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... A X A X … X A is often denoted by Ak. Functions and relations • A relation from set A to set B is a subset of A X B. • A function (from A to B) is a relation R in which for every i, there is a unique j such that* is in R.
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... A X A X … X A is often denoted by Ak. Functions and relations • A relation from set A to set B is a subset of A X B. • A function (from A to B) is a relation R in which for every i, there is a unique j such that

Advanced Topics in Mathematics – Logic and Metamathematics Mr

... 1. Consider the following theorem: Suppose n is an integer larger than 1 and n is not prime. Then 2n 1 is not prime. (a) Identify the hypotheses and conclusion of the theorem. Are the hypotheses true when n = 6? What does the theorem tell you in this instance? Is it right? (b) What can you conclud ...

... 1. Consider the following theorem: Suppose n is an integer larger than 1 and n is not prime. Then 2n 1 is not prime. (a) Identify the hypotheses and conclusion of the theorem. Are the hypotheses true when n = 6? What does the theorem tell you in this instance? Is it right? (b) What can you conclud ...

Decision problem

... The Complexity Class NP An algorithm that chooses (by a really good guess!) some number of non-deterministic bits during its execution is called a non-deterministic algorithm We say that an algorithm A non-deterministically accepts a string x if there exists a choice of nondeterministic bits that l ...

... The Complexity Class NP An algorithm that chooses (by a really good guess!) some number of non-deterministic bits during its execution is called a non-deterministic algorithm We say that an algorithm A non-deterministically accepts a string x if there exists a choice of nondeterministic bits that l ...

- people.vcu.edu

... Complexity is the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties. It requires that we find distinctly different ways of interacting with systems. Distinctly different in the sense that when we make successful models ...

... Complexity is the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties. It requires that we find distinctly different ways of interacting with systems. Distinctly different in the sense that when we make successful models ...

Chapter 0 - Ravikumar - Sonoma State University

... Example 2: Show that the set of all binary strings of length n can be arranged in a way that every adjacent string differs in exactly one bit position, and further the first and the last string also differ in exactly one position. For n = 2, one such is 00, 01, 11, 10. ...

... Example 2: Show that the set of all binary strings of length n can be arranged in a way that every adjacent string differs in exactly one bit position, and further the first and the last string also differ in exactly one position. For n = 2, one such is 00, 01, 11, 10. ...

full text (.pdf)

... next step, then there is always a pebble on a nal state. Now we proceed to the formal proof of the correctness of this construction. Theorem 1. The following are equivalent: (i) The rule (2) is relationally valid. (ii) The rule (2) is derivable in PHL. (iii) The automaton accepts all strings. Proof ...

... next step, then there is always a pebble on a nal state. Now we proceed to the formal proof of the correctness of this construction. Theorem 1. The following are equivalent: (i) The rule (2) is relationally valid. (ii) The rule (2) is derivable in PHL. (iii) The automaton accepts all strings. Proof ...

Methods of Proof for Boolean Logic

... To prove Q from a disjunction, prove it from each disjunct separately. “There are irrational numbers b,c such that bc is rational”. 22 is either rational or irrational. 1. If rational, then take b=c= 2, known to be irrational. ...

... To prove Q from a disjunction, prove it from each disjunct separately. “There are irrational numbers b,c such that bc is rational”. 22 is either rational or irrational. 1. If rational, then take b=c= 2, known to be irrational. ...

Day04-FunctionsOnLanguages_DecisionProblems - Rose

... sequence of inference steps each of which was constructed using a sound inference rule. • A set of inference rules R is complete iff, given any set A of axioms, all statements that are entailed by A can be proved by applying the rules in R. ...

... sequence of inference steps each of which was constructed using a sound inference rule. • A set of inference rules R is complete iff, given any set A of axioms, all statements that are entailed by A can be proved by applying the rules in R. ...

Methods of Proof for Boolean Logic

... known to be a logical consequence of some already proven sentences, then you may assert Q in your proof. 2. Each step should be significant and easily understood (this is where ...

... known to be a logical consequence of some already proven sentences, then you may assert Q in your proof. 2. Each step should be significant and easily understood (this is where ...

Welcome to CS 39 - Dartmouth Computer Science

... • The proof you have just seen is one of the most profound results in the theory of computing. • Make sure you understand it. • Try to explain the proof to a friend. ...

... • The proof you have just seen is one of the most profound results in the theory of computing. • Make sure you understand it. • Try to explain the proof to a friend. ...

02-03-RegularHandout

... Regular languages are closed under intersection. See Supplementary Materials—Review of Mathematical Concepts for more formal definitions of these terms. Examples of Regular Languages L( a*b* ) = L( (a b) ) = L( (a b)* ) = L( (ab)*a*b*) = L = {w {a,b}* : |w| is even} L = {w {a,b}* : w contai ...

... Regular languages are closed under intersection. See Supplementary Materials—Review of Mathematical Concepts for more formal definitions of these terms. Examples of Regular Languages L( a*b* ) = L( (a b) ) = L( (a b)* ) = L( (ab)*a*b*) = L = {w {a,b}* : |w| is even} L = {w {a,b}* : w contai ...

