Alonzo Church

This article is about the mathematician and logician. For the president of the University of Georgia, see Alonzo S. Church.

File:Alonzo Church.jpg

Alonzo Church (1903-1995)

Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science.

Life

Alonzo Church was born in Washington, DC, he received a bachelor's degree from Princeton University in 1924, completing his Ph.D. there in 1927, under Oswald Veblen. After a postdoc at Göttingen, he taught at Princeton, 1929–1967, and at the University of California, Los Angeles, 1967–1990.

Church's doctoral students were an extraordinarily accomplished lot, including C. Anthony Anderson, Martin Davis, Leon Henkin, John George Kemeny, Stephen Kleene, Michael O. Rabin, Hartley Rogers, Jr, J. Barkley Rosser, Dana Scott, Raymond Smullyan, and Alan Turing. See [1].

He died in 1995 and was buried in Princeton Cemetery.

Mathematical work

Church is best known for the following accomplishments:

• His proof that Peano arithmetic and first-order logic are undecidable. The latter result is known as Church's theorem.
• His articulation of what has come to be known as Church's thesis.
• He was the founding editor of the Journal of Symbolic Logic, editing its reviews section until 1979.
• His creation of the lambda calculus.

The lambda calculus emerged in his famous 1936 paper showing the existence of an "undecidable problem." This result preempted Alan Turing's famous work on the halting problem which also demonstrated the existence of a problem unsolvable by mechanical means. He and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative "mechanical processes for computation." This resulted in the Church-Turing thesis.

The lambda calculus influenced the design of the LISP programming language and functional programming languages in general. The Church encoding is named in his honor.

Church-Turning thesis

In computability theory the Church–Turing thesis (also known as Church's thesis, Church's conjecture and Turing's thesis) is a hypothesis about the nature of computers, such as a digital computer or a human with a pencil and paper following a set of rules. The thesis claims that any calculation that is possible can be performed by an algorithm running on a computer, provided that sufficient time and storage space are available. The thesis cannot be mathematically proven; it is sometimes proposed as a physical law or as a definition.

Informally the Church–Turing thesis states that our notion of algorithm can be made precise and computers can run those algorithms. Furthermore, a computer can theoretically run any algorithm; in other words, all ordinary computers are equivalent to each other in terms of theoretical computational power, and it is not possible to build a calculation device that is more powerful than the simplest computer (a Turing machine). Note that this formulation of power disregards practical factors such as speed or memory capacity; it considers all that is theoretically possible, given unlimited time and memory.

The thesis, named after Alonzo Church and Alan Turing, was first proposed by Church in 1934, then referencing the class of λ-definable functions, but didn't gain acceptance until Turing defined the equivalent but much more convincing Turing-computable functions.

Formal statement

The thesis can be stated as:

"Every 'function which would naturally be regarded as computable' can be computed by a Turing machine."

The term effectively calculable is commonly used for a function that is naturally regarded as computable, for example one for which an algorithm is provided. Due to the vagueness of the concept of effective calculability, the Church–Turing thesis cannot formally be proven. "Disproof" would be possible only if humanity found ways of building hypercomputers whose results are accepted as "computable"; as such, the thesis, although referencing mathematical objects, is properly in the realm of science rather than mathematics.

Any non-interactive computer program can be translated into a Turing machine, and any Turing machine can be translated into any Turing complete programming language, so the thesis is equivalent to saying that any Turing complete programming language is sufficient to express any algorithm. This also means that any solvable problem can therefore be reduced to previously solved problems (the Turing machine instruction set) or by definition is unsolvable.

Variations of the thesis exist; for example, the Physical Church–Turing thesis (PCTT) states:

"Every function that can be physically computed can be computed by a Turing machine."

Another variation is the Strong Church–Turing Thesis (SCTT), which is not due to Church or Turing, but rather was realized gradually in the development of complexity theory. It states (cf. Bernstein, Vazirani 1997):

"Any 'reasonable' model of computation can be efficiently simulated on a probabilistic Turing machine."

