Specifying and Verifying Fault-Tolerant Systems
... SpecParams. The module is the basic unit of a TLA+ specification. It is a collection of declarations, definitions, assumptions, and theorems. The import statement imports the contents of the modules FiniteSets and Reals. This statement has almost the same effect as inserting the text of these modules i ...
... SpecParams. The module is the basic unit of a TLA+ specification. It is a collection of declarations, definitions, assumptions, and theorems. The import statement imports the contents of the modules FiniteSets and Reals. This statement has almost the same effect as inserting the text of these modules i ...
Full text
... appears at the end of the tree, but not initially. This will always be the case since the equations in (4) are satisfied by the trivial or zero solution. Although P 0 * 0, the p-pair (0, 0) is permissible after the first step. D The following question arises immediately. How do we use such a tree to ...
... appears at the end of the tree, but not initially. This will always be the case since the equations in (4) are satisfied by the trivial or zero solution. Although P 0 * 0, the p-pair (0, 0) is permissible after the first step. D The following question arises immediately. How do we use such a tree to ...
(formal) logic? - Departamento de Informática
... use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuitionist, a mathematical object (such as the solution of an equation) does not exist unless a finite construction (algorithm) can be given ...
... use methods of classical logic (as proofs by contradiction). However the philosophy behind intuitionistic logic is appealing for a computer scientist. For an intuitionist, a mathematical object (such as the solution of an equation) does not exist unless a finite construction (algorithm) can be given ...
ON THE BITS COUNTING FUNCTION OF REAL NUMBERS 1
... In this article, we will first prove related results when m is allowed to be an irrational number (in Theorem 1) via a study of the function Bn (x) = #{j ≤ n : xj = 1} where n ≥ 0 and x = (x−p · · · x−1 x0 . x1 x2 x3 · · · )2 is the binary expansion of x ≥ 0. The case where x is a rational number of ...
... In this article, we will first prove related results when m is allowed to be an irrational number (in Theorem 1) via a study of the function Bn (x) = #{j ≤ n : xj = 1} where n ≥ 0 and x = (x−p · · · x−1 x0 . x1 x2 x3 · · · )2 is the binary expansion of x ≥ 0. The case where x is a rational number of ...
ON THE PRIME NUMBER LEMMA OF SELBERG
... did not involve sharpening of Selberg’s lemma per se, and they did not explicitly study the relationship of its error term to that of the error in the Prime Number Theorem either. Remark 2.6. The sharpest version of (2) is due to V. Nevanlinna [14]: x ...
... did not involve sharpening of Selberg’s lemma per se, and they did not explicitly study the relationship of its error term to that of the error in the Prime Number Theorem either. Remark 2.6. The sharpest version of (2) is due to V. Nevanlinna [14]: x ...
printable
... • Prove that there are some problems that cannot be solved • Show that there are some problems that (are believed to) require an exponential amount of time to solve (NPComplete) • Examine some strategies for dealing with these problems • Along the way, learn how to model computation mathematically, ...
... • Prove that there are some problems that cannot be solved • Show that there are some problems that (are believed to) require an exponential amount of time to solve (NPComplete) • Examine some strategies for dealing with these problems • Along the way, learn how to model computation mathematically, ...
Sequentiality by Linear Implication and Universal Quantification
... of it into linear logic which is both correct and complete, thus fully relating the two formalisms. Computing in SMR is in the logic programming style: a goal of first order atoms (agents) has to be reduced to empty through backchaining by clauses, thus producing a binding for variables. Goals are o ...
... of it into linear logic which is both correct and complete, thus fully relating the two formalisms. Computing in SMR is in the logic programming style: a goal of first order atoms (agents) has to be reduced to empty through backchaining by clauses, thus producing a binding for variables. Goals are o ...
Document
... Theorem: Every n ∈ ℕ is the sum of distinct powers of two. Proof: By strong induction. Let P(n) be “n is the sum of distinct powers of two.” We prove that P(n) is true for all n ∈ ℕ. As our base case, we prove P(0), that 0 is the sum of distinct powers of 2. Since the empty sum of no powers of 2 is ...
... Theorem: Every n ∈ ℕ is the sum of distinct powers of two. Proof: By strong induction. Let P(n) be “n is the sum of distinct powers of two.” We prove that P(n) is true for all n ∈ ℕ. As our base case, we prove P(0), that 0 is the sum of distinct powers of 2. Since the empty sum of no powers of 2 is ...
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.