Rice's theorem - AbsoluteAstronomy.com

Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide

Decision problem

In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

whether an algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

computes a partial function with that property. Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none, and an effective decision method is called general if it decides correctly for every algorithm.
The theorem is named after Henry Gordon Rice

Henry Gordon Rice

Henry Gordon Rice is a logician and mathematician best known as the author of Rice's theorem, which he proved in his doctoral dissertation of 1951 at Syracuse University. He was also a Professor of Mathematics at the University of New Hampshire. After 1960 he was employed by Computer Sciences...

, and is also known as the Rice-Myhill-Shapiro theorem after Rice, John Myhill

John Myhill

John R. Myhill was a mathematician, born in 1923. He received his Ph.D. from Harvard University under Willard Van Orman Quine in 1949. He was professor at SUNY Buffalo from 1966 until his death in 1987...

, and Norman Shapiro

Norman Shapiro

Norman Z. Shapiro is an American mathematician,who is the co-author of the Rice–Shapiro theorem.Shapiro spent the summer of 1954 at Bell Laboratories in Murray Hill, New Jerseywhere, in collaboration withKarel de Leeuw,Ed Moore, and...

Introduction

Another way of stating Rice's theorem that is more useful in computability theory

Computability theory (computer science)

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science...

follows.

Let S be a set of languages

Formal language

A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...

that is nontrivial, meaning

there exists a Turing machine that recognizes a language in S
there exists a Turing machine that recognizes a language not in S

Then, it is undecidable

Decision problem

to determine whether the language decided by an arbitrary Turing machine

Turing machine

A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...

lies in S.

In practice, this means that there is no machine that can always decide whether the language of a given Turing machine has a particular nontrivial property. Special cases include the undecidability of whether a Turing machine accepts a particular string, whether a Turing machine recognizes a particular recognizable language, and whether the language recognized by a Turing machine could be recognized by a nontrivial simpler machine, such as a finite automaton

Finite state machine

A finite-state machine or finite-state automaton , or simply a state machine, is a mathematical model used to design computer programs and digital logic circuits. It is conceived as an abstract machine that can be in one of a finite number of states...

.

It is important to note that Rice's theorem does not say anything about those properties of machines or programs which are not also properties of functions and languages. For example, whether a machine runs for more than 100 steps on some input is a decidable property, even though it is non-trivial. Implementing exactly the same language, two different machines might require a different number of steps to recognize the same input. Similarly, whether a machine has more than 5 states is a decidable property. Where a property is of the kind that either of the two machines may or may not have it, while still implementing exactly the same language, the property is of the machines and not of the language, and Rice's Theorem does not apply.

Using Rogers

Hartley Rogers, Jr

Hartley Rogers, Jr. is a mathematician who has worked in recursion theory, and who is currently a professor in the Mathematics Department of the Massachusetts Institute of Technology. The Rogers equivalence theorem is named after him....

' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming language

Programming language

A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

s: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

As an example, consider the following variant of the halting problem

Halting problem

In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...

: Take the property a partial function F has if F
is defined for argument 1. It is obviously non-trivial, since there are partial functions that are defined for 1 and others that are
undefined at 1. The 1-halting problem is the problem of deciding of any algorithm whether it defines a function with this property,
i.e., whether the algorithm halts on input 1. By Rice's theorem, the 1-halting problem is undecidable.

Formal statement

Let

be a Gödel number

Gödel number

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...

ing of the computable function

Computable function

Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...

s; a map from the natural numbers to the class

of unary (partial) computable functions.
Denote by

the

th (partial) computable function.

We identify each property that a computable function may have with the subset of

consisting of the functions with that property.
Thus given a set

, a computable function

has property F if and only if

. For each property

there is an associated decision problem

Decision problem

of determining, given e , whether

.

Rice's theorem states that the decision problem

is decidable

Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which terminates after a finite amount of time and correctly decides whether or not a given number belongs to the set....

(also called recursive or computable) if and only if

Examples

According to Rice's theorem, if there is at least one computable function in a particular class C of computable functions and another computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable. For example, Rice's theorem shows that each of the following sets of computable functions is undecidable:

The class of computable functions that return 0 for every input, and its complement.
The class of computable functions that return 0 for at least one input, and its complement.
The class of computable functions that are constant, and its complement.

Proof by Kleene's recursion theorem

A corollary to Kleene's recursion theorem

Kleene's recursion theorem

In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions...

states that for every Gödel number

Gödel number

ing

of the computable function

Computable function

s and every computable function

, there is an index

such that

returns

. (In the following, we will say that

"returns"

if either

, or both

and

are undefined.) Intuitively,

is a quine, a function that returns its own source code (Gödel number), except that rather than returning it directly,

passes its Gödel number to

and returns the result.

Let

be a set of computable functions such that

. Then there are computable functions

and

. Suppose that the set of indices

such that

is decidable; then, there exists a function

that returns

otherwise. By the corollary to the recursion theorem, there is an index

such that

returns

. But then, if

, then

is the same function as

, and therefore

; and if

, then

, and therefore

. In both cases, we have a contradiction.

Proof sketch

Suppose, for concreteness, that we have an algorithm for examining a program p and determining infallibly whether p is an implementation of the squaring function, which takes an integer d and returns d². The proof works just as well if we have an algorithm for deciding any other nontrivial property of programs, and will be given in general below.

The claim is that we can convert our algorithm for identifying squaring programs into one which identifies functions that halt. We will describe an algorithm which takes inputs a and i and determines whether program a halts when given input i.

