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Lambda calculus | A Wisdom Archive on Lambda calculus | | Lambda calculus A selection of articles related to Lambda calculus | |
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lambda calculus
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| ARTICLES RELATED TO Lambda calculus | | | |  Lambda calculus: Encyclopedia II - Halting problem - Relationship with Gödel's incompleteness theoremThe concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that a complete, consistent and sound axiomatization of all statements about natural numbers is unachievable. The "sound" part is the weakening: it means that we require ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Relationship with Gödel's incompleteness theorem |
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| | | | | | Lambda calculus: Encyclopedia II - Functional programming - Comparison with imperative programming Functional programming can be contrasted with imperative programming. Functional programming appears to be missing several constructs often (though incorrectly) considered essential to an imperative language such as C or Pascal. For example, in strict functional programming, there is no explicit memory allocation and no explicit variable assignment. However, these operations occur automatically when a function is invoked: memory allocation occurs to create space for the parameters and the return value, and assignment occurs to copy the param ...
See also:Functional programming, Functional programming - Introduction, Functional programming - History, Functional programming - Comparison with imperative programming, Functional programming - Functional programming languages, Functional programming - Higher-order functions, Functional programming - Speed and space considerations, Functional programming - Functional languages Read more here: » Functional programming: Encyclopedia II - Functional programming - Comparison with imperative programming |
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| | | | | | | Lambda calculus: Encyclopedia II - Functional programming - Comparison with imperative programming Functional programming can be contrasted with imperative programming. Functional programming appears to be missing several constructs often (though incorrectly) considered essential to an imperative language such as C or Pascal. For example, in strict functional programming, there is no explicit memory allocation and no explicit variable assignment. However, these operations occur automatically when a function is invoked: memory allocation occurs to create space for the parameters and the return value, and assignment occurs to copy the param ...
See also:Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages Read more here: » Functional programming: Encyclopedia II - Functional programming - Comparison with imperative programming |
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| | | | | | | Lambda calculus: Encyclopedia II - Functional programming - Higher-order functions A powerful mechanism used in functional programming is the notion of higher-order functions. Functions are higher-order when they can take other functions as arguments, and/or return functions as results. (The differential operator in calculus is a common example of a function that maps a function to a function.)
Higher-order functions are closely related to first-class functions, in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two i ...
See also:Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages Read more here: » Functional programming: Encyclopedia II - Functional programming - Higher-order functions |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Sketch of proof The proof proceeds by reductio ad absurdum. We start with assuming that there is a function halt(a, i) that returns true if the algorithm represented by the string a halts when given as input the string i, and returns false otherwise. (The existence of the universal Turing machine proves that every possible algorithm corresponds to at least one such string.) Given this algorithm we can construct another algorithm trouble(s) as follows:
function trouble(string s)
if halt(s, s) = false
r ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Sketch of proof |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Formalization of the halting problem In his original proof Turing formalized the concept of algorithm by introducing Turing machines. However, the result is in no way specific to them; it applies equally to any other model of computation that is equivalent in its computational power to Turing machines, such as Markov algorithms, Lambda calculus, Post systems or register machines.
What is important is that the formalization allows a straightforward mapping of algorithms to some data type that the algorithm can operate upon. For example, if the formalism lets algori ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Formalization of the halting problem |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Recognizing partial solutions No program can solve the halting problem. There are programs that give correct answers for some instances of it, and run forever on all other instances. A program that returns answers for some instances of the halting problem might be called a partial halting solver (PHS). Can we recognize a correct PHS when we see it? Let the PHS recognition problem be this: given a PHS, determine whether it returns only correct answers. This problem sounds like it might be easier than the halting problem itself. It is not. It is ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Recognizing partial solutions |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Can humans solve the halting problem? It might seem like humans could solve the halting problem. After all, a programmer can often look at a program and tell whether it will halt. It is useful to understand why this cannot be true. For simplicity, we will consider the halting problem for programs with no input, which is also undecidable.
