Is there always an algorithm? An introduction to computability theory. Computer Science Department Seminar 9th October 2009 Rick Thomas 1
Equations If we want to solve the quadratic equation ax 2 + bx + c = 0 , we know that the solutions are − b + √ and − b − √ b 2 − 4 ac b 2 − 4 ac . 2 a 2 a Question: Can we do the same for cubic equations? What about quartic equations? Higher degrees? For cubic equations the first advances were by Scipione dal Ferro (1465-1526): he could solve equations of the form x 3 + mx = n (where m and n are positive). Just before his death he revealed his method to his student Antonio Fior. 2
Cubic Equations Nicolo of Brescia (known as “Tartaglia”) managed to solve equations of the form x 3 + mx 2 = n . Fior challenged Tartaglia to a public contest. Eight days before the problems were to be collected, Tartaglia found the general method for solving all cubics. For more information about the topics discussed here, see the MacTutor History of Mathematics archive: http://www-history.mcs.st-and.ac.uk/ 3
Nicolo Tartaglia (1500 - 1557) 4
Tartaglia and Cardano News of Tartaglia’s victory reached Girolamo Cardano in Milan. Cardano invited Tartaglia to visit him; Tartaglia eventually revealed his method, but made Cardano promise to keep it secret until Tartaglia had published it himself. In 1545 Cardano published Ars Magna , the first Latin treatise on algebra, containing Tartaglia’s method. 5
Girolamo Cardano (1501 - 1576) 6
Quartic Equations Cardano then encouraged his own student, Lodovico Ferrari (1522 - 1565), to examine quartic equations. Ferrari managed to solve the quartic. Cardano published the method in Ars Magna. But . . . In 1823 Niels Abel proved that there is no such method using the given ingredients for the degree five case (the quintic). 7
Niels Abel (1802 - 1829) 8
The General Situation Evariste Galois then gave a general method for deciding whether or not a given degree n equation admits a solution of this type: “On the condition that an equation be soluble by radicals.” This research essentially introduced the idea of a “group” into Mathematics. 9
Evariste Galois (1811 - 1832) 10
Algorithms David Hilbert’s famous address “Mathematical Problems” was delivered to the Second International Congress of Mathematicians in Paris in 1900. One question was the issue as to whether Mathematics could be finitely “mechanized”. Can we create a machine into which one feeds a statement about Mathematics and which delivers an answer: true or false? Given a formal system, can we produce a machine that would eventually output all the “truths” of that system? 11
David Hilbert (1862 - 1943) 12
Hilbert’s Tenth Problem A Diophantine Equation is a polynomial equation where we insist that the solutions be integers. “ Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.” 13
G¨ odel’s Incompleteness Theorem A set of rules is complete if, for any statement in the relevant language, either that statement or its negation is provable from the rules. A set of rules is consistent if there is no statement so that both the statement and its negation are provable from the rules. Any sufficiently rich formal system cannot be both consistent and complete. In particular, for any sufficiently rich formal system, there is a statement that is true but not provable in the system. 14
Kurt G¨ odel (1906 - 1978) 15
A Mechanical Approach? Around the same time, various people were working on formalizing what we might mean by a “mechanical method.” There were various models proposed, such as the lambda calculus (by Alonzo Church) . . . • If x is a variable, then x ∈ Λ ; • If x is a variable and M ∈ Λ , then ( λ x . M ) ∈ Λ ; • If M , N ∈ Λ , then ( MN ) ∈ Λ . . . . and partial recursive functions (by Stephen Kleene) . . . • Functions built up from certain elementary functions by composition, primitive recursion and minimization. 16
Alonzo Church (1903 - 1995) 17
Stephen Kleene (1909 - 1994) 18
The Turing Machine Alan Turing considered a very simple mechanized device. We have input, memory, a finite collection of states and transitions. Input goes here Finite state control This is used as a work tape # 19
Alan Turing (1912 - 1954) 20
Church-Turing Thesis Church-Turing Thesis. Any algorithmic problem for which there exists an algorithm that can be programmed in some programming language on some computer is also solvable by a Turing machine. Turing’s fundamental paper was part of the inspiration for the first computers, and strongly influenced other researchers such as John von Neumann. 21
John von Neumann (1903 - 1957) 22
Longevity of the Model Where a calculator on the ENIAC is equipped with 18 , 000 vacuum tubes and weighs 30 tons, computers in the future may have only 1 , 000 vacuum tubes and perhaps weigh 1 . 5 tons. Popular Mechanics, March 1949 However, the Turing Machine models the limits of the power of computers today (just as it did in the 1940’s). We can show that there is no algorithm to decide the following question. INPUT: A Turing machine (program) M and an input α . QUESTION: Does the machine (program) M halt on input α ? 23
Halting Problem Suppose we had a program H as in the figure below that would accept a (listing of a) program M and the input α for M , and decide whether or not M would halt with input α . AAAAA Y M halts with input α AAAAA H M , α AAAAA N M doesn't halt with input α 24
Halting Problem We could then use H to construct another program J as in the figure below which would go into an infinite loop if M halted with input α and would terminate otherwise. ∞ loop M halts with input α AAAAA J M , α AAAAA N AAAAA M doesn't halt with input α 25
Halting Problem If J existed, we could then construct a simpler program K as in the figure below which performs in the same way as J but with α equal to (the listing of) M . AAAAA AAAAA ∞ loop M halts with input M AAAAA K M AAAAA N M doesn't halt with input M 26
Halting Problem We can now input (the listing of) K into K as in the figure below. ∞ loop K halts with input K AAAAA K K AAAAA N K doesn't halt with input K Now comes the fun bit! 27
Halting Problem ∞ loop K halts with input K AAAAA K K AAAAA N K doesn't halt with input K This gives a contradiction: we now have that K halts with input K if it doesn’t and K doesn’t halt with input K if it does! So the original machine H can’t exist, i.e. the halting problem is unsolvable. 28
Undecidability It is important to appreciate what the term “undecidable” means. To say that a problem is undecidable means that there is no way, even given unlimited resources (in terms of time and space), that the problem can be decided by algorithmic means. Some more examples of undecidable problems: • Given a program P , is there any input for which P terminates? • Given two programs P and Q , are they equivalent (i.e. do they produce the same output for any given input)? 29
Languages We can think of the valid terms derived from some axiomatic system as being a language . If we have some finite set of symbols, then some strings of symbols will represent valid terms and some will not. A language that is decidable is often said to be “recursive”: A language L is recursive if there is a Turing machine that, given an input α , will output “yes” if α ∈ L and “no” if α �∈ L . It is important to distinguish this from the concept of a language being recursively enumerable : A language L is recursively enumerable if there is a Turing machine that, given an input α , will output “yes” if α ∈ L . 30
Diophantine Equations If we have a finite axiomatic system then, in general, the set of valid terms will be recursively enumerable. However, it may well not be recursive. Let us reconsider Hilbert’s tenth problem: “ Determination of the solvability of a Diophantine equation. Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.” 31
Diophantine Equations It is not difficult to see that the set of solutions to a Diophantine equation is recursively enumerable. We call such a set Diophantine . Building on work of Julia Robinson, Martin Davis and Hilary Putnam, in 1970 Yuri Matiyasevich showed that the converse is also true (i.e. that every recursively enumerable set is Diophantine). As a result, we see that Hilbert’s tenth problem is unsolvable. 32
Julia Robinson (1919 - 1985) and Yuri Matiyasevich (1947 - ) 33
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