Theories with quantifier elimination

We now give five examples of theories that admit quantifier elimination and are complete. These examples are used in the rest of this work. The first example is the theory of nontrivial divisible ordered abelian groups.

38 Chapter 2. Preliminaries Definition 2.100. Let Log= (0,+,⩽) be the language of ordered groups. The theory of non-trivial divisible ordered abelian groups, denotedTh_doag consists of theLog-sentences expressing the axioms of ordered abelian groups (associativity of addition, commutativity of addition, zero as the neutral element, existence of an inverse, total order axioms, compatibility of order and addition), the axiom of nontriviality∃y, y̸= 0, and an infinite series of axioms defining divisi-bility: for every n⩾2,Th_doag contains a sentence ∀y₁,∃y₂, y₁ =y₂+y₂+· · ·+y₂ (where the addition in takenntimes).

The theory of of nontrivial divisible ordered abelian groups admits quantifier elimination.

Theorem 2.101 ([Mar02, Corollary 3.1.17]). The theoryTh_doag of nontrivial divisible ordered abelian groups admits quantifier elimination and is complete.

As a corollary, we see that if Γ is a nontrivial divisible ordered abelian group, then its definable sets have a particularly simple structure. More precisely, we make the following definition.

Definition 2.102. We say that a subsetS_⊂Γⁿ is abasic semilinear set if it is of the form S={g∈Γⁿ:∀i= 1, . . . , p, fi(g)> h⁽ⁱ⁾,∀i=p+ 1, . . . , q, fi(g) =h⁽ⁱ⁾},

where f_i ∈ Z[X₁, . . . , X_n] are homogeneous linear polynomials with integer coefficients and h⁽ⁱ⁾∈Γ. We say thatS issemilinear if it is a finite union of basic semilinear sets.

Lemma 2.103. A setS_⊂Γⁿ is definable in Log if and only if it is semilinear.

Sketch of the proof. It is easy to see that every semilinear set is definable. To prove the converse, note that a union of two semilinear sets is semilinear. Furthermore, an intersection of two basic semilinear sets is basic semilinear. Hence, by the fact that intersection distributes over union, an intersection of two semilinear sets is semilinear. Moreover, the complement of a basic semilinear set is semilinear and hence the complement of any semilinear set is semilinear. Furthermore, one can prove by induction that if ϕ is an Log-term with a nonempty set of free variables y₁, . . . , y_n, then there exist m₁, . . . , m_n ∈ N such that ϕ(y) = m₁y₁ +· · ·+m_ny_n for any y ∈ Γⁿ. Hence, any set defined by an atomicLog-formula is of the form {y ∈ Γⁿ:f(y) ⩽h} or {y ∈ Γⁿ:f(y) = h}, where f ∈ Z[X1, . . . , Xn] is a homogeneous linear polynomial with integer coefficients and h ∈Γ. In particular, these sets are semilinear. Hence, sets defined by negations, conjunctions, and disjunctions of atomic formulas are semilinear. Moreover, one can prove by induction that every quantifier-free formula can be written in the form

∨p k=1

q_k

∧

ℓ=1

ϕ_k,ℓ,

where every ϕk,ℓ is an atomic formula or its negation. Therefore, every set defined by a quantifier-free formula is semilinear. The claim follows from Theorem 2.101.

Example 2.104. IfΓ =R, then every basic semilinear set is a relative interior of a polyhedron.

Note that the converse is not true. For example, the set {y ∈R²:y1+y2 > π} is semilinear, but{y ∈R²:√

2y₁+y₂ >0} is not. IfΓ =Q, then basic semilinear sets correspond precisely to relatively open rational polyhedra.

2.6. Model theory 39 In this work, we use divisible ordered abelian groups which arise as value groups of nonar-chimedean real closed fields. Since the valuation map may evaluate to −∞, we need to deal with divisible ordered abelian groups with bottom element.

