Rice, Condoleezza, 1954-, U.S. government official and educator, b. Birmingham, Ala. A political scientist who has specialized in Russian and E European studies, Rice has been a professor at Stanford Univ. since 1981. From 1989 to 1991 she was an adviser on Soviet and E European affairs on President George H. W. Bush's National Security Council. Subsequently, she served (1993-99) as Stanford's provost. During the 2000 presidential campaign she was George W. Bush's foreign policy adviser, and in 2001 she became President Bush's national security adviser—the first woman and second African American (after Colin Powell) to hold the post. A member of the president's inner circle, she has been an advocate of U.S. military power, a supporter of the Iraq invasion (see Persian Gulf Wars), and a spokeswoman for the administration's assertive foreign policy. She served (2005-9) as secretary of state during Bush's second term, succeeding Colin Powell. Her books include The Gorbachev Era (1986, with A. Dallin) and Germany Unified and Europe Transformed (1995, with P. Zelikow).

See biographies by A. Felix (2002), M. Mabry (2007), and E. Bumiller (2008); J. Mann, Rise of the Vulcans: The History of Bush's War Cabinet (2004); G. Kessler, The Confidante (2007).

Rice, Elmer, 1892-1967, American dramatist, b. New York City, LL.B. New York Law School, 1912. After the success of his first play, On Trial (1914), he turned his interests to the theater. Rice's first major contribution to the American stage was The Adding Machine (1923), an expressionistic play satirizing man in the machine age. Street Scene (1929; operatic version by Kurt Weill, 1947), one of his most compassionate works, is a realistic drama of tenement life in New York. His plays of the 1930s—including Counsellor-at-Law (1931), We, the People (1933), and Between Two Worlds (1934)—continued to express his social and political views. Although Dream Girl (1945), a romantic comedy, was a huge success, his later plays for the most part lack the power of his early works. He was also the author of novels and of essays, some of which were published as The Living Theatre (1959). During the 1930s Rice was regional director of the N.Y. Federal Theater project.

See his autobiography Minority Report (1963); A. F. Palmieri, Elmer Rice: A Playwright's Vision of America (1980).

Rice, Jerry Lee, 1962-, American football player, b. Crawford, Miss. Winning national attention while at the otherwise obscure Mississippi Valley State College, Itta Bena, Miss., Rice subsequently played professionally with the San Francisco 49ers (1985-2001), the Oakland Raiders (2001-2004), and the Seattle Seahawks (2004). One of the game's most durable players, he became the NFL's oldest ever wide receiver and one of its greatest players. At his retirement he held career records for receptions (1,549), receiving yards (22,895), touchdowns (208), and receiving touchdowns (197) during the regular season and the season record for receiving yards (1,848). Rice was rookie of the year for the 1985 season, most valuable player for 1987, Super Bowl most valuable player in 1989, and NFL player of the year for 1990 and 1997, and helped the 49ers win three Super Bowls (1989-90, 1995).

See his Rice (with M. Silver, 1996).

rice, cereal grain (Oryza sativa) of the grass family (Graminae), probably native to the deltas of the great Asian rivers—the Ganges, the Chang (Yangtze), and the Tigris and Euphrates. The plant is an annual, from 2 to 6 ft (61-183 cm) tall, with a round, jointed stem; long, pointed leaves; and edible seeds borne in a dense head on separate stalks. Wild rice is obtained from a different grass plant.

Cultivation and Harvesting

Methods of growing differ greatly in different localities, but in most Asian countries the traditional hand methods of cultivating and harvesting rice are still practiced. The fields are prepared by plowing (typically with simple plows drawn by water buffalo), fertilizing (usually with dung or sewage), and smoothing (by dragging a log over them). The seedlings are started in seedling beds and, after 30 to 50 days, are transplanted by hand to the fields, which have been flooded by rain or river water. During the growing season, irrigation is maintained by dike-controlled canals or by hand watering. The fields are allowed to drain before cutting.

Rice when it is still covered by the brown hull is known as paddy; rice fields are also called paddy fields or rice paddies. Before marketing, the rice is threshed to loosen the hulls—mainly by flailing, treading, or working in a mortar—and winnowed free of chaff by tossing it in the air above a sheet or mat.

In the United States and in many parts of Europe, rice cultivation has undergone the same mechanization at all stages of cultivation and harvesting as have other grain crops. Rice was introduced to the American colonies in the mid-17th cent. and soon became an important crop. Although U.S. production is less than that of wheat and corn, rice is grown in excess of domestic consumption and has been exported, mainly to Europe and South America. Chief growing areas of the United States are in California, Mississippi, Texas, Arkansas, and Louisiana. The world's leading rice-producing countries are China, India, Indonesia, Bangladesh, and Thailand. Total annual world production is more than half a billion metric tons.

Importance of Rice as a Food

It has been estimated that half the world's population subsists wholly or partially on rice. Ninety percent of the world crop is grown and consumed in Asia. American consumption, although increasing, is still only about 25 lb (11 kg) per person annually, as compared with 200 to 400 lb (90-181 kg) per person in parts of Asia. Rice is the only major cereal crop that is primarily consumed by humans directly as harvested, and only wheat and corn are produced in comparable quantity. Plant breeders at the International Rice Research Institute in the Philippines, attempting to keep pace with demand from a burgeoning world population, have repeatedly developed improved varieties of "miracle rice" that allow farmers to increase crop yields substantially.