The algorithm is simple: we construct a new program t which (1) temporarily ignores its input while it tries to
execute program a on input i, and then, if that halts, (2) returns the square of its input. Clearly, t is a function for computing squares if and only if step (1) halts. Since we've assumed that we can infallibly identify program for computing squares, we can determine whether t is such a program, and therefore whether program a halts on input i. Note that we needn't actually execute t; we need only decide whether it is a squaring program, and, by hypothesis, we know how to do this.

t(n) {
a(i)
return n×n
}

This method doesn't depend specifically on being able to recognize functions that compute squares; as long as some program can do what we're trying to recognize, we can add a call to a to obtain our t. We could have had a method for recognizing programs for computing square roots, or programs for computing the monthly payroll, or programs that halt when given the input "Abraxas", or programs that commit array bounds errors; in each case, we would be able to solve the halting problem similarly.

Formal proof

For the formal proof, algorithms are presumed to define partial functions over strings

String (computer science)

In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....

and are themselves represented by strings. The partial function computed by the algorithm represented by a string a is denoted F_a. This proof proceeds by reductio ad absurdum

Reductio ad absurdum

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

: we assume that there is a non-trivial property that is decided by an algorithm, and then show that it follows that we can decide the halting problem

Halting problem

In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...

, which is not possible, and therefore a contradiction.

Let us now assume that P(a) is an algorithm that decides some non-trivial property of F_a. Without loss of
generality we may assume that P(no-halt) = "no", with no-halt being the representation of an algorithm that never halts. If this is not true, then this will hold for the negation of the property. Since P decides a non-trivial property, it follows that there is a string b that represents an algorithm and P(b) = "yes". We can then define an algorithm H(a, i) as follows:

1. construct a string t that represents an algorithm T(j) such that

T first simulates the computation of F_a(i)
then T simulates the computation of F_b(j) and returns its result.

2. return P(t).

We can now show that H decides the halting problem:

Assume that the algorithm represented by a halts on input i. In this case F_t = F_b and, because P(b) = "yes" and the output of P(x) depends only on F_x, it follows that P(t) = "yes" and, therefore H(a, i) = "yes".

Assume that the algorithm represented by a does not halt on input i. In this case F_t = F_no-halt, i.e., the partial function that is never defined. Since P(no-halt) = "no" and the output of P(x) depends only on F_x, it follows that P(t) = "no" and, therefore H(a, i) = "no".

Since the halting problem is known to be undecidable, this is a contradiction and the assumption that there is an algorithm P(a) that decides a non-trivial property for the function represented by a must be false.

Rice's theorem and index sets

Rice's theorem can be succinctly stated in terms of index sets:

Let be a class of partial recursive functions with index set
Index set (recursion theory)
In the field of recursion theory, index sets describe classes of partial recursive functions, specifically they give all indices of functions in that class according to a fixed enumeration of partial recursive functions .-Definition:...

. Then is recursive if and only if or .

where

is the set of natural numbers, including zero.

An analogue of Rice's theorem for recursive sets

One can regard Rice's theorem as asserting the impossibility of effectively deciding for any recursively enumerable set
whether it has a certain nontrivial property.
In this section, we give an analogue of Rice's theorem for recursive sets, instead of recursively enumerable sets.
Roughly speaking, the analogue says that if one can effectively determine for any recursive set whether it has a certain property,
then finitely many integers determine whether a recursive set has the property.
This result is analogous to the original Rice's theorem because both assert that a property is "decidable"
only if one can determine whether a set has that property by examining for at most finitely many
(for no

, for the original theorem), if

belongs to the set.

Let

be a class (called a simple game and thought of as a property) of recursive sets.
If

is a recursive set, then for some

, computable function

is the characteristic function of

. We call

a characteristic index for

.
(There are infinitely many such

.)
Let's say the class

is computable if there is an algorithm (computable function) that decides
for any nonnegative integer

(not necessarily a characteristic index),

if is a characteristic index for a recursive set belonging to , then the algorithm gives "yes";
if is a characteristic index for a recursive set not belonging to , then the algorithm gives "no".

A set

extends a string

of 0's and 1's
if for any

(the length of

),
the

th element of

is 1 if

; is 0 otherwise.
For example,

extends string

.
A string

is winning determining if any recursive set extending

belongs to

.
A string

is losing determining if no recursive set extending

belongs to

.

We can now state the following analogue of Rice's theorem
(Kreisel, Lacombe, and Shoenﬁeld, 1959,
Kumabe and Mihara, 2008):

A class

of recursive sets is computable
if and only if there are a recursively enumerable set

of losing determining strings
and a recursively enumerable set

of winning determining strings such that
any recursive set extends a string in

.

This result has been applied to foundational problems in computational social choice (more broadly, algorithmic game theory

Algorithmic game theory

Algorithmic game theory is an area in the intersection of game theory and algorithm design, whose objective is to design algorithms in strategic environments. Typically, in Algorithmic Game Theory problems, the input to a given algorithm is distributed among many players who have a personal...

).
For instance, Kumabe and Mihara (2008, 2008)
apply this result to an investigation of the Nakamura number

Nakamura number

In cooperative game theory and social choice theory, the Nakamura number measures the degree of rationalityof preference aggregation rules , such as voting rules....

s for simple games in cooperative game theory

Cooperative game

In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...

and social choice theory

Social choice theory

Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...

Introduction

Formal statement

Examples

Proof by Kleene's recursion theorem

Proof sketch

Formal proof

Rice's theorem and index sets

An analogue of Rice's theorem for recursive sets

See also