Model Theory and Exponentiation David Marker odel theory is a branch of math- interested in showing that there is an algorithm ematical logic in which one studies for deciding first-order sentences about the real mathematical structures by con- numbers and hence showing the decidability of sidering the first-order sentences elementary Euclidean geometry. true of those structures and the Tarski’s decidability of the real numbers Msets definable in those structures by first-order stands in stark contrast to Gödel’s undecid- formulas. ability of the natural numbers. The great free- The fields of real and complex numbers have dom we have to solve polynomial equations in long served as motivating examples for model the real numbers insures that the real solution theorists. Many model theoretic concepts arose sets of systems of polynomial equations are by abstracting classical algebraic phenomena to topologically well behaved. In the natural num- a more general setting (see, for example, [29]). bers such solution sets can behave very badly. In the past five years significant progress has This wild behavior allows considerable latitude been made in the other direction. Model theoretic for coding complicated combinatorial phenom- methods have been used to develop new in- ena, while, as we will see below, in the reals we sights into real analytic geometry. In particular, cannot even define the integers. this has led to a generalization of semialgebraic The natural language for studying the real and subanalytic geometry to a setting in which numbers is the language of ordered rings one studies global exponentiation on the reals. = +, , ,<,0,1 . In this language the basic - L { − · } L Tarski’s Theorem formulas are of the form p(v1,...,vn)= q(v ,...,v ) and p(v ,...,v )<q(v ,...,v ) The logical study of the field of real numbers 1 n 1 n 1 n where p, q Z[X1,...,Xn]for some n N. The began with the work of Tarski. These results ∈ ∈ set of all -formulas is obtained by taking the were announced by Tarski in [33], but publica- L basic -formulas and closing under the Boolean tion was interrupted by the war, and the proofs L connectives , , and quantifiers and . For did not appear until [34]. Tarski was primarily ∧ ∨ ¬ ∃ ∀ example, (1) yx=y2 y David Marker is professor of mathematics at the Uni- ∃ − versity of Illinois at Chicago. His e-mail address is (2) x y (x = y2 x + y2 =0) [email protected]. ∀ ∃ ∨ This paper is based on an invited lecture presented in (3) w (x w 2 + xy =0 w<3y 4) October 1994 at the AMS Central regional meeting in ∃ − ∧ − Stillwater, Oklahoma. are legitimate -formulas. We say that an oc- L Partially supported by NSF grants DMS-9306159 and currence of a variable in a formula is free if it is INT-9224546, and an AMS Centennial Fellowship. not inside the scope of a quantifier; otherwise, JULY 1996 NOTICES OF THE AMS 753 we say it is bound. For example, in (1) x is a free Step 1: (Elimination of Quantifiers) If variable and y is bound, while in (2) both x and φ(v1,...,vn) is an -formula, then there is an L y are bound. If a formula has free variables, we equivalent quantifier-free -formula L often denote the formula in a way that explic- (v1, . , vn). (Here by equivalent we mean that itly shows which variables are free. For example, φ holds of a = (a1, . , an) if and only if holds n we might denote (3) by φ(x, y). We think of a for- of a for all a R .) Moreover, there is an algo- ∈ mula φ(x1, . , xn) as expressing a relation that rithm that transforms φ to . holds among the variables x1, . , xn. For exam- Note that quantifier-free formulas are just fi- ple, nite Boolean combinations of p(x1, . , xn) = q(x , . , x ) and p(x ,...,x ) > q(x ,...,x ) (4) w xw 2 + yw + z = 0 1 n 1 n 1 n ∃ where p, q Z[X1, . , Xn] for some n N. ∈ ∈ expresses the relationship “x, y, z are coeffi- We are familiar with some natural examples cients of a solvable quadratic polynomial”, while of quantifier elimination. For example, (4) is equivalent to the quantifier-free formula 2 2 (5) x y ux + vxy + zy 0 2 ∀ ∀ ≥ y 4xz 0. − ≥ expresses that u, v, z are the coefficients of a Also, while the formula in free variables positive semidefinite binary quadratic form. vi,j : 1 i, j n that expresses that the ma- A formula is a sentence if it has no free vari- { ≤ ≤ } trix (vi,j ) has an inverse involves existential ables. If is an ordered ring, then any -sen- A L quantifiers, it is equivalent to the quantifier- tence φ will either be true or false for . For A free formula that asserts that the determinant example, (2) is true for the field of real numbers of (vi,j ) is nonzero. R but false for the field of rational numbers Q. Tarski’s original proof relied on techniques Formulas that are not sentences may hold for from elimination theory and Sturm’s algorithm. some assignments of the free variables but not Other proofs are based on model-theoretic ideas for others. For example, in Q (4) holds for (1,5,6) (see, for example, [24]) or cylindric decomposi- but fails for (1,0,-2). Of course, (4) holds for (1,0,- tion (see, for example, [2]). 