Lexicographic Ordering

... The bit patterns representing a character can be interpreted as an unsigned integer and so the natural order of numbers can be used to order the characters. Collating sequence. The collating sequence of a character set is the order of the underlying bit representation. c1

... The bit patterns representing a character can be interpreted as an unsigned integer and so the natural order of numbers can be used to order the characters. Collating sequence. The collating sequence of a character set is the order of the underlying bit representation. c1

chap1sec7 - University of Virginia, Department of Computer

... logical equivalences. This is table 1 on page 89 of the text. Take a look at this table and compare it to our table for logical equivalences. You will find them to be strikingly similar. This is not a coincidence. Every set operation that we introduced could be expressed in set builder notation usin ...

... logical equivalences. This is table 1 on page 89 of the text. Take a look at this table and compare it to our table for logical equivalences. You will find them to be strikingly similar. This is not a coincidence. Every set operation that we introduced could be expressed in set builder notation usin ...

Comments on the use of the pumping lemma for regular languages

... If any y transversal are still needed to process w0, these are included in the z partition. w1 is the original unmodified w. By construction we chose w such that |w| m, and must transverse the loop due to the pigeon hole principle. The explicit “y loop” in w is taken only once in w because of the ...

... If any y transversal are still needed to process w0, these are included in the z partition. w1 is the original unmodified w. By construction we chose w such that |w| m, and must transverse the loop due to the pigeon hole principle. The explicit “y loop” in w is taken only once in w because of the ...

MAT 300 Mathematical Structures

... analyze the statement of the theorem, determining what the hypotheses and the conclusion are. The hypotheses are statements that we assume are true, and the conclusion is the statement that we must prove. Example 1. We will prove the following theorem: Theorem 2. Let a, b ∈ R. If 0 < a < b then a2 < ...

... analyze the statement of the theorem, determining what the hypotheses and the conclusion are. The hypotheses are statements that we assume are true, and the conclusion is the statement that we must prove. Example 1. We will prove the following theorem: Theorem 2. Let a, b ∈ R. If 0 < a < b then a2 < ...

Beautifying Gödel - Department of Computer Science

... He is not concerned with what mathematical object is referred to by 0÷0 . He is likewise willing to accept |– “)+x=(” as a sentence of TT , though it is not a theorem. We defined |– as a predicate on strings, so that |– s is a sentence no matter what string s may be. Similarly Gödel, in his formal p ...

... He is not concerned with what mathematical object is referred to by 0÷0 . He is likewise willing to accept |– “)+x=(” as a sentence of TT , though it is not a theorem. We defined |– as a predicate on strings, so that |– s is a sentence no matter what string s may be. Similarly Gödel, in his formal p ...

Recursive Enumerable

... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...

... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...

The Surprise Examination Paradox and the Second Incompleteness

... Berry’s paradox: consider the expression “the smallest positive integer not definable in under eleven words.” This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is ...

... Berry’s paradox: consider the expression “the smallest positive integer not definable in under eleven words.” This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is ...

The Surprise Examination Paradox and the Second Incompleteness

... in under eleven words”. This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is defined to be the length (in bits) of the shortest computer program that outputs x (an ...

... in under eleven words”. This expression defines that integer in under eleven words. To formalize Berry’s paradox, Chaitin uses the notion of Kolmogorov complexity. The Kolmogorov complexity K(x) of an integer x is defined to be the length (in bits) of the shortest computer program that outputs x (an ...

Document

... General phenomenon: can’t tell a book by its cover and you can’t tell what a program does just by its code... Rice’s Theorem: In general there is no way to tell anything about the input/output (I/O) behavior of a program P just given it code! ...

... General phenomenon: can’t tell a book by its cover and you can’t tell what a program does just by its code... Rice’s Theorem: In general there is no way to tell anything about the input/output (I/O) behavior of a program P just given it code! ...

MATHEMATICAL NOTIONS AND TERMINOLOGY

... What makes some problems computationally hard and others easy? • First, by understanding which aspect of the problem is at the root of the difficulty. • Second, you may be able to settle for less than a perfect solution to the problem. • Third, some problems are hard only in the worst case situation ...

... What makes some problems computationally hard and others easy? • First, by understanding which aspect of the problem is at the root of the difficulty. • Second, you may be able to settle for less than a perfect solution to the problem. • Third, some problems are hard only in the worst case situation ...

Automata, Languages, and Programming

... it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 ∈ · · · ∈ pn b = bpn ∈ qb = bq, where q is any program built from atomic actions p1 , . . . , pn . I ...

... it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 ∈ · · · ∈ pn b = bpn ∈ qb = bq, where q is any program built from atomic actions p1 , . . . , pn . I ...

Second-Order Logic and Fagin`s Theorem

... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...

... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...

In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov–Chaitin complexity, algorithmic entropy, or program-size complexity) of an object, such as a piece of text, is a measure of the computational resources needed to specify the object. It is named after Andrey Kolmogorov, who first published on the subject in 1963.For example, consider the following two strings of 32 lowercase letters and digits:abababababababababababababababab4c1j5b2p0cv4w1x8rx2y39umgw5q85s7The first string has a short English-language description, namely ""ab 16 times"", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 32 characters.More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings, like the abab example above, whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem.