem, we refer the reader to Yong Cheng [2]. It T means S is interpretable in T but it is unknown whether S interprets T follows from the idea behind the proof of Theorem 4.7 of [2] that the interpretability degrees of computably enumerable essentially undecidable theories have infima. Hence, if there exists a minimal essentially undecidable theory, then that theory is also the minimum essentially undecidable theory.

Summary
We have formulated essentially undecidable theories WD ⊂ C ⊂ D and WBT ⊂ BT ⊂ BTQ (see Fig. 3). The theories WD, WBT and R are mutually interpretable while the theories D, BTQ and Q are mutually interpretable. The theories WD, WBT and BTQ have minimal essentially undecidable axiomatizations, but it not clear whether the same is true of D. Both WD and D are purely universally axiomatised.