UNIT-DISTANCE PRESERVING THEOREM IS LOGICALLY NON-TRIVIAL by Olga Kosheleva and Vladik Kreinovich1 Computer Science Department University of Texas at El Paso, El Paso, TX 79968 email
[email protected] and
[email protected] Abstract. If we express the statement of the unit-distance preserving theorem (that every mapping from Rn to Rn that preserves unit distance is linear) in terms of quantifiers and logical connectives, then we get a formula that is very close to formulas from elementary geometry. A. Tarski has proved that elementary geometry is trivial in the sense that this theory has a deciding algorithm, i.e., an algorithm that, given a formula, decides whether it is true or not (actually, Tarski’s algorithm returns either the proof of a given formula, or a counterexample to it). So, the natural question is: can we extend Tarski’s algorithm to a class of statements that contain unit-distance preserving theorem? Our answer to that question is “no”: for the simplest generalization of elementary geometry that contains the unit-distance preserving theorem, no deciding algorithm is possible. This class also contains Alexandrov’s theorem that every mapping Rn → Rn that preserves Minkowski causality (i.e., future cones in the sense of special relativity) is linear. 1. UNIT-DISTANCE PRESERVING THEOREM It is well known that for n > 1, a function f : Rn → Rn that preserves unit distance is an isometry (and is thus linear). This result was first proven in (Beckman et al 1953); for references to further results, see (Greenwell et al 1993) and (Kuz’minykh 1993).
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This work was partially supported by NSF grant No. CDA– 9015006 and NASA grant No. NAG 9-757.
2. THE STATEMENT OF THIS THEOREM IS CLOSE TO THE LANGUAGE OF ELEMENTARY GEOMETRY In 1951, A. Tarski proved his famous decidability result. To describe this result, let us consider formulas of the following type: • We start with variables x, y, z, ..., that run over real numbers, and with two constants: 0 and 1. • From these variables and constants, we can form expressions by applying addition and multiplication. For example, x · x + y · y is an expression in this sense. • From expressions t, t0 , ..., we can form elementary formulas of the type t = t0 , t 6= t0 , t > t0 , t < t0 , t ≤ t0 , and t ≥ t0 . For example, x · x + y · y = z · z is an elementary formula in our language. • From elementary formulas, we can form formulas by applying logical connectives & (“and”), ∨ (“or”), → (“implies”), ¬ (“not”), and quantifiers ∀x and ∃x. The resulting formulas are called formulas of elementary geometry, because, as Tarksi has shown, practically all statements from elementary geometry can be easily formulated in this language (with Cartesian coordinates of the points and coefficients of the straight lines and circles as variables x, y, z, ...). In particular, several geombinatoric problems can be described in this language; for a recent example, see, e.g., (Johnson 1994). Tarski has proved (Tarski 1951; see also Seidenberg 1954), that the set of all these statements is decidable in the following sense: there exists an algorithm that, given a formula from this language, decides whether this formula is true or not. Therefore, from purely logical viewpoint, elementary geometry is trivial in the sense that for such problems, there is no need to prove theorems: there is an algorithm that solves each of these problems. From the practical viewpoint, the field may not be exactly trivial (because Tarski’s algorithm may take a very long time to run), but theoretically, it is trivial. Tarksi’s theorem has been successfully applied to several geombinatoric problems (see, e.g., (Johnson 1994), (Kosheleva et al
1994)). The formulation of the unit-distance preserving theorem is not a formula from elementary geometry, because in addition to quantifiers over real numbers this formulation requires a quantifier over arbitrary functions from Rn to Rn : ∀f ((∀x1 , ..., xn , y1 , ..., yn ((x1 − y1 )2 + ... + (xn − yn )2 ) = 1 → (f1 (x1 , ..., xn ) − f1 (y1 , ..., yn ))2 + ...+ (fn (x1 , ..., xn ) − fn (y1 , ..., yn ))2 = 1) → ∃A11 , ...Ann , b1 , ..., bn ∀x1 , ..., xn (f1 (x1 , ..., xn ) = A11 x1 + A12 x2 + ... + A1n xn + b1 & ... & fn (x1 , ..., xn ) = An1 x1 + ... + Ann xn + bn )). Comment. In this formula, x2 can be viewed as an abbreviation for x · x. 3. A NATURAL QUESTION: ARE SUCH STATEMENTS STILL DECIDABLE (LOGICALLY TRIVIAL)? Main problem. A natural question is: do such statements (with one additional quantifier over functions) still form a decidable theory? Comment: some reasonable extensions of elementary geometry are decidable. The very fact that we are talking about an extension of elementary geometry does not necessarily make this theory undecidable: for example, another extension, that is obtained by adding “quantifiers” “for countably many (x1 , ..., xm )”, turned out to be decidable (Rapp 1985). A geombinatoric application of this extension is described in (Kosheleva et al 1989) and (Kosheleva et al 1994).