The word 'efficiently' here means up to polynomial-time reductions. The Strong Church–Turing Thesis, then, posits that all 'reasonable' models of computation yield the same class of problems that can be computed in polynomial time. Assuming the conjecture that probabilistic polynomial time (BPP) equals deterministic polynomial time (P), the word 'probabilistic' is optional in the Strong Church–Turing Thesis.

If quantum computers are physically possible, they could invalidate the Strong Church–Turing Thesis, since it is also conjectured that quantum polynomial time (BQP) is larger than BPP. In other words, there are efficient quantum algorithms that perform tasks that are not known to have efficient probabilistic algorithms; for example, factoring integers.

History

One of the important problems for logicians in the 1930s was David Hilbert's Entscheidungsproblem, which asked if there was a mechanical procedure for separating mathematical truths from mathematical falsehoods.

In the course of studying the problem, Alonzo Church and his student Stephen Kleene introduced the notion of λ-definable functions, and were able to prove that several large classes of functions frequently encountered in number theory were λ-definable. Armed with this evidence, Church proposed to Kurt Gödel that one should think of the class of "effectively computable" functions (i.e., functions computable by some concrete algorithm) as the λ-definable functions, and claimed to solve the Entscheidungsproblem by proving that there was no λ-definable function separating truths from falsehoods. Kurt Gödel, however, was not convinced that this was true, calling the proposal "thoroughly unsatisfactory".

Then, in his 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" Alan Turing tried to capture the notion of "effective computability", with the introduction of Turing machines. He proposed, like Church and Kleene before him, that his formal notion of mechanical computing agent was really the correct one. However, this time Gödel was convinced, writing about Alan Turing's machines:

"That this really is the correct definition of mechanical computability was established beyond any doubt by Turing."

By this point, it was known that the classes of functions defined by λ-calculus and Turing machines coincided (Turing 1936, 263ff), so the two proposals were essentially identical. However, although Church's claim predated Turing's, it was Turing who, in the opinions of Gödel and others, finally gave a convincing argument for why these functions really contained all functions that one would be inclined to call "effectively computable", and the thesis was gaining acceptance.

In his 1943 paper Recursive Predicates and Quantifiers (reprinted in The Undecidable, p. 255) Stephen Kleene proposed his "THESIS I":

"This heuristic fact [general recursive functions are effectively calculable]...led Church to state the following thesis [Kleene's footnote 22]. The same thesis is implicit in Turing's description of computing machines [Kleene's footnote 23].

"THESIS I. Every effectively calculable function (effectively decidable predicate) is general recursive [Kleene's italics]

"Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it..." (Kleene in Undecidable, p. 274)

Kleene's footnote 22 references the paper by Alonzo Church and his footnote 23 references the paper by Alan Turing. He goes on to note that:

"...the thesis has the character of an hypothesis — a point emphasized by Post and by Church" [his footnote 24, The Undecidable, p. 274. His references are to Post's paper (1936) and to Church's paper Formal definitions in the theory of ordinal numbers, Fund. Math. vol 28 (1936) pp.11-21 (see ref. #2, p. 286, Undecidable)].

Success of the thesis

Since that time, many other formalisms for describing effective computability have been proposed. The three most commonly quoted are the recursive functions, the lambda calculus, and the Turing machine. Stephen Kleene (1952) adds to the list the functions " reckonable in the system S₁" of Kurt Gödel 1936, and Emil Post's (1943, 1946) "canonical [also called normal] systems" (cf Kleene (1952) p. 320). Since Kleene (1952) the various register machines, the various Turing machine-like models such as the Post-Turing machine, combinatory logic, and Markov algorithms have been added to the list. Gurevich adds the pointer machine model of Kolmogorov and Uspensky (1953, 1958): "...they just wanted to ... convince themselves that there is no way to extend the notion of computable function." (Gurevich (1988) p. 2). Gandy (1980) proposed four principles "the formulation [of which] is quite abstract, and can be applied to all kinds of automata and to algebraic systems. It is proved that if a device satisfies the principles then its successive states form a computable sequence." (Gurevich (2000), p. 4).

