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... equivalent notions of computability – Church for software, Turing for hardware. Their ideas were used to make precise the insights of Brouwer from 1900 that mathematics is based on fundamental human intuitions about numbers and human computation. This use of logic was able to shed light on the long ...
... equivalent notions of computability – Church for software, Turing for hardware. Their ideas were used to make precise the insights of Brouwer from 1900 that mathematics is based on fundamental human intuitions about numbers and human computation. This use of logic was able to shed light on the long ...
Humans, Computer, and Computational Complexity
... This first example, Turing’s Halting Problem, is a problem of practical as well as theoretical significance. Suppose you are trying to write a computer program. The code quickly becomes too complicated to keep it all in your mind at once. There are too many conditions and too many loops. You want t ...
... This first example, Turing’s Halting Problem, is a problem of practical as well as theoretical significance. Suppose you are trying to write a computer program. The code quickly becomes too complicated to keep it all in your mind at once. There are too many conditions and too many loops. You want t ...
HISTORY OF LOGIC
... Computability Theory • Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. • Developed to Recursion. • Computation Complexity Theory, was also characterized in logical terms as a result of investigations into descriptive complexity. ...
... Computability Theory • Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. • Developed to Recursion. • Computation Complexity Theory, was also characterized in logical terms as a result of investigations into descriptive complexity. ...
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... Since the language only provides two function symbols (all others would be an abbreviation for combinations of these) there are only four substitution axioms. This means that the theory Q is finitely axiomatizable. ...
... Since the language only provides two function symbols (all others would be an abbreviation for combinations of these) there are only four substitution axioms. This means that the theory Q is finitely axiomatizable. ...
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... An interesting consequence of Church's Theorem is that rst-order logic is incomplete (as a theory), because it is obviously consistent and axiomatizable but not decidable. This, however, is not surprising. Since there is an unlimited number of models for rst-order logic, there are plenty of rst-o ...
... An interesting consequence of Church's Theorem is that rst-order logic is incomplete (as a theory), because it is obviously consistent and axiomatizable but not decidable. This, however, is not surprising. Since there is an unlimited number of models for rst-order logic, there are plenty of rst-o ...
ppt
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
... statements are true, what other statements can you also deduce are true? • If I tell you that all men are mortal, and Socrates is a man, what can you deduce? ...
Extended Analog Computer and Turing machines - Hektor
... computation that come from different behaviors of computation processes in time have been considered. The first family contains models of computation on real numbers but in discrete time. These are, for example, the Analog Recurrent Neural Network [2] and the machines of Blum-Shub-Smale [3]. The sec ...
... computation that come from different behaviors of computation processes in time have been considered. The first family contains models of computation on real numbers but in discrete time. These are, for example, the Analog Recurrent Neural Network [2] and the machines of Blum-Shub-Smale [3]. The sec ...
NP Complexity
... compute the square, thus this type of reduction is Turing reduction – Further note that knowing squaring we can compute multiplication thus multiplication and squaring are equally hard! ...
... compute the square, thus this type of reduction is Turing reduction – Further note that knowing squaring we can compute multiplication thus multiplication and squaring are equally hard! ...
RoadMap
... • Moreover, formal logic provides us with a language to express statements, as well as various procedures to test whether some statement follows from other statements ...
... • Moreover, formal logic provides us with a language to express statements, as well as various procedures to test whether some statement follows from other statements ...
This Sentence is Wrong - Vienna Summer of Logic 2014
... “All Cretans are liars”, said Epimenides, a Cretan. But this means that his statement must be a lie too. But then it is false that Cretans are liars and the statement must be true. So what now? Sentences which say something about themselves can lead to paradoxes. The mathematician and logician Kurt ...
... “All Cretans are liars”, said Epimenides, a Cretan. But this means that his statement must be a lie too. But then it is false that Cretans are liars and the statement must be true. So what now? Sentences which say something about themselves can lead to paradoxes. The mathematician and logician Kurt ...
