Notes on Writing Proofs
... we do not attempt to prove but rather which we accept as given. By a proposition, we mean a statement that is either true or false (but not both). Euclid and Aristotle thought of axioms as propositions which were “obviously” true. One of Euclid’s axioms, for example, was “It shall be possible to dra ...
... we do not attempt to prove but rather which we accept as given. By a proposition, we mean a statement that is either true or false (but not both). Euclid and Aristotle thought of axioms as propositions which were “obviously” true. One of Euclid’s axioms, for example, was “It shall be possible to dra ...
The Semantic Complexity of some Fragments of English
... (The depth of any expression , denoted % , is the maximum level of nesting of function-symbols in , with non-functional expressions assigned depth 0.) Consider the familiar A-ordering defined by setting if and only if: (i) % % , (ii) Vars Vars , and ...
... (The depth of any expression , denoted % , is the maximum level of nesting of function-symbols in , with non-functional expressions assigned depth 0.) Consider the familiar A-ordering defined by setting if and only if: (i) % % , (ii) Vars Vars , and ...
Monadic Second-Order Logic with Arbitrary Monadic Predicates⋆
... For the sake of readability, we often define predicates as P = (Pn )n∈N with Pn ⊆ {0, 1}n . In such case we can see P as a language over {0, 1}, which contains exactly one word for each length. Also, we often define predicates P = (Pn )n∈N with Pn ∈ An for some finite alphabet A. This is not formall ...
... For the sake of readability, we often define predicates as P = (Pn )n∈N with Pn ⊆ {0, 1}n . In such case we can see P as a language over {0, 1}, which contains exactly one word for each length. Also, we often define predicates P = (Pn )n∈N with Pn ∈ An for some finite alphabet A. This is not formall ...
Circuit principles and weak pigeonhole variants
... It is unknown over S21 whether sPHP (PV ) implies iPHP (PV ), which is why an analogous result does not follow immediately from Jeřábek’s result. As far as the authors know, it is open whether RSA is vulnerable to quasi-polynomial local search attacks; the main problem with breaking RSA using such ...
... It is unknown over S21 whether sPHP (PV ) implies iPHP (PV ), which is why an analogous result does not follow immediately from Jeřábek’s result. As far as the authors know, it is open whether RSA is vulnerable to quasi-polynomial local search attacks; the main problem with breaking RSA using such ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
... (the biconditional as a whole) is indeed tautologically or at least linguistically true because the set description is a legitimate naming of a collective individual object whose members satisfy the relevant predicate. Considered a singular proposition a set description can only lead to inconsistenc ...
... (the biconditional as a whole) is indeed tautologically or at least linguistically true because the set description is a legitimate naming of a collective individual object whose members satisfy the relevant predicate. Considered a singular proposition a set description can only lead to inconsistenc ...
Note 2 - inst.eecs.berkeley.edu
... a proof consists of a sequence of logical deductions: Simple steps that apply the rules of logic. This results in a sequence of statements where each successive statement is necessarily true if the previous statements were true. This property is enforced by the rules of logic: Each statement follows ...
... a proof consists of a sequence of logical deductions: Simple steps that apply the rules of logic. This results in a sequence of statements where each successive statement is necessarily true if the previous statements were true. This property is enforced by the rules of logic: Each statement follows ...
Note 2 - EECS: www-inst.eecs.berkeley.edu
... a proof consists of a sequence of logical deductions: Simple steps that apply the rules of logic. This results in a sequence of statements where each successive statement is necessarily true if the previous statements were true. This property is enforced by the rules of logic: Each statement follows ...
... a proof consists of a sequence of logical deductions: Simple steps that apply the rules of logic. This results in a sequence of statements where each successive statement is necessarily true if the previous statements were true. This property is enforced by the rules of logic: Each statement follows ...
on the Complexity of Quantifier-Free Fixed-Size Bit-Vector
... 0 and is incremented by 1 in each clock cycle up to a value of n. When the counter reaches a value of n, it does not change anymore and the output of the atomic Sequential Circuit is set to the same value as the output in the previous cycle. A counter like this can be realized with rlog2 pnqs gates, ...
... 0 and is incremented by 1 in each clock cycle up to a value of n. When the counter reaches a value of n, it does not change anymore and the output of the atomic Sequential Circuit is set to the same value as the output in the previous cycle. A counter like this can be realized with rlog2 pnqs gates, ...
Advanced Topics in Theoretical Computer Science
... Let T be the conjunction of all implication formulas mentioned above. As M has finitely many transitions and the alphabet is finite, this conjunction is finite as well, and thus a formula of first order logic. ...
... Let T be the conjunction of all implication formulas mentioned above. As M has finitely many transitions and the alphabet is finite, this conjunction is finite as well, and thus a formula of first order logic. ...