All these systems have been shown to compute the same functions as Turing machines; systems like this are called Turing-complete. Because all these different attempts of formalizing the concept of algorithm have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S₁":

"It may also be shown that a function which is computable ['reckonable'] in one of the systems S_i, or even in a system of transfinite type, is already computable [reckonable] in S₁. Thus the concept 'computable' [ 'reckonable' ] is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined" (translation of Gödel (1936) by Davis in The Undecidable p. 83, differing in the use of the word 'reckonable' in the translation in Kleene (1952) p. 321)

However, the thesis is a definition and not a theorem, and hence cannot be proved true. The physical version could, however, be disproved if a method could be exhibited which is universally accepted as being an effective algorithm but which cannot be performed on a Turing machine.

In the early twentieth century, mathematicians often used the informal phrase effectively computable, so it was important to find a good formalization of the concept. Modern mathematicians instead use the well-defined term Turing computable (or computable for short). Since the undefined terminology has faded from use, the question of how to define it is now less important.

The success of the Church–Turing thesis prompted supertheses that extend the thesis, including the strong Church–Turing thesis mentioned earlier.

Philosophical implications

The Church–Turing thesis has been alleged to have some profound implications for the philosophy of mind. There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of hypercomputation. When applied to physics, the thesis has several possible meanings:

1. The universe is equivalent to a Turing machine or is weaker; thus, computing non-recursive functions is physically impossible. This has also been termed the strong Church–Turing thesis (not to be confused with the previously mentioned SCTT) and is a foundation of digital physics.
2. The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer. For example, a universe in which physics involves real numbers, as opposed to computable reals, might fall into this category.
3. The universe is a hypercomputer, and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all quantum mechanical events are Turing-computable, although it is known that rigorous models such as quantum Turing machines are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.) John Lucas and, more famously, Roger Penrose have suggested that the human mind might be the result of quantum hypercomputation, although there is no scientific evidence for this proposal.

There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept.

Non computable functions

One can formally define functions that are not computable. For example, there are functions on natural numbers that produce values, usually very large values, that cannot be computed. The most famous such function is the busy beaver. To make things simple, let us say that it describes the largest amount of work that a Turing machine can produce with limited resources (i.e., no more than n states). At best, researchers can give lower bounds for the busy beaver function for the smallest values of n: 2, 3, 4, 5, and (painfully) 6 states.

The basic problem is that to calculate the nth busy beaver number, you have to run every possible n-state Turing machine and see which one does the most work before stopping. To make matters worse, many Turing machines never stop at all — and for some of those it's impossible to know in advance whether they will stop or not. This last connection to the halting problem makes computing the busy beaver function theoretically impossible, rather than merely impractical.