5.8.2 Unsolvable Problems
... Proof To prove that LC is not decidable, we assume that it is decidable by the TM MC and show that this implies the existence of a TM MH that decides LH , which has been shown previously not to exist. Thus, MC cannot exist. We consider two cases, the first in which B∗ is in not C and the second in w ...
... Proof To prove that LC is not decidable, we assume that it is decidable by the TM MC and show that this implies the existence of a TM MH that decides LH , which has been shown previously not to exist. Thus, MC cannot exist. We consider two cases, the first in which B∗ is in not C and the second in w ...
decidable
... that, given an encoding of TM T, halts and says “yes” if T halts on blank tape, but M fails to halt if T fails to halt on blank tape – The passing problem is semidecidable if there is a Turing machine M that, given an encoding of a student A, halts and says “yes” if A passes the course, but fails to ...
... that, given an encoding of TM T, halts and says “yes” if T halts on blank tape, but M fails to halt if T fails to halt on blank tape – The passing problem is semidecidable if there is a Turing machine M that, given an encoding of a student A, halts and says “yes” if A passes the course, but fails to ...
slides - Center for Collective Dynamics of Complex Systems (CoCo)
... – An axiomatic system defines an infinitely large network of all possible logical expressions (nodes) connected by logical relationships – Truth values are states of nodes and propagates through inference – Incompleteness theorems say that the final attractor of this network must be non-stationary i ...
... – An axiomatic system defines an infinitely large network of all possible logical expressions (nodes) connected by logical relationships – Truth values are states of nodes and propagates through inference – Incompleteness theorems say that the final attractor of this network must be non-stationary i ...
Foundations of Boundedly Rational Choices and Satisficing
... express our mathematics in a way that is as free as possible from philosophical concepts. We might in the end find ourselves agreeing with him about set theory. It is unnecessary.” Harold Edwards: Kronecker's Algorithmic Mathematics, The Mathematical Intelligencer, Vol. 31, Number 2, Spring, p. 14; ...
... express our mathematics in a way that is as free as possible from philosophical concepts. We might in the end find ourselves agreeing with him about set theory. It is unnecessary.” Harold Edwards: Kronecker's Algorithmic Mathematics, The Mathematical Intelligencer, Vol. 31, Number 2, Spring, p. 14; ...
1996TuringIntro
... adequacy of the Turing Test when interpreted as an operational definition of intelligence. However he regards this as a serious misinterpretation with unfortunate consequences, suggesting that it has led researchers in Artificial Intelligence to put far too much emphasis on the imitation of human pe ...
... adequacy of the Turing Test when interpreted as an operational definition of intelligence. However he regards this as a serious misinterpretation with unfortunate consequences, suggesting that it has led researchers in Artificial Intelligence to put far too much emphasis on the imitation of human pe ...
KaplanPsyOrf322S05Presentation
... “Imagine that you carry out the steps in a program for answering questions in a language you do not understand. I do not understand Chinese, so I imagine that I am locked in a room with a lot of boxes of Chinese symbols (the database), I get small bunches of Chinese symbols passed to me (questions i ...
... “Imagine that you carry out the steps in a program for answering questions in a language you do not understand. I do not understand Chinese, so I imagine that I am locked in a room with a lot of boxes of Chinese symbols (the database), I get small bunches of Chinese symbols passed to me (questions i ...
ppt
... we have enough information to find it. However, actually getting the answer from the inputs is not feasible due to the complexity of the problem. This is not like asking “will the human race exist in 5,000 years” – we do not have enough information to answer that ...
... we have enough information to find it. However, actually getting the answer from the inputs is not feasible due to the complexity of the problem. This is not like asking “will the human race exist in 5,000 years” – we do not have enough information to answer that ...
Universal language Decision problems Reductions Post`s
... 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. Theorem (Matiyasevich 1970) Hilbert’ ...
... 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. Theorem (Matiyasevich 1970) Hilbert’ ...