Brown rice has a greater food value than white, since the outer brown coatings contain the proteins and minerals; the white endosperm is chiefly carbohydrate. As a food rice is low in fat and (compared with other cereal grains) in protein. The miracle rices have grains richer in protein than the old varieties. In the East, rice is eaten with foods and sauces made from the soybean, which supply lacking elements and prevent deficiency diseases. Elsewhere, especially in the United States, rice processing techniques have produced breakfast and snack foods for retail markets. Deficient in gluten, rice cannot be used to make bread unless its flour is mixed with flour made from other grains.

Other Uses

For feeding domestic animals, the bran, meal, and chopped straw are useful, especially when mixed with the polishings or given with skim milk. The polishings are also an important source of furfural and other chemurgic products. The straw, which is soft and fine, is plaited in East Asia for hats and shoes, and the hulls supply mattress filling and packing material. Laundry starch is manufactured from the broken grain, which is also used by distillers. A distilled liquor called arrack is sometimes prepared from a rice infusion, and in Japan the beverage sake is brewed from rice. Rice paper is made from a plant of the ginseng family.

History of Rice Cultivation

Rice has been cultivated in China since ancient times and was introduced to India before the time of the Greeks. Chinese records of rice cultivation go back 4,000 years. In classical Chinese the words for agriculture and for rice culture are synonymous, indicating that rice was already the staple crop at the time the language was taking form. In several Asian languages the words for rice and food are identical. Many ceremonies have arisen in connection with planting and harvesting rice, and the grain and the plant are traditional motifs in Oriental art. Thousands of rice strains are now known, both cultivated and escaped, and the original form is unknown.

Rice cultivation has been carried into all regions having the necessary warmth and abundant moisture favorable to its growth, mainly subtropical rather than hot or cold. The crop was common in West Africa by the end of the 17th cent. It is thought that slaves from that area who were transported to the Carolinas in the mid-18th cent. introduced the complex agricultural technology, thus playing a key part in the establishment of American rice cultivation. Their labor then insured a flourishing rice industry. Modern culture makes use of irrigation, and a few varieties of rice may be grown with only a moderate supply of water.


Rice is classified in the division Magnoliophyta, class Liliopsida, order Cyperales, family Gramineae.


See Food and Agricultural Organization, Rice (annual); D. H. Grist, Rice (6th ed. 1986); J. A. Carney, Black Rice: The African Origins of Rice Cultivation in the Americas (2001).

In computer science, Rice's theorem named after Henry Gordon Rice (also known as The Rice-Myhill-Shapiro theorem after Rice and John Myhill) states that, for any non-trivial property of partial functions, there exists at least one algorithm for which it is undecidable whether the algorithm computes a partial function with this property. Here, a property of partial functions is called trivial if it holds for all partial computable functions or for none.


Another way of stating this problem that is more useful in computability theory is this: suppose we have a set of languages S. Then the problem of deciding whether the language of a given Turing machine is in S is undecidable, provided that there exists a Turing machine that recognizes a language in S and a Turing machine that recognizes a language not in S. Effectively this means that there is no machine that can always correctly 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.

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 when it is non-trivial. Implementing exactly the same language, two different machines might require a different number of steps to recognize the same input. Where a property is of the kind that 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.

Similarly, whether a machine has more than 5 states is a decidable property. On the other hand, the statement that "No modern general-purpose computer can solve the general problem of determining whether a program is virus free" is a consequence of Rice's Theorem because, while a statement about computers, it can be reduced to a statement about languages.

Using Rogers' characterization of acceptable programming systems, this result may essentially be generalized to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs. This is one explanation of the difficulty of debugging.

As an example, consider the following variant of the halting problem: 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 phicolon mathbb{N} to mathbf{P}^{(1)} be a Gödel numbering of the computable functions; a map from the natural numbers to the class of unary partial computable functions.

We identify each property that a computable function may have with the subset of mathbf{P}^{(1)} consisting of the functions with that property. Thus given a set F subseteq mathbf{P}^{(1)}, a computable function phi_e has property F if and only if phi_e in F. For each property F subseteq mathbf{P}^{(1)} there is an associated decision problem D_F of determining, given e , whether phi_e in F.

Rice's theorem states that the decision problem D_F is decidable if and only if F = emptyset or F = mathbf{P}^{(1)}.


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 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) {
     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 and are themselves represented by strings. The partial function computed by the algorithm represented by a string a is denoted Fa. This proof proceeds by reductio ad absurdum: 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, 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 Fa. 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 Fa(i)
* then T simulates the computation of Fb(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 Ft = Fb and, because P(b) = "yes" and the output of P(x) depends only on Fx, 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 Ft = Fno-halt, i.e., the partial function that is never defined. Since P(no-halt) = "no" and the output of P(x) depends only on Fx, 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 mathcal{C} be a class of partial recursive functions with index set C. Then C is recursive if and only if C is empty, or C is all of omega.

where omega is the set of natural numbers, including zero.

See also


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