4. MAIN RESULT: THIS EXTENSION IS LOGICALLY NON-TRIVIAL Definition 1. For every n ≥ 1, let us define a ∀f −extension of elementary geometry as follows: • We start with variables x, y, z, ..., that run over real numbers, and with two constants: 0 and 1. • From these variables and constants, we can form expressions by applying addition, multiplication, and n function symbols f1 , ..., fn . For example, x · x + y · y and f1 (x1 , ..., xn ) are expression in this sense. • From expressions t, t0 , ..., we can form elementary formulas of the type t = t0 , t 6= t0 , t > t0 , t < t0 , t ≤ t0 , and t ≥ t0 . • From elementary formulas, we can form geometric formulas by applying logical connectives & (“and”), ∨ (“or”), → (“implies”), ¬ (“not”), and quantifiers ∀x and ∃x. • By a formula from the ∀f −extension of elementary geometry, we mean the formula of the type ∀f F , where F is a geometric formula. Comment. For example, the statement of the unit-preserving theorem is a formula from the ∀f −extension of elementary geometry. PROPOSITION 1. For every n ≥ 1, the ∀f −extension of elementary geometry is undecidable. Comment. So, the natural extension of elementary geometry that contains the statement of the unit-distance preserving theorem is undecidable (logically non-trivial). Can this non-triviality be due to the fact that in addition to the quantifier over f , we allow (in this definition) arbitrary number of quantifiers over real numbers? In other words, is it possible that for the small number of quantifiers, we will get a decidable theory? The answer is again “no”: PROPOSITION 2. For every n ≥ 1, the set of all formulas of the type ∀f ∃x1 ...∃x13 Q with quantifier-free Q is undecidable. Since the statement of the unit-distance preserving theorem contains (for n ≥ 3) more than 13 quantifiers, we cannot get a decidable class by restricting this number.
5. ANOTHER GEOMETRIC RESULTS FROM THE SAME LOGICALLY NON-TRIVIAL CLASS OF STATEMENTS This same class of formulas (∀f −extended elementary geometry) for which we have proved that it is logically non-trivial, contains another interesting theorem: namely, Alexandrov’s theorem that every mapping Rn → Rn that preserves Minkowski causality (i.e., future cones in the sense of special relativity) is linear (Alexandrov 1950, Alexandrov et al 1953; see also Zeeman 1964). Actually, Alexandrov has proven several results (about mappings that preserve open cones; about mappings that preserve closed cones; about mappings that preserve double cones, etc), and each of these results can be reformulated in this language. For example, the result about double cones can be reformulated as follows: ∀f (∀x1 , ..., xn , y1 , ..., yn ((x1 − y1 )2 ≥ (x2 − y2 )2 + ... + (xn − yn )2 → (f1 (x1 , ..., xn ) − f1 (y1 , ..., yn ))2 ≥ (f2 (x1 , ..., xn ) − f2 (y1 , ..., yn ))2 + ...+ (fn (x1 , ..., xn ) − fn (y1 , ..., yn ))2 ) → ∃A11 , ...Ann , b1 , ..., bn ∀x1 , ..., xn (f1 (x1 , ..., xn ) = A11 x1 + A12 x2 + ... + A1n xn + b1 & ... & fn (x1 , ..., xn ) = An1 x1 + ... + Ann xn + bn )). Comment. Several generalizations of this result have been proposed; see, e.g., (Kuz’minykh 1975), (Kuz’minykh 1976), (Kosheleva et al 1979), (Kreinovich 1994). Some of them can also be described in this language (e.g., the one from (Kreinovich 1994)). 6. PROOFS Proof of Proposition 1. It is sufficient to prove the result for n = 1. If we prove that even for n = 1, the theory is undecidable,
then it is undecidable for n ≥ 2: indeed, for every formula ∀f F (f ) that contains a reference to the function f of one variable, we can form a formula ∀f (∀x1 , ..., xn (f (x1 , ..., xn ) = f (x1 , 0, ..., 0) → F ∗ ), where F ∗ is obtained from F if, instead of every expression of the type f (t), we substitute f1 (t, 0, ..., 0). Then, this new formula is true iff F is true for every function of one variable. Therefore, the impossibility to decide whether formulas ∀f F are true for functions of one variable leads to the impossibility to decide whether an arbitrary formula ∀f F is true for functions of n variables. So, let us prove the result for n = 1. In this proof, we will use the well-known G¨odel’s incompleteness theorem according to which arithmetic is undecidable. In other words, the set