Definition 2.105. We denote by Logb := (0,−∞,+,⩽) the language of ordered groups with bottom element. The theory of nontrivial divisible ordered abelian groups with bottom ele-ment, denoted Th_doagb, consists of the axioms of divisible ordered abelian groups (cf. Defini-tion 2.100),⁴ the nontriviality axiom ∃y,(y ̸= 0∧y ̸= −∞), and the axioms that extend the addition and order to−∞, namely ∀y, −∞+y=−∞ and ∀y, y⩾−∞.

Using Theorem 2.101, we can show that Th_doagb admits quantifier elimination and is com-plete.

Theorem 2.106. The theoryThdoagb admits quantifier elimination and is complete. Moreover, anyLogb-formulaϕ(y₁, . . . , y_m)withm⩾1is equivalent to a quantifier-free formula of the form

∨

Σ⊂[m]

((∀σ∈Σ, y_σ ̸=−∞⁾ ∧ (∀σ /∈Σ, y_σ =−∞) ∧ ψ_Σ⁾, (2.10)

where the disjunction goes over all subsets of[m], and everyψ_Σ is a quantifier-freeLog-formula such that Fvar(ψΣ)⊂ {yσ}σ∈Σ.

The proof of the above result is easy but technical. We have put it in Appendix B.1 for the sake of completeness. As in the previous case, quantifier elimination allows us to describe the class of definable sets. To do so, we need to observe that if Γ ∪ {−∞} is a nontrivial divisible ordered abelian group with bottom element, then it has a natural stratification.

Definition 2.107. If x ∈ (Γ ∪ {−∞})ⁿ, then we define its support as the set of its finite coordinates, {k ∈ [n] :x_k ̸= −∞}. Given a nonempty subset K ⊂ [n], and a set S _⊂ (Γ ∪ {−∞})ⁿ, we define thestratum ofSassociated withK, denotedS_{K ⊂}Γ^K, as the subset ofΓ^K formed by the projection of the points x∈Swith support K. More formally, a point y∈Γ^K belongs to S_K if the point x ∈(Γ ∪ {−∞})ⁿ defined asx_k =y_k for all k ∈K and x_k = −∞

otherwise belongs toS.

Lemma 2.108. A subset S_⊂(Γ ∪ {−∞})ⁿ is definable in Logb if and only if all strata of S are semilinear.

Again, the technical proof of this claim can be found in Appendix B.1. The next theory that is used in this work is the theory of real closed fields.

Definition 2.109. The theory of real closed fields, denotedTh_rcf, is a theory in the language of ordered rings Lor := (0,1,+,·,⩽).⁵ It consists of the usual axioms of ordered fields, the axiom

∀x1,(x1⩾0→ ∃x2, x1 =x2·x2)that governs the existence of square roots, and an infinite set of axioms that states the fact that every polynomial of an odd degree has a root. In other words, for everyn⩾1,Thrcf contains the axiom∀x0, . . . ,∀x2n,∃y, y²ⁿ⁺¹+x2ny²ⁿ+· · ·+x1y+x0 = 0.

4With one change required by the presence of −∞: the existence of an inverse should be formulated as

∀y,(y̸=−∞ → ∃z,(z ̸=−∞ ∧y+z= 0)). The assumption that z ̸=−∞is not needed (by the nontriviality axiom) but it simplifies the presentation.

5We assume that+and·have type(1,1,1)and that⩽has type(1,1). It it clear that any ordered filed is an Lor-structure with the natural interpretation of the symbols.

40 Chapter 2. Preliminaries A classical result proved by Tarski states that the theory of real closed fields admits quantifier elimination and is complete.

Theorem 2.110 ([Mar02, Theorem 3.3.15 and Corollary 3.3.16]). The theory Th_rcf of real closed fields admits quantifier elimination and is complete.

As previously, ifK is a real closed field, then quantifier elimination allows us to characterize the class of definable sets.

Lemma 2.111. A setS ⊂Kⁿ is definable in Lor if and only if it is semialgebraic.