See also

• Church-Turing thesis
• Church-Turing-Deutsch principle
• Higher-order logic

References

ISBN links support NWE through referral fees

• Bernstein, E. & Vazirani, U. Quantum complexity theory, SIAM Journal on Computing 26(5) (1997) 1411?1473
• Andreas Blass and Yuri Gurevich (2003), Algorithms: A Quest for Absolute Definitions, Bulletin of European Association for Theoretical Computer Science 81, 2003. Includes an excellent bibliography of 56 references.
• Church, Alonzo. 1932, "A set of Postulates for the Foundation of Logic", Annals of Mathematics, second series, 33, 346-366.
• --------, 1936, "An Unsolvable Problem of Elementary Number Theory", American Journal of Mathematics, 58, 345-363.
• --------, 1936, "A Note on the Entscheidungsproblem", Journal of Symbolic Logic, 1, 40-41.
• --------, 1941, The Calculi of Lambda-Conversion, Princeton: Princeton University Press.
• --------, 1996. Introduction to mathematical logic. Princeton landmarks in mathematics and physics. Princeton, N.J.: Princeton University Press. ISBN 0691029067 ISBN 9780691029061
• Martin Davis editor, The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, Raven Press, New York, 1965. All the original papers are here including those by Gödel, Church, Turing, Rosser, Kleene, and Post mentioned in this section. Valuable commentary by Davis prefaces most papers.
• Robin Gandy, 1980, Church's Thesis and the Principles for Mechanisms, reprinted in H.J. Barwise, H.J. Keisler and K. Kunen, eds., (1980), The Kleene Symposium, North-Holland Publishing Company, pp. 123-148.
• Gödel, K., 1934, On Undecidable Propositions of Formal Mathematical Systems, lecture notes taken by Kleene and Rosser at the Institute for Advanced Study, reprinted in Davis, M. (ed.) 1965, The Undecidable, New York: Raven Press.
• Gödel, K., 1936, "On The Length of Proofs", reprinted in Davis, M. (ed.) 1965, The Undecidable, New York: Raven Press (pp.82-83), "Translated by the editor from the original article in Ergenbnisse eines mathematishen Kolloquiums, Heft 7 (1936) pp. 23-24." Cited by Kleene (1952) as "Über die Lāange von Beweisen", in Ergebnisse eines math. Koll, etc.
• Yuri Gurevich, 1988, On Kolmogorov Machines and Related Issues, Bulletin of European Assoc. for Theor. Comp. Science, Number 35, June 1988, 71-82.
• Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pages 77-111. Includes bibliography of 33 sources.
• Herbrand, J., 1932, "Sur la non-contradiction de l'arithmétique", Journal fur die reine und angewandte Mathematik, 166, 1-8.
• Hofstadter, Douglas R., Gödel, Escher, Bach: an Eternal Golden Braid, Chapter 17.
• Kleene, S.C., 1935, "A Theory of Positive Integers in Formal Logic", American Journal of Mathematics, 57, 153-173, 219-244.
• Kleene, S.C., 1936, "Lambda-Definability and Recursiveness", Duke Mathematical Journal 2, 340-353.
• Knuth, Donald E.,The Art of Computer Programming, Second Edition, Volume 1/Fundamental Algorithms, Addison-Wesley, 1973.
• Markov, A.A., 1960, "The Theory of Algorithms", American Mathematical Society Translations, series 2, 15, 1-14. A. A. Markov (1954) Theory of algorithms. Also: [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e. Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 60-51085.]
• Pour-El, M.B. & Richards, J.I., 1989, Computability in Analysis and Physics, Springer Verlag.
• Robert Soare, (1995-6), Computability and Recursion, Bulletin of Symbolic Logic 2 (1996), 284—321.
• Turing, A.M., 1936, "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, Series 2, 42 (1936-37), pp.230-265.

Sources and external links

• John J. O'Connor and Edmund F. Robertson. Alonzo Church at the MacTutor archive
• H B Enderton, In memoriam: Alonzo Church
• Alonzo Church at the Mathematics Genealogy Project
• "Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo Rota. Contains a section on Church at Princeton.
• Interview of Church about his time at Princeton
• Archived papers
• Detailed info on the Church–Turing Hypothesis (Stanford Encyclopedia of Philosophy)

General Philosophy Sources

• Stanford Encyclopedia of Philosophy. Retrieved July 14, 2007.
• The Internet Encyclopedia of Philosophy. Retrieved July 14, 2007.
• Philosophy Sources on Internet EpistemeLinks. Retrieved July 14, 2007.
• Guide to Philosophy on the Internet. Retrieved July 14, 2007.
• Paideia Project Online. Retrieved July 14, 2007.
• Project Gutenberg. Retrieved July 14, 2007.

Academic Genealogy
Notable teachers	Notable students
Oswald Veblen	C. Anthony Anderson Peter Andrews George Alfred Barnard Martin Davis Leon Henkin David Kaplan John George Kemeny Stephen Kleene John McCarthy (computer scientist) Michael O. Rabin Hartley Rogers, Jr J. Barkley Rosser Dana Scott Raymond Smullyan Alan Turing

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