pdf
... Church and Turing in 1936 laid the foundations for computer science by defining 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 numb ...
... Church and Turing in 1936 laid the foundations for computer science by defining 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 numb ...
Chapter 1, Part I: Propositional Logic
... We write this as p⇔q or as p≡q where p and q are compound propositions. Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree. This truth table show ¬p ∨ q is equivalent to p → q. p ...
... We write this as p⇔q or as p≡q where p and q are compound propositions. Two compound propositions p and q are equivalent if and only if the columns in a truth table giving their truth values agree. This truth table show ¬p ∨ q is equivalent to p → q. p ...
Resources - CSE, IIT Bombay
... Tautologies are formulae whose truth value is always T, whatever the assignment is ...
... Tautologies are formulae whose truth value is always T, whatever the assignment is ...
(Jed Liu's solutions)
... • ∼ ψ. Using T (∼ ψ) and F (∼ ψ) derives F (ψ) and T (ψ), respectively. Since ψ has degree n, by the induction hypothesis, this branch can be further expanded to contain atomic conjugates. • ψ1 ∧ ψ2 , ψ1 ∨ ψ2 , or ψ1 ⊃ ψ2 . We can derive: F (ψ1 ∨ ψ2 ) F (ψ1 ⊃ ψ2 ) T (ψ1 ∧ ψ2 ) F (ψ1 ) T (ψ1 ) T (ψ1 ...
... • ∼ ψ. Using T (∼ ψ) and F (∼ ψ) derives F (ψ) and T (ψ), respectively. Since ψ has degree n, by the induction hypothesis, this branch can be further expanded to contain atomic conjugates. • ψ1 ∧ ψ2 , ψ1 ∨ ψ2 , or ψ1 ⊃ ψ2 . We can derive: F (ψ1 ∨ ψ2 ) F (ψ1 ⊃ ψ2 ) T (ψ1 ∧ ψ2 ) F (ψ1 ) T (ψ1 ) T (ψ1 ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... the two-sorted language with one sort for numbers and one sort for strings as the preferred language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded ari ...
... the two-sorted language with one sort for numbers and one sort for strings as the preferred language for the theory. This