The Equational Compatibility Problem and One-Variable Variants Hilbert's

The real line: the equational compatibility problem and one-variable variants Hilbert’s 10th Problem

George F. McNulty

Abstract. Walter Taylor proved recently that there is no algorithm for determining of a finite set of equations whether it is topologically compatible with the real numbers R in the sense that it has a model with universe R and with basic operations which are all continuous with respect to the usual topology of the real line. Taylor’s account used operations symbols suitable for the theory of rings together with three unary operation symbols intended to name trigonometric functions supplemented finally by a countably infinite list of constant symbols. In this note we show how to reduce this list of constant symbols to a single constant symbol.

Introduction

Let T be a topological space and let Γ be a set of equations. We will say that Γ is compatible with T if there is an algebra T with universe T whose basic operations are continuous such that Γ is true in T. By a signature we mean a system of operation symbols each equipped with a natural number to specify its rank (i.e. its number of arguments). Let σ be a countable computable signature. The Equational Compatibility Problem for T and σ Find an algorithm for determining of any finite set Γ of equations of signature σ whether Γ is compatible with T. We say that this compatibility problem is algorithmically unsolvable if no such algorithm exits. Walter Taylor proved in [13] that the equational compatibility problem is algorithmically unsolvable for the real line R and the signature of ring theory enhanced by three additional operation symbols of rank 1 and a countably infinite number of constant symbols. He notes that this countably infinite list of constant symbols can be replaced by finitely many. In fact, Taylor devised a method for establishing the algorithmic unsolvability of equational compatibility problems. His method has two parts: a topological finite determination theorem and the algorithmic unsolvability of an appropriate variant of Hilbert’s 10th Problem. Suppose that T is a topological space, that σ and σ0 are signatures with σ0 an expansion of σ, and that T is an algebra of signature σ with universe T and for which all the basic operations are continuous. We will say that the topological algebra T is finitely determined with respect to σ0 provided there is a finite set Σ of equations of signature σ0 which is compatible with an expansion of T and up to isomorphism (simultaneously algebraic and topological) T

1 2 GEORGE F. MCNULTY is the only topological algebra of signature σ which can be expanded to a topological model of Σ. We also say that T is finitely determined by Σ. For the second element of Taylor’s method we formulate: Hilbert’s 10th Problem in n variables for the algebra T Find an algorithm for determining of any finite set Γ of equations in no more than n variables whether Γ has a solution in T. Taylor’s method is predicated on establishing for a given topological algebra T of signature σ that it is finitely determined with respect to some σ0 by some set Σ of equations and that Hilbert’s 10th Problem in n variables for T is not solvable algorithmically. Then the algorithmic unsolvability of the Equational Compatibility Problem for T and σ00 can be deduced from these two results as follows. Expand the 0 00 signature σ by n new constant symbols c0, c1,..., cn−1 to obtain the signature σ and for each finite set Γ of equations of signature σ in the variables x0, x1, . . . , xn−1 0 associate the set Γ of equations by substituting the constant symbol ci for the variable xi, for each i < n. Then 0 Γ has a solution in T if and only if Σ ∪ Γ is compatible with T. This reduces Hilbert’s 10th Problem in n variables for the algebra T to the equational compatibility problem for T and σ00. Walter Taylor carried out this method for the real line. Taylor’s proof is based on the topological algebra ∗ ∗ R = hR, +, ·, −, 0, 1, sin i ∗ π where sin (χ) := sin( 2 χ) for all real numbers χ, and where R is given the usual topology on the real line. We will reserve σ to denote the signature of this algebra. ∗ π We also need two more unary operations on R, namely cos (χ) := cos( 2 χ) and λ(χ) := pcos∗(sin∗(χ)) for all real numbers χ. We use σ0 to denote the expansion of σ by two addition unary operation symbols. To distinguish the operation symbols of a signature from the basic operations of an algebra, we use boldface for the symbols. So our operation symbols will be +, ·, −, 0, 1, sin∗, cos∗, λ. Many familiar equations hold in R∗ and its expansion by cos∗ and λ. Among them, Taylor distinguished the following set Σ: x + (y + z) ≈ (x + y) + z x · (y · z) ≈ (x · y) · z x + y ≈ y + x x · y ≈ y · x x + 0 ≈ x x · 1 ≈ x x + (−x) ≈ 0 x · (y + z) ≈ x · y + x · z cos∗(x + y) ≈ cos∗(x) · cos∗(y) − sin∗(x) · sin∗(y) sin∗(1) ≈ 1 sin∗(x + y) ≈ sin∗(x) · cos∗(y) + cos∗(x) · sin∗ y cos∗(0) ≈ 1 cos∗(sin∗(x)) ≈ (λ(x))2 cos∗(1) ≈ 0 In [13], Taylor proved TAYLOR’S ALGORITHMIC UNSOLVABILITY OF TOPOLOGICAL R-COMPATIBILITY 3