2) in R. The first-order theory of R, T h(R), is the The second step is just an easy observation. set of all -sentences that are true about R. L Many important properties of R can be expressed Step 2: Quantifier-free sentences are finite Boolean as -sentences. For example, (2) expresses that combinations of ones of the form m = n and L for any x either x or x is a square. For each n m > n where m and n are integers. These can − be decided easily. the sentence Thus, to decide if a sentence φ is true in R, a, b w0, . , wn [(a < b ∀ ∀ ∧ n n we first transform it into a quantifier-free sen- i i tence and then test to see if the quantifier-free ( wia )( wib ) < 0) → sentence is true. iX=0 iX=0 n While Tarski’s result proves the decidability i x(a < x < b wix = 0)] of the theory of the real numbers, the algorithm ∃ ∧ iX=0 obtained is far from feasible. The best-known al- gorithm for eliminating quantifiers is doubly expresses that the sign change property holds exponential in the number of quantifier blocks for polynomials of degree n. One sees that all [17], and it is known that even deciding first- of the axioms for real closed ordered fields (see order sentences for the ordered group (R, +, <) [21]) are contained in T h(R). On the other hand, is exponentially hard [14]. there is no first-order sentence that expresses It is also worth noting that for any formula the completeness of the ordering of R. Indeed, φ(v1, . , vn), the formula (v1,...,vn) ob- we cannot hope to characterize the real numbers tained by quantifier elimination is actually equiv- up to isomorphism by T h(R), as the Löwenheim- alent to φ(v1, . , vn) in any other real closed Skolem theorem (see [3]) insures there will be field as well. Since the decision made in Step 2 models of T h(R) of every infinite cardinality. depends only on Z, it follows that a sentence φ Tarski’s Theorem. The first-order theory of the is true in R if and only if it is true in every real closed field if and only if it is true in some real field of real numbers is decidable. That is, there closed field. This means that the axioms for real is an algorithm that will decide on input φ closed fields completely axiomatize T h(R). whether or not φ is a sentence true of R. In [35] Tarski asked if his results could be ex- The proof of Tarski’s theorem breaks into tended to include the exponential function. x two steps. Namely, let exp = e . Is the first-order L L ∪ { } 754 NOTICES OF THE AMS VOLUME 43, NUMBER 7 exp-theory of R decidable?1 We will give a par- to define the integers. Since we can define Z, L tial answer to Tarski’s problem in Section 4. Gödel’s results show that this theory is far from One might hope to generalize the arguments decidable. used for the ordered field, but serious difficul- ties arise. First, quantifier elimination fails. In [5], Definable Sets and o-Minimality We say that X Rn is definable using if there van den Dries showed that the formula ⊆ L is an -formula φ(x1, . , xn, y1, . , ym) and y > 0 w (wy = x z = yew ) L b1, . , bm R such that ∧ ∃ ∧ ∈ is not equivalent to a quantifier-free exp-for- n L X = (a1, . , an) R : mula. { ∈ φ(a1, . , an, b1, . , bm) holds . There are rather silly ways to extend the lan- } guage so that we do have elimination of quan- For example, let φ(x1, x2, y) be the formula tifiers, but the trick is to extend the language so 2 2 2 2 u (x1 u) + (x2 u ) < y . that we have quantifier elimination and the quan- ∃ − − tifier-free formulas are still natural and mean- Substituting the real number π for the variable ingful. y, we get the definable set 2 Even if we could eliminate quantifiers, there (x1, x2) : the distance from (x1, x2) to the is no obvious uniform algorithm for deciding { π quantifier-free sentences like parabola y = x2 is less than .