Sketch of the proof. By induction, if ϕ(x₁, . . . x_n) is an Lor-term with a nonempty set of free variables, then there exists a polynomial P ∈N[X₁, . . . , X_n]with natural coefficients such that ϕ(x) = P(x) for all x ∈ Kⁿ. Therefore, a set definable by an atomic formula is of the form {x∈Kⁿ:P(x) = 0} or{x ∈Kⁿ:P(x) ⩾0}, whereP ∈K[X1, . . . , Xn]is a polynomial with coefficients inK. The rest of the proof is the same as the proof of Lemma 2.103 (with the word

“semilinear” replaced by “semialgebraic”).

The final two theories used in this work concern valued fields. First, we present a quan-tifier elimination result in henselian valued fields with angular component. This result was obtained by Pas [Pas89, Pas90a] and based on the work of Denef [Den84, Den86]. Suppose that (K, Γ,k,val,ac) is a valued field with a fixed residue field and an angular component.

Furthermore, we suppose that both K and k have characteristic 0. We start by defining a three-sorted language of such fields.

Definition 2.112. We define the Denef–Pas language as LPas := (LK,LΓ,Lk,val,ac), where LK is the language of rings associated withK,LK := (0_K,1_K,+_K,·K),Lk is the language of rings associated withk,Lk := (0_k,1k,+k,·k),LΓ := (0_Γ,−∞Γ,+Γ,⩽Γ)denotes the language of ordered groups with bottom element associated withΓ, andval,acare two function symbols.⁶ In this way, any valued field with angular component is an LPas-structure. We now define the associated theory.

Definition 2.113. The theory of henselian valued fields with angular component in equicharac-teristic0, denotedTh_Pas, is a theory in the Denef–Pas languageLPas. It consists of the axioms of fields (forK andk), axioms of ordered abelian groups with bottom element (forΓ∪{−∞}), the axioms specifying that val is a valuation (see Definition 2.60), the axioms specifying that ac is an angular component (see Definition 2.66), and infinite sequences of axioms describing the facts that K and k have characteristic 0, and an infinite sequence of axioms describing henselianity (see Definition 2.72).

Remark 2.114. We point out that the axioms mentioned in the definition above can be written as sentences in Th_Pas. We leave this as an exercise. The most problematic are the axioms of henselianity. However, as noted in Remark 2.67, the definition of the residue map is already implied in the definition of angular component, because the map res(x) defined as ac(x) if val(x) = 0 and res(x) = 0 otherwise, is a residue map. Therefore, one can write the infinite sequence of axioms describing henselianity using only the symbols forac and val.

6The types of symbols are given in a natural way: we putI ={1,2,3},C1 :={0K,1K},C2 :={0Γ,−∞Γ}, C3:={0k,1k}. Then,valhas type(1,2),achas type(1,3),+khas type(3,3,3)and so on.

2.6. Model theory 41 Pas [Pas89] has shown that ThPas admits elimination of K-quantifiers, i.e., the quantifiers that act on the variables associated with K. To state this theorem, we use the following notation. Ifθis aLPas-formula, then we denote it byθ(X, Y, Z), whereX= (x₁, . . . , x_n1)(resp.

Y = (y1, . . . , yn2), Z = (z1, . . . , zn3)) is the sequence of free variables of ϕ associated with the sort K (resp. Γ,k). We have the following theorem.

Theorem 2.115 ([Pas89]). Any LPas-formulaθ(X, Y, Z) is equivalent inTh_Pas to a formula of the form

∨m i=1

(

ϕ_i⁽val(f_i1(X)), . . . ,val(f_iki(X)), Y⁾∧ψ_i⁽ac(f_i(ki+1)(X)), . . . ,ac(f_ili(X)), Z⁾⁾, (2.11) where, for every i= 1, . . . , m, fi1, . . . , fili ∈Z[X] are polynomials with integer coefficients, ϕi

is an LΓ-formula, and ψ_i is an Lk-formula.

Remark 2.116. We point out that the original statement of this theorem given by Pas [Pas89, Theorem 4.1] is slightly weaker. We took our formulation from [CLR06, Theorem 4.2]. Never-theless, the proof of Pas actually proves the theorem stated above.

We use Theorem 2.115 to prove a quantifier elimination result in real closed valued fields.