Taylor’s Finite Determination Theorem for R∗. The topological algebra R∗ is finitely determined with respect to σ0 by the set Σ of equations. and Hilbert’s 10th Problem for R∗. There is a natural number n so that Hilbert’s 10th Problem in n variables for R∗ is algorithmically unsovlable. As Hilbert originally framed the problem (for the ring of integers), there was no finite bound on the number of variables under consideration. The resolution of Hilbert’s 10th Problem actually showed that the recursively enumerable (i.e. listable) sets and the Diophantine sets coincide. From this it follows that there must be a finite bound for Hilbert’s 10th Problem over the integers. This bound will be inherited automatically in Taylor’s result for R∗. The best known bound is n = 9 due to Matiyasevich—a detailed exposition of this result can be found in [6]. Actually, Taylor proved a stronger form of his Finite Determination Theorem, namely that cos∗ is also determined by Σ. The set Σ does not determine, in the topological sense, the interpretation of λ(x) as pcos∗(sin∗(χ)) for all real numbers χ, since choosing the negative square root, for appropriate reals, works equally well. However, adding the two equations λ(x + 2) ≈ λ(x) λ(0) ≈ 1 to Σ forces the selection of the nonnegative square root in all cases. In section 8.1 of [13], Taylor presents alternatives for R∗ and for the set Σ. The alternative basic operations still include the ring operations but the trigonometric functions are replaced by functions which are piecewise rational. It is not clear that the methods we consider below can be extended to this case. In essence, Hilbert’s 10th Problem was formulated for the ring of integers. It was shown to be algorithmically unsolvable by way of a proof laid out in the work of Martin Davis, Hilary Putnam, and Julia Robinson and completed by Yuri Matiya- sevich. Notable references here are the 1950 abstract of Davis [2] (see also [3]), the 1958 paper of Martin Davis and Hilary Putnam [4], the 1960 abstract of Julia Robinson [12], the groundbreaking 1961 work of Davis, Putnam, and Robinson [5], and finally Yuri Matiyasevich’s 1970 achievement [7]. A fully detailed and high readable exposition of this result and related matters was given in 1993 by Yuri Matiyasevich [10]. On the other hand, Hilbert’s 10th Problem has long been known to have a positive solution, as a result of Tarski’s decision method [14], when formulated for the ring of real numbers or the ring of complex numbers. The situation arising from the expansion of the ring of reals (or of complex numbers) by various exponential and trigonometric functions has attracted considerable attention. In particular, Matiyasevich [10] has been able to show that Hilbert’s 10th Problem in 1 variable for the algebra hR, +, ·, −, 0, 1, sini is algorithmically unsolvable. This result of Matiyasevich depends substantially on earlier work of Daniel Richardson [11] in 1968, B. F. Caviness [1] in 1970, and Paul Wang [15] from 1974. Indeed, Wang’s result differs from that of Matiyasevich essentially only because Wang needed to include π as a constant of the algebra. 4 GEORGE F. MCNULTY