of all formulas that are built from elementary ones (t = t0 , etc, where t and t0 are obtained from variables n1 , ..., nk , ... and constants by addition and multiplication) with quantifiers over integers ni is undecidable. So, to prove that our extension of elementary geometry is undecidable, let us prove that an arbitrary arithmetic formula can be reformulated in this language. The only difference between arithmetic formulas and formulas from elementary geometry is that in elementary geometry, variables run over arbitrary real numbers, while in arithmetic, they can only run over integers only. So, to express arithmetic formulas in our language, we will express the property “x is an integer” in this language. To do that, let us consider the following formula f (1) = 1 & ∀x(0 ≤ x < 1 → f (x) = 0) & ∀x(f (x + 1) = f (x) + 1). (1) It is easy to see that this formula determines the unique function: f (x) = bxc. For this function f , a real number x is an integer iff f (x) = x. Therefore, we can express the statement “x is an integer” as ∀f (F1 (f ) → f (x) = x), where by F1 , we denoted the formula (1). So, we can express ∀nA as ∀x(f (x) = x) → A), and
∃nA as ∃x(f (x) = x & A). Hence, an arbitrary arithmetic formula ∀n1 ...∃n2 ... can be reformulated as follows: ∀f (F1 (f ) → ∀x1 (f (x1 ) = x1 → ...∃x2 (f (x2 ) = x2 &...)). This is a formula from the ∀f −extension of elementary geometry. So, if there was an algorithm to decide all formulas from this extension, then we would thus have an algorithm to decide arithmetic, which contradicts to G¨odel’s result. Q.E.D. Proof of Proposition 2. In this proof, we will use the result from (Matiyasevich 1970), (Matiyasevich et al 1974), and (Davis et al 1976) that no algorithm is possible to solve Diophantine equations with 13 variable, i.e., no algorithm can decide whether a formula ∃n1 , ..., n13 P with a quantifier-free P is true or not (this result solved tenth Hilbert’s problem (Hilbert 1902)). For such formulas, the reduction described in the proof of Proposition 1 leads to the following equivalent formula from the ∀f −extension of elementary geometry: ∀f (F1 (f ) → ∃x1 , ..., x13 (f (x1 ) = x1 & ... & f (x13 ) = x13 & P )). This formula contains 15 quantifiers (13 existential plus two in the definition of F1 ), and we need 13. So, to prove our proposition, we need to simplify this formula: • First, F1 (f ) can be reformulated as ∀xA(f, x), where A(f, x) stands for f (1) = 1 & (0 ≤ x < 1 → f (x) = 0) & (f (x + 1) = f (x) + 1). • Second, since P → Q is equivalent to ¬P ∨ Q for arbitrary formulas P and Q, the formula ∀xA(f, x) → ∃x1 , ..., x13 (f (x1 ) = x1 & ... & f (x13 ) = x13 & P ) is equivalent to ∃x¬A(f, x) ∨ ∃x1 , ..., x13 (f (x1 ) = x1 & ... & f (x13 ) = x13 & P ),
and this formula, in its turn, is equivalent to the following formula with only 13 quantifiers: ∃x1 , ..., x13 (¬A(f, x1 ) ∨ (f (x1 ) = x1 & ... & f (x13 ) = x13 & P )). So, undecidability of formulas from Matiyasevich’s theorem proves that the set of all formulas of the type ∀f ∃x1 ...∃x13 Q with quantifier-free Q is also undecidable. Q.E.D. 7. CONCLUSION AND AN OPEN PROBLEM In this paper, we have shown that: • The unit-distance preserving theorem can be formulated in a language that is close to the logically trivial language of elementary geometry, • In spite of this closeness, the natural extension of the (logically trivial) language of elementary geometry that contains the statement of the unit-distance preserving theorem is logically non-trivial. • On the other hand, other extensions of elementary geometry are known to be decidable (logically trivial). So, the natural open problem is: To find an interesting decidable (logically trivial) extension of elementary geometry that would include unit-distance preserving theorem (and maybe Alexandrov’s theorem). REFERENCES Alexandrov, A. D. On Lorentz transformations, Uspekhi Math. Nauk, 1950, Vol. 5, No. 3 (37), p. 187 (in Russian). Alexandrov, A. D., and Ovchinnikova, V. V. Remarks on the foundations of special relativity, Leningrad University Vestnik, 1953, No. 11, pp. 94–110 (in Russian). Beckman, F.S., and Quarles, D. A. Jr., On isometries of Euclidean space, Proc. Amer. Math. Soc., 1953, Vol. 4, pp. 810–815.
