REGULAR COST FUNCTIONS, PART I: LOGIC AND ALGEBRA
... using two dual forms of automata is an important concept in the theory of regular cost functions. The study of MSO+U has been pursued in several directions. Indeed, the general problem of the satisfaction of MSO+U is a challenging open problem. One partial result concerns the decision of WMSO+U (the ...
... using two dual forms of automata is an important concept in the theory of regular cost functions. The study of MSO+U has been pursued in several directions. Indeed, the general problem of the satisfaction of MSO+U is a challenging open problem. One partial result concerns the decision of WMSO+U (the ...
Approximate equivalence relations.
... R◦n = R◦n−1 ◦ R. Assume R(b) is finite, and |R◦3 (a)|/|R(b)| ≤ k for a, b ∈ G. Then there exists a symmetric, reflexive relation S such that S ◦m ⊂ R◦4 , and for all a ∈ G outside an -slice U , |S(a)| ≥ Ok,m (1)|R(a)|. Moreover S is 0-definable, uniformly in (G, R), in a language with cardinality c ...
... R◦n = R◦n−1 ◦ R. Assume R(b) is finite, and |R◦3 (a)|/|R(b)| ≤ k for a, b ∈ G. Then there exists a symmetric, reflexive relation S such that S ◦m ⊂ R◦4 , and for all a ∈ G outside an -slice U , |S(a)| ≥ Ok,m (1)|R(a)|. Moreover S is 0-definable, uniformly in (G, R), in a language with cardinality c ...
(pdf)
... Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat’s Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. A ...
... Prime numbers are especially important for random number generators, making them useful in many algorithms. The Fermat Test uses Fermat’s Little Theorem to test for primality. Although the test is not guaranteed to work, it is still a useful starting point because of its simplicity and efficiency. A ...
MATH 115, SUMMER 2012 LECTURE 5 Last time:
... - defined congruence - listed a bunch of properties, similar to “=”, except that we can’t always “cancel” - defined complete residue system mod m - one representative per residue class - forgot to mention: if a ≡ b mod m, then (a, m) = (b, m). 1. Reduced Residue Systems and the φ-function Since we c ...
... - defined congruence - listed a bunch of properties, similar to “=”, except that we can’t always “cancel” - defined complete residue system mod m - one representative per residue class - forgot to mention: if a ≡ b mod m, then (a, m) = (b, m). 1. Reduced Residue Systems and the φ-function Since we c ...
Computability and Incompleteness
... Three themes are developed in this course. The first is computability, and its flip side, uncomputability or unsolvability. The informal notion of a computation as a sequence of steps performed according to some kind of recipe goes back to antiquity. In Euclid, one finds algorithmic procedures for c ...
... Three themes are developed in this course. The first is computability, and its flip side, uncomputability or unsolvability. The informal notion of a computation as a sequence of steps performed according to some kind of recipe goes back to antiquity. In Euclid, one finds algorithmic procedures for c ...
pdf version
... Since a, 2a, 3a,. . . (p − 1)a are all distinct mod p, then mod p they must just be 1, 2,. . . ,p − 1, possibly in some other order, so (a)(2a)(3a) · · · (p − 1)a ≡ (1)(2) · · · (p − 1) (mod p), that is [ap−1 − 1](1)(2) · · · (p − 1) ≡ 0 (mod p). ...
... Since a, 2a, 3a,. . . (p − 1)a are all distinct mod p, then mod p they must just be 1, 2,. . . ,p − 1, possibly in some other order, so (a)(2a)(3a) · · · (p − 1)a ≡ (1)(2) · · · (p − 1) (mod p), that is [ap−1 − 1](1)(2) · · · (p − 1) ≡ 0 (mod p). ...
Full text
... Theorem 5 allows immediate computation of the asymptotic proportion of l's (and 0ss) in Fibonacci sequences (see [4]). Indeed, denoting by p, q and p, q, respectively, the asymptotic proportions of 0!s and lfs in Ck and Bk, ...
... Theorem 5 allows immediate computation of the asymptotic proportion of l's (and 0ss) in Fibonacci sequences (see [4]). Indeed, denoting by p, q and p, q, respectively, the asymptotic proportions of 0!s and lfs in Ck and Bk, ...
Interpreting and Applying Proof Theories for Modal Logic
... The identity axioms, the basic structural rules, the logical rules ¬L, ¬R, ∧L and ∧R, the modal rules L and R, plus the classical structural rules of weakening, contraction and cut, form the Display proof system for the basic normal modal logic K. In order to obtain display calculi for other modal ...
... The identity axioms, the basic structural rules, the logical rules ¬L, ¬R, ∧L and ∧R, the modal rules L and R, plus the classical structural rules of weakening, contraction and cut, form the Display proof system for the basic normal modal logic K. In order to obtain display calculi for other modal ...
introduction to proofs
... some nonzero number of chips. If the second player removes the same number of chips from the other pile then we have reduced to a game with equal-sized piles, each with k or fewer chips, which the inductive hypothesis says is a win for the second player. Exercise 1.17 We shall use induction to prove ...
... some nonzero number of chips. If the second player removes the same number of chips from the other pile then we have reduced to a game with equal-sized piles, each with k or fewer chips, which the inductive hypothesis says is a win for the second player. Exercise 1.17 We shall use induction to prove ...
Paper - Department of Computer Science and Information Systems
... set of equations axiomatising the variety of Boolean algebras with operators and additional equations corresponding the axioms of L. A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissib ...
... set of equations axiomatising the variety of Boolean algebras with operators and additional equations corresponding the axioms of L. A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissib ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.