It is inviting to attempt the modiﬁcation of the path to Taylor’s theorem along the lines suggested by the work of Richardson, Caviness, Wang, and Matiyasevich with the reward that we shall only need one additional constant symbol. We take up the invitation. There are two essentially diﬀerent approaches. We could attempt to adapt Taylor’s Finite Determination Theorem to apply to sin rather than sin∗ or we could adapt the arguments of Wang, Caviness, and Richardson to apply to sin∗ rather than to sin. We take the second approach. We establish ∗ Theorem 0. Hilbert’s 10th Problem in 1 variable for hR, +, ·, −, 0, 1, sin i is algorithmically unsolvable. As shown in Section 4, we can apply Taylor’s method to obtain

Theorem 1. The equational compatibility problem for the real line R with signature σ0, expanded by one additional constant symbol, is algorithmically unsolvable. The remainder of the paper is primarily devoted to proving Theorem 0. I would like to thank Walter Taylor for his careful reading of this paper and for his helpful advice.

1. The methods of Richardson, Wang, and Matiyasevich

Richardson’s 1968 paper contains a number of interesting results, among which we draw attention to the two which are directly relevant here. Richardson’s First Theorem. There is a natural number n so that Hilbert’s 10th † Problem in n variables for R = hR, +, ·, −, 0, 1, exp, sin, ln 2, π, qiq∈Q is algorithmically unsolvable. Let ρ be the signature appropriate to the algebra R† over the reals that occurs in the statement of Richardson’s First Theorem. Richardson’s Second Theorem. There is no algorithm which, given a term t(x) of signature ρ in no more than one variable, will determine if there is a real number † r such that tR (r) is negative. Daniel Richardson’s work was accomplished before Matiyasevich completed the resolution of Hilbert’s 10th Problem. Instead, Richardson used the result of Davis, Putnam, and Robinson [5] that the version of Hilbert’s 10th Problem over the integers for exponential Diophantine equations is algorithmically unsolvable. For this reason, Richardson considered an algebra over the reals which included among its basic operations the natural exponential function and, as a distinguished constant, the natural logarithm of 2. The latter was needed since the Davis-Putnam-Robinson exponential Diophantine equations involved 2x and Richardson’s method invoked partial derivatives. The other basic operations Richardson demanded over the reals were the ring operations of addition, multiplication, negation, 0, and 1, the sine function, π as a distinguished constant, and a distinguished constant for each rational. A careful reading of Richardson’s work [11] shows that all those rationals (apart from 1) need not be distinguished as constants. B. F. Caviness [1], also TAYLOR’S ALGORITHMIC UNSOLVABILITY OF TOPOLOGICAL R-COMPATIBILITY 5 not yet aware of Matiyasevich’s work, points out that in the event that Hilbert’s 10th Problem is algorithmically unsolvable (as it indeed is), then the exponential function and ln 2 can be dropped as basic operations from Richardson’s results. Caviness also simpliﬁed some parts of Richardson’s argument. By 1974, the time of Paul Wang’s paper [15], the resolution of Hilbert’s 10th Problem by Matiyasevich was known. Let ? R = hR, +, ·, −, 0, 1, sin, π, qiq∈Q. Wang was able to show Wang’s Theorem. Hilbert’s 10th Problem in 1 variable for R? is algorithmically unsolvable. In section 9.2 of his book Matiyasevich [10] presents an exposition of a version of these results. In the process he drops the rationals as distinguished constants and, more surprisingly, is also able to drop π as well, leaving only the ring operations and the sine function. Moreover, Matiyasevich gives a readable account with full details. However, it is Wang’s Theorem to which we will appeal most directly. Except at one important point, the way in which π and the sine function enter into Richardson’s original argument, and into its subsequent adaptations by Caviness and Wang, is in expressions like " # 2 2 X 2 F (x0, . . . , xn−1) = (n + 1) p (x0, . . . , xn−1) + (sin πxi)ki(x0, . . . , xn−1) −1 i 0, there is a real number η so that |ei(η) − χi| < ε for all i < m. 6 GEORGE F. MCNULTY

