Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.
For some classes of expressions (generated by other primitives than in Richardson's theorem), there exist algorithms that can determine whether an expression is zero.[1]
Deciding whether an expression A in E represents a function that is nonnegative everywhere
If E includes also the expression |x| (representing the absolute value function), deciding whether an expression A in E represents a function that is zero everywhere
If E includes an expression B representing a function whose antiderivative has no representative in E, deciding whether an expression A in E represents a function whose antiderivative can be represented in E. (Example: has an antiderivative in the elementary functions if and only if a = 0.)
Extensionsedit
After Hilbert's tenth problem was solved in 1970, B. F. Caviness observed that the use of ex and ln 2 could be removed.[3] Wang later noted that under the same assumptions under which the question of whether there was x with A(x) < 0 was insolvable, the question of whether there was x with A(x) = 0 was also insolvable.[4]
Miklós Laczkovich removed also the need for π and reduced the use of composition.[5] In particular, given an expression A(x) in the ring generated by the integers, x, sin xn, and sin(x sin xn) (for n ranging over positive integers), both the question of whether A(x) > 0 for some x and whether A(x) = 0 for some x are unsolvable.
By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.
See alsoedit
Constant problem – Problem of deciding whether an expression equals zero
^ Dan Richardson and John Fitch, 1994, "The identity problem for elementary functions and constants", Proceedings of the international symposium on Symbolic and algebraic computation, pp. 85–290.
^Richardson, Daniel (1968). "Some Undecidable Problems Involving Elementary Functions of a Real Variable". Journal of Symbolic Logic. 33 (4): 514–520. JSTOR 2271358. Zbl 0175.27404.
^Caviness, B. F. (1970). "On Canonical Forms and Simplification". Journal of the ACM. 17 (2): 385–396. doi:10.1145/321574.321591.
^Wang, P. S. (1974). "The undecidability of the existence of zeros of real elementary functions". Journal of the Association for Computing Machinery. 21 (4): 586–589. doi:10.1145/321850.321856.
^Laczkovich, Miklós (2003). "The removal of π from some undecidable problems involving elementary functions". Proc. Amer. Math. Soc. 131 (7): 2235–2240. doi:10.1090/S0002-9939-02-06753-9.
richardson, theorem, mathematics, establishes, undecidability, equality, real, numbers, defined, expressions, involving, integers, displaystyle, exponential, sine, functions, proved, 1968, mathematician, computer, scientist, daniel, richardson, university, bat. In mathematics Richardson s theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers p ln 2 displaystyle ln 2 and exponential and sine functions It was proved in 1968 by mathematician and computer scientist Daniel Richardson of the University of Bath Specifically the class of expressions for which the theorem holds is that generated by rational numbers the number p the number ln 2 the variable x the operations of addition subtraction multiplication composition and the sin exp and abs functions For some classes of expressions generated by other primitives than in Richardson s theorem there exist algorithms that can determine whether an expression is zero 1 Contents 1 Statement of the theorem 2 Extensions 3 See also 4 References 5 Further reading 6 External linksStatement of the theorem editRichardson s theorem can be stated as follows 2 Let E be a set of expressions that represent R R displaystyle mathbb R to mathbb R nbsp functions Suppose that E includes these expressions x representing the identity function ex representing the exponential functions sin x representing the sin function all rational numbers ln 2 and p representing constant functions that ignore their input and produce the given number as output Suppose E is also closed under a few standard operations Specifically suppose that if A and B are in E then all of the following are also in E A B representing the pointwise addition of the functions that A and B represent A B representing pointwise subtraction AB representing pointwise multiplication A B representing the composition of the functions represented by A and B Then the following decision problems are unsolvable Deciding whether an expression A in E represents a function that is nonnegative everywhere If E includes also the expression x representing the absolute value function deciding whether an expression A in E represents a function that is zero everywhere If E includes an expression B representing a function whose antiderivative has no representative in E deciding whether an expression A in E represents a function whose antiderivative can be represented in E Example eax2 displaystyle e ax 2 nbsp has an antiderivative in the elementary functions if and only if a 0 Extensions editAfter Hilbert s tenth problem was solved in 1970 B F Caviness observed that the use of ex and ln 2 could be removed 3 Wang later noted that under the same assumptions under which the question of whether there was x with A x lt 0 was insolvable the question of whether there was x with A x 0 was also insolvable 4 Miklos Laczkovich removed also the need for p and reduced the use of composition 5 In particular given an expression A x in the ring generated by the integers x sin xn and sin x sin xn for n ranging over positive integers both the question of whether A x gt 0 for some x and whether A x 0 for some x are unsolvable By contrast the Tarski Seidenberg theorem says that the first order theory of the real field is decidable so it is not possible to remove the sine function entirely See also editConstant problem Problem of deciding whether an expression equals zero Elementary function Mathematical function Tarski s high school algebra problem Mathematical problemReferences edit Dan Richardson and John Fitch 1994 The identity problem for elementary functions and constants Proceedings of the international symposium on Symbolic and algebraic computation pp 85 290 Richardson Daniel 1968 Some Undecidable Problems Involving Elementary Functions of a Real Variable Journal of Symbolic Logic 33 4 514 520 JSTOR 2271358 Zbl 0175 27404 Caviness B F 1970 On Canonical Forms and Simplification Journal of the ACM 17 2 385 396 doi 10 1145 321574 321591 Wang P S 1974 The undecidability of the existence of zeros of real elementary functions Journal of the Association for Computing Machinery 21 4 586 589 doi 10 1145 321850 321856 Laczkovich Miklos 2003 The removal of p from some undecidable problems involving elementary functions Proc Amer Math Soc 131 7 2235 2240 doi 10 1090 S0002 9939 02 06753 9 Further reading editPetkovsek Marko Wilf Herbert S Zeilberger Doron 1996 A B A K Peters p 212 ISBN 1 56881 063 6 Archived from the original on 2006 01 29 External links editWeisstein Eric W Richardson s theorem MathWorld Retrieved from https en wikipedia org w index php title Richardson 27s theorem amp oldid 1214894189, wikipedia, wiki, book, books, library,