Algebraic Proof Systems
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
Towards NP−P via Proof Complexity and Search
... Note that the theorem states more than it is difficult to search for a short proof; instead, it is already hard to determine whether such a proof exists (assuming P 6= NP ). The proof of this theorem by [4] uses a reduction from the Minimum Monotone Circuit Satisfying Assignment problem [42]. 1−o(1) ...
... Note that the theorem states more than it is difficult to search for a short proof; instead, it is already hard to determine whether such a proof exists (assuming P 6= NP ). The proof of this theorem by [4] uses a reduction from the Minimum Monotone Circuit Satisfying Assignment problem [42]. 1−o(1) ...
Module 31
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
Module 31
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
Model theory makes formulas large
... lower bound for both the classical Łoś-Tarski theorem and its variants for classes of finite forests and all classes of finite structures that contain all trees (but not for classes of finite structures of bounded degree). We prove a further lower bound that is concerned with the classical Feferman ...
... lower bound for both the classical Łoś-Tarski theorem and its variants for classes of finite forests and all classes of finite structures that contain all trees (but not for classes of finite structures of bounded degree). We prove a further lower bound that is concerned with the classical Feferman ...
Module 31
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
... – In all possible derivations of x, we have shown that x belongs to (L(G1))* – Thus, we have proven the inductive case ...
Document
... Lemma: A minor theorem used as a stepping-stone to proving a major theorem. Corollary: A minor theorem proved as an easy consequence of a major theorem. Conjecture: A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory: The set of all ...
... Lemma: A minor theorem used as a stepping-stone to proving a major theorem. Corollary: A minor theorem proved as an easy consequence of a major theorem. Conjecture: A statement whose truth value has not been proven. (A conjecture may be widely believed to be true, regardless.) Theory: The set of all ...
CSI 2101 / Rules of Inference (§1.5)
... Definition: An integer n is even iff ∃ integer k such that n = 2k Definition: An integer n is odd iff ∃ integer k such that n = 2k+1 Definition: Let k and n be integers. We say that k divides n (and write k | n) if and only if there exists an integer a such that n = ka. Definition: An integer n is p ...
... Definition: An integer n is even iff ∃ integer k such that n = 2k Definition: An integer n is odd iff ∃ integer k such that n = 2k+1 Definition: Let k and n be integers. We say that k divides n (and write k | n) if and only if there exists an integer a such that n = ka. Definition: An integer n is p ...
author`s
... In 55, we show, by analyzing our proof of the automata-theoreticcharacterization of spectra,that many (ah?) spectraare the spectrum of a sentence which has at most one model of each finite cardinality. In $6, we make use of the automata-theoreticcharacterizationof spectra to show that if spectraare ...
... In 55, we show, by analyzing our proof of the automata-theoreticcharacterization of spectra,that many (ah?) spectraare the spectrum of a sentence which has at most one model of each finite cardinality. In $6, we make use of the automata-theoreticcharacterizationof spectra to show that if spectraare ...
3.3 Inference
... introduced the variables m and n. We used only well-known consequences of the fact that they were in the universe of even numbers in our proof. Thus we felt justified in asserting that what we concluded about m and n is true for any pair of integers. We might say that we were treating m and n as gene ...
... introduced the variables m and n. We used only well-known consequences of the fact that they were in the universe of even numbers in our proof. Thus we felt justified in asserting that what we concluded about m and n is true for any pair of integers. We might say that we were treating m and n as gene ...
Section 3.6: Indirect Argument: Contradiction and Contraposition
... Proof. Suppose the negation of the theorem. Specifically, let P denote the set of prime integers. Then the negation would be ∃a, b, c ∈ P, a2 + b2 = c2 . We shall prove this by considering the different parities of a, b and c. (New contradiction proof) First assume that a and b are both odd. Then a2 ...
... Proof. Suppose the negation of the theorem. Specifically, let P denote the set of prime integers. Then the negation would be ∃a, b, c ∈ P, a2 + b2 = c2 . We shall prove this by considering the different parities of a, b and c. (New contradiction proof) First assume that a and b are both odd. Then a2 ...
Slide 1
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
Post Systems in Programming Languages Pr ecis 1 Introduction
... A term is provable in a Post system if we can nd a proof of it. The set of words provable from a Post system forms the language derived by the system. Sometimes it is necessary to consider the language derived by a Post system to be the set of strings from a subset of the signs which are provable. ...
... A term is provable in a Post system if we can nd a proof of it. The set of words provable from a Post system forms the language derived by the system. Sometimes it is necessary to consider the language derived by a Post system to be the set of strings from a subset of the signs which are provable. ...
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: ...
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 ...
Kolmogorov complexity
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.