Let (K, Γ,k,val,ac) be valued field such that K is real closed, val is nontrivial and convex, and residue field and angular component are fixed. We point out that, by Theorem 2.75, k is real closed. As previously, we start by defining the associated language and theory.

Definition 2.117. We define the language of real closed valued fields as Lrcvf := (LK,LΓ,Lk,val,ac),

where LK is the language of ordered rings associated with K, LK := (0_K,1_K,+_K,·K,⩽K), Lk is the language of ordered rings associated with k, Lk := (0_k,1_k,+_k,·k,⩽k), LΓ :=

(0_Γ,−∞Γ,+_Γ,⩽Γ) denotes the language of ordered groups with bottom element associated withΓ, andval,acare two function symbols.⁷

Definition 2.118. The theory of real closed valued fields, denoted Thrcvf, is a theory in the language Lrcvf. It consists of the axioms of real closed fields for K, the axioms of ordered abelian groups with bottom element for Γ ∪ {−∞}, the axioms of ordered fields for k, the axioms specifying thatval is a nontrivial and convex valuation, and the axioms specifying that ac is an angular component.

Remark 2.119. We point out that the axioms of Th_rcvf imply thatkis real closed. Indeed, the axioms of acimply that k is a residue field and hence it is real closed by Theorem 2.75.

Using the quantifier elimination results presented in this section, we can prove that the theory of real closed valued fields admits quantifier elimination and is complete.

Theorem 2.120. The theory Thrcvf admits quantifier elimination and is complete. Moreover, anyLrcvf-formulaθ(X, Y, Z) is equivalent in Th_rcvf to a formula of the form

∨m i=1

( ϕi

(val(fi1(X)), . . . ,val(f_iki(X)), Y⁾∧ψi

(ac(f_i,ki+1(X)), . . . ,ac(f_ili(X)), Z⁾⁾,

where, for every i= 1, . . . , m, f_i1, . . . , f_ili ∈Z[X] are polynomials with integer coefficients, ϕ_i is a quantifier-free LΓ-formula, and ψi is a quantifier-free Lk-formula.

7The types of symbols are given in a natural way as in Definition 2.112.

42 Chapter 2. Preliminaries Sketch of the proof. Let θ(X, Y, Z) denote anyLrcvf-formula. We recall (Lemma 2.12) that the order⩽in any real closed field can be defined asx₁ ⩽x₂ ⇐⇒ ∃x₃, x₂ =x²₃+x₁. This enables us to eliminate the symbols ⩽. More precisely, by a simple induction, θ(X, Y, Z) is equivalent inThrcvf to someLPas-formulaθ(X, Y, Zˆ ). Moreover, by Theorem 2.75,(K, Γ,val)is henselian.

Furthermore, since K and k are ordered, they have characteristic zero. This enables us to apply Theorem 2.115. As a result, θ(X, Y, Z)ˆ is equivalent in Th_rcvf to a formula of the form (2.11). Then, we apply Theorem 2.106 and Theorem 2.110 to eliminate the quantifiers in the formulasϕi andψi. This shows the last part of the statement. In the case whereθis a sentence, the formulas ϕ_i and ψ_i are also sentences. The completeness results in Theorem 2.106 and Theorem 2.110 applied to each subformula ϕi and ψi allow to prove that either θ or ¬θ is a logical consequence of Th_rcvf.

Remark 2.121. We note that the proof of Theorem 2.120 is constructive, in the sense that there exists an algorithm that, given a Lrcvf-formula, outputs an equivalent quantifier-free formula of the form stated in Theorem 2.120. Indeed, the proof first uses the Denef–Pas quantifier elimination to eliminate quantifiers acting over K and the algorithm to do so is given in the proof of Pas [Pas89]. Second, we eliminate the quantifiers in the residue field k by using quantifier elimination in real closed fields (see [BPR06, Chapter 14] for a detailed discussion about the algorithms for this task). Third, we eliminate the quantifiers in the value group Γ∪ {−∞}. An algorithm for quantifier elimination inΓ is given in [FR75] and it can be easily extended to Γ ∪ {−∞}(using the proof of Theorem 2.106).