Richardson’s Approximate Decoding Lemma. Let h(x) = x sin x and g(x) = x sin x3. Then hh(gi(x)) | i is a natural numberi = hh(x), h(g(x), h(g(g(x)))),... i is a system of approximate decoding functions. A very short argument, given by Wang in [15], is needed to deduce Wang’s The- orem from the Richardson-Caviness Inequality Lemma and Richardson’s Approxi- mate Decoding Lemma. What is needed to prove our Theorem 0 is a modiﬁcation of Richardson’s Approximate Decoding Lemma.

2. How to make a system of approximate decoding functions

Lemma. Let h(x) = x sin∗ x and g(x) = x sin∗ x3. Then

hei(x) | i is a natural numberi := hh(x), h(g(x), h(g(g(x)))),... i is a system of approximate decoding functions.

For each i < n we let ei(x) be the obvious term that denotes the corresponding ∗ decoding function ei in the algebra R . Proof. The claim below is key to proving the Lemma. Claim. Given any real numbers χ and ψ and any ε > 0, there is a real number η such that |h(η) − χ| < ε g(η) = ψ

Proof of the Claim. It is harmless to suppose that ε < 1. Now ﬁx an integer k so large that 6(4k − 1)2 4 > and (2k + 1)π + 1 ε 4k − 1 > |χ| and 4k − 2 > |ψ| Consider the closed interval [4k − 1, 4k + 1]. Notice π π π sin∗(4k − 1) = sin (4k − 1) = sin(2πk − ) = sin(− ) = −1 2 2 2 π π π sin∗(4k + 1) = sin (4k + 1) = sin(2πk + ) = sin( ) = 1 2 2 2

So sin∗(x) maps [4k − 1, 4k + 1] onto [−1, 1]. Hence, the image of the interval [4k − 1, 4k + 1] under the map h(x) must include the interval [−4k + 1, 4k + 1]. This means that there is η0 in [4k − 1, 4k + 1] so that h(η0) = χ, since 4k − 1 > |χ| entails that −4k + 1 < χ < 4k + 1. TAYLOR’S ALGORITHMIC UNSOLVABILITY OF TOPOLOGICAL R-COMPATIBILITY 7

By continuity, h will map any sufficiently small interval about η0 into the interval about χ of radius ε. Our argument depends on finding out how small sufficiently small must be. After a bit of reverse engineering, we take ε δ = . (2k + 1)π + 1

Let ν be any element of [η0 − δ, η0 + δ]. According to the Mean Value Theorem, we pickη ˆ in [η0 − δ, η0 + δ] so that 0 |h(ν) − h(η0)| ≤ |h (ˆη)|δ Now just observe

|h(ν) − χ| = |h(ν) − h(η0)| ≤ |h0(ˆη)|δ π π π = sin( ηˆ) + ηˆcos( ηˆ) δ 2 2 2 π ≤ 1 + (η + δ) δ 2 0 π ε ε ≤ 1 + 4k + 1 + 2 (2k + 1)π + 1 (2k + 1)π + 1 π π ε ≤ 1 + 2kπ + + 2 (4k + 2)π + 2 (2k + 1)π + 1 π π ε < 1 + 2kπ + + 2 2 (2k + 1)π + 1 ε = (1 + 2kπ + π) (2k + 1)π + 1 ε = ((2k + 1)π + 1) (2k + 1)π + 1 = ε This means |h(ν) − χ| < ε for any ν in [η0 − δ, η0 + δ]. So it remains to ﬁnd η in [η0 − δ, η0 + δ] so that g(η) = ψ. Now the cubing 3 3 function maps [η0 − δ, η0 + δ] onto [(η0 − δ) , (η0 + δ) ]. Also, observe 3 3 2 3 (η0 + δ) − (η0 − δ) = 6η0δ + 2δ ≥ 6(4k − 1)2δ ε = 6(4k − 1)2 (2k + 1)π + 1 6(4k − 1)2 = ε (2k + 1)π + 1 4 > ε = 4 ε 8 GEORGE F. MCNULTY