To "solve" the halting problem means to be able to look at any program and tell whether it halts. It is not enough to be able to look at some programs and decide. Humans may also not be able to solve the halting problem, due ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Can humans solve the halting problem? |
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| | | | | Lambda calculus: Encyclopedia II - List of mathematics categories - Mathematicians categories
List of mathematics categories - 0–9.
10th century mathematicians -- 11th century mathematicians -- 12th century mathematicians -- 13th century mathematicians -- 14th century mathematicians -- 15th century mathematicians -- 16th century mathematicians -- 17th century mathematicians -- 18th century mathematicians -- 19th century mathematicians -- 20th century mathematicians -- 21st century mathematicians -- 5th century mathema ...
See also:List of mathematics categories, List of mathematics categories - Mathematics categories, List of mathematics categories - 0–9, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - J, List of mathematics categories - K, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - O, List of mathematics categories - P, List of mathematics categories - Q, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - U, List of mathematics categories - V, List of mathematics categories - W, List of mathematics categories - Z, List of mathematics categories - Mathematicians categories, List of mathematics categories - 0–9, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - J, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - P, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - U, List of mathematics categories - V, List of mathematics categories - W, List of mathematics categories - Mathematics-related categories, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - K, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - O, List of mathematics categories - P, List of mathematics categories - Q, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - V, List of mathematics categories - W Read more here: » List of mathematics categories: Encyclopedia II - List of mathematics categories - Mathematicians categories |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Common pitfalls Many students, upon analyzing the above proof, ask whether there might be an algorithm that can return a third option, such as "undecidable." This reflects a misunderstanding of decidability. It is easy to construct one algorithm that always answers "halts" and another that always answers "doesn't halt." For any specific program and input, one of these two algorithms answers correctly, even though nobody may know which one. The difficulty of the halting problem lies not in particular programs, but in the requ ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Common pitfalls |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Importance and consequences The importance of the halting problem lies in the fact that it is the first problem to be proved undecidable. Subsequently, many other such problems have been described; the typical method of proving a problem to be undecidable is with the technique of reduction. To do this, the computer scientist shows that if a solution to the new problem was found, it could be used to decide an undecidable problem (by transforming instances of the undecidable problem into instances of the new problem). Since we already know that no method can decid ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Importance and consequences |
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| | | | | | Lambda calculus: Encyclopedia II - Functional programming - History Lambda calculus, invented by Alonzo Church in the 1930s, provides a theoretical framework for describing functions and their evaluation. Though it is a mathematical abstraction rather than a programming language, lambda calculus forms the basis of almost all functional programming languages today.
The first computer-based functional programming language was Information Processing Language (IPL), developed by Newell, Shaw, and Simon at RAND Corporation for the JOHNNIAC computer in the mid-1950s. A much-improved functional programming l ...
See also:Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages Read more here: » Functional programming: Encyclopedia II - Functional programming - History |
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| | | | | Lambda calculus: Encyclopedia II - Halting problem - Formal statement One possible way of formally stating the halting problem is as follows:
Given a Gödel numbering of the computable functions,
with the Cantor pairing function,
the set is called the halting set.
The problem of deciding whether the halting set is recursive or not is called the halting problem. As the set is recursively enumerable the halting problem is not solvable by a computable function.
Alternative equivalent formulations, for inst ...
See also:Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes Read more here: » Halting problem: Encyclopedia II - Halting problem - Formal statement |
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| | | | | | Lambda calculus: Encyclopedia II - Lambda calculus - Undecidability of equivalence There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which the unsolvability could be proven. Of course, in order to do so, the notion of algorithm has to be cleanly defined; Church used a definition via recursive functions, which is now known to ...
See also:Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages Read more here: » Lambda calculus: Encyclopedia II - Lambda calculus - Undecidability of equivalence |
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