Davis, M., Matiyasevich, Yu. V., and Robinson, J. Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution, In: Mathematical developments arising from Hilbert’s problems, Proceedings of Symposia in Pure Mathematics, Vol. 28, American Math. Society, Providence, RI, 1976, Part 2, pp. 323– 378. Greenwell, D., and Johnson, P. D. Jr., Functions which preserve unit distance, Geombinatorics, 1993, Vol. 2, No. 3, pp. 61–64. Hilbert, D. Mathematical Problems, lecture delivered before the International Congress of Mathematics in Paris in 1900, translated in Bull. Amer. Math, Soc., 1902, Vol. 8, pp. 437–479. Johnson, P. D. Jr., The choice number of the plane, Geombinatorics, 194, Vol. 3, No. 4, pp. 122–128. Kosheleva, O. M.; Kreinovich, V., and Vroegindewey, P. G. An extension of a theorem of A. D. Alexandrov to a class of partially ordered fields, Proceedings of the Royal Academy of Science of Netherlands, 1979, Vol. 82(3), Series A, September 21, pp. 363– 376. Kosheleva, O. M. and Kreinovich, V. Algorithmic problems of combinatorial geometry. Center for New Informational Technology “Informatika”, Technical Report, Leningrad, 1989 (in Russian). Kreinovich, V. Approximately measured causality implies the Lorentz group: Alexandrov-Zeeman result made more realistic, International Journal of Theoretical Physics, 1994, Vol. 33, No. 8, pp. 1761–1775. Kreinovich, V. and Kosheleva, O. M. An application of logic to combinatorial geometry: how many tetrahedra are equidecomposable with a cube?, Mathematical Logic Quarterly, 1994, Vol. 40, No. 1, pp. 31–34. Kuz’minykh, A. V. Characterization of Lorentz transformations, Soviet Math. Dokl., 1975, Vol. 16, No. 6, pp. 1626–1628.
Kuz’minykh, A. V. Minimal condition determining the Lorentz transformations, Siberian Mathematical Journal, 1976, Vol. 17. No. 6, pp. 968–972. Kuz’minykh, A. V. About a problem concerning functions which preserve unit distance, Geombinatorics, 1994, Vol. 3, No. 4, pp. 137–139. Matiyasevich, Yu. V. Enumerable sets are diophantine, Soviet Math. Doklady, 1970, Vol. 11, pp. 354–357. Matiyasevich, Yu. V., and Robinson, J. Reduction of an arbitrary Diophantine equation to one in 13 unknowns, Acta Arithmetica, 1974, Vol. 27, pp. 521–553. Rapp, A. The ordered field of real numbers and logics with Malitz quantifiers, The Journal of Symbolic Logic, 1985, Vol. 50, No. 2, pp. 380–389. Seidenberg, A. A new decision method for elementary algebra, Annals of Math., 1954, Vol. 60, pp. 365–374. Tarski, A. A decision method for elementary algebra and geometry, 2nd ed., Berkeley and Los Angeles, 1951, 63 pp. Zeeman, E. C. Causality implies the Lorentz group, Journal of Mathematical Physics, 1964, Vol. 5, pp. 490–493.