3 Therefore, as ν ranges over [η0 − δ, η0 + δ] we ﬁnd that ν ranges over an interval of π 3 length at least 4. In turn, this means that 2 ν ranges over an interval of length at ∗ 3 least 2π. Consequently, as ν ranges over [η0 − δ, η0 + δ] we conclude that sin (ν ) = π 3 sin( 2 ν ) takes on all values between −1 and 1. ∗ 3 Then g(ν) = ν sin (ν ) has to take on all values between δ−η0 and η0 −δ. Recall that 4k −1 ≤ η0. Since we have |ψ| < 4k −2, we know that ψ will lie between δ −η0 and η0 − δ and we can pick η in [η0 − δ, η0 + δ] so that g(η) = ψ and |h(η) − χ| < ε, as desired. This completes the proof of the Claim. Our whole line of reasoning in support of the Lemma can now be concluded by a straightforward induction on the natural number m to the eﬀect that for all reals χ0, . . . , χm−1 and every ε > 0, there is ηm so that

|ei(ηm) − χi| < ε for all i < m. The base step holds vacuously. Here is the inductive step. Suppose reals χ0, . . . , χm are given along with ε > 0. The inductive hypothesis applied to the system χ1, . . . , χm yields ηm so that

|ei(ηm) − χi| < ε for all i with 1 ≤ i ≤ m.

Use the Claim to obtain ηm+1 so that |h(ηm+1) − χ0| < ε and g(ηm+1) = ηm. It follows that

|e0(ηm+1) − χ0| = |h(ηm+1) − χ0| < ε

|e1(ηm+1) − χ1| = |e0(g(ηm+1)) − χ1| = |e0(ηm) − χ1| < ε

|e2(ηm+1) − χ2| = |e1(g(ηm+1)) − χ2| = |e1(ηm) − χ2| < ε . .

3. Just one unknown: The proof of Theorem 0

We repeat here Wang’s argument, with one small change (as Wang allowed the 1 rational number 2 ). Let n be a natural number so that Hilbert’s 10th Problem in n variables over the integers is algorithmically unsolvable. With each polynomial p(x0, . . . , xn−1) with integer coeﬃcients we will associate a term G(x) in the signature of R∗ so that p(x0, . . . , xn−1) ≈ 0 has a solution in the integers if and only if G(x) ≈ 0 has a solution in the real numbers.

Moreover, there will be an algorithm which upon input of p(x0, . . . , xn−1) will output the associated G(x). In this way Hilbert’s 10th Problem in one variable for TAYLOR’S ALGORITHMIC UNSOLVABILITY OF TOPOLOGICAL R-COMPATIBILITY 9

R∗ will be reduced to Hilbert’s 10th Problem in n variables for the ring of integers, and Theorem 0 will be established. First, let H(x0, . . . , xn−1) be the term " # 2 2 X ∗ 2 2(n + 1) p (x0, . . . , xn−1) + (sin 2xi) ki(x0, . . . , xn−1) −1, i

G(x) = H(e0(x), e1(x),..., en−1(x)) = 2F (e0(x),..., en−1(x)) +1 . Now it follows from the Lemma by the continuity of H that

H(b0, . . . , bn−1) < 0 for some b0, . . . , bn−1 ∈ R if and only if

H(e0(η), . . . , en−1(η)) < 0 for some η ∈ R. A similar equivalence prevails with the inequalities going in the other direction. First, suppose that a0, . . . , an−1 are integers so that p(a0, . . . , an−1) = 0. Then F (a0, . . . , an−1) = −1, which implies that H(a0, . . . , an−1) = −1 as well. Conse- quently, G(µ) is negative for some real number µ. On the other hand it is easy π π to see that H( 4 ,..., 4 ) must be positive. So G(ν) must be positive for some real number ν, by the Lemma and the continuity of H. By the Intermediate Value Theorem there must be a real number η so that G(η) = 0. Conversely, suppose that G(η) = 0. Then there are real numbers β0, . . . , βn−1 so that 0 = 2F (β0, . . . , βn−1) + 1. This means that F (β0, . . . , βn−1) is negative. By the Richardson-Caviness Inequality Lemma, we conclude that p(x0, . . . , xn−1) ≈ 0 has a solution in the integers. That’s all that needed to be proved.

4. A proof of Theorem 1

Let σ00 be the signature with operation symbols +, ·, −, 0, 1, sin∗, cos∗, λ, c where c is a new constant symbol. We reduce Hilbert’s 10th Problem in one variable for R∗ to the equational compatibility problem for R and σ00. According to The- orem 0 the former problem is algorithmically unsolvable, hence the compatibility problem must also be algorithmically unsolvable. Let Γ be any ﬁnite set of equations of the signature of R∗ so that x is the only variable to occur in Γ. Let Γ0 result from Γ by substituting the constant symbol c for the variable x. We contend that ∗ 0 Γ has a solution in R if and only if Σ ∪ Γ is compatible with R. 10 GEORGE F. MCNULTY

The left-to-right implication is immediate from the natural interpretations for the operation symbols. The right-to-left implication is immediate since Σ determines R∗ according to Taylor’s Finite Determination Theorem.

References [1] B. F. Caviness,On canonical forms and simplification. Journal of the ACM, 17 (1970) 385–396. [2] Martin Davis, Arithmetical problems and recursively enumerable predicates (abstract), Journal of Symbolic Logic, 15 (1950) 77–78. [3] Martin Davis, Arithmetical problems and recursively enumerable predicates, Journal of Symbolic Logic, 18 (1953) 33–41. [4] Martin Davis and Hilary Putnam, Reductions of Hilbert’s Tenth Problem. Journal of Symbolic Logic, 23 (1958) 183–187. [5] Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential Diophantine equations. Annals of Mathematics, Second Series, 74 (1961) 425–436. [6] James P. Jones, Universal Diophantine equations. Journal of Symbolic Logic, 47 (1982) 549–571. [7] Yuri V. Matiyaseviˇc, Enumerable sets are Diophantine. Soviet Mathematics Doklady, 11 (1970) 354–358. [8] Yuri V. Matiyaseviˇc, Diophantine representation of recursively enumerable predicates. In Actes du CongrèsInternational des Mathématiciens (Nice 1970), 1 (1971) 235–238, Gauthier-Villars, Paris. [9] Yuri V. Matiyaseviˇc, Some purely mathematical results inspired by mathematical logic. In Logic, Foundations of Mathematics, and Computability Theory, vol. 1 of Proceedings of the Fifth International congress of Logic, Methodology and Philosophy of Science, (Robert E. Butts and Jaakko Hintikka, eds) (1977) 121–127, D. Reidel Publishing Company, Dordrecht, Holland. [10] Yuri V. Matiyasevich, Hilbert’s Tenth Problem, Foundations of Computing Series, MIT Press, Cambridge, Massachusetts, 1993. [11] Daniel Richardson, Some undecidable problems involving elementary functions of a real variable. Journal of Symbolic Logic,33 (1968) 514–520. [12] Julia Robinson, The undecidability of exponential Diophantine equations. Notices of the American Mathematical Society, 7 (1960) p. 75. [13] Walter Taylor, Equations on real intervals, Algebra Universalis, to appear. [14] Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley and Los Angeles, 1951. [15] Paul S. Wang, The undecidability of the existence of zeros of real elementary functions. Journal of the Association of Computing Machines, 21 (1974) 586–589.

Department of Mathematics University of South Carolina Columbia, SC 29208 USA E-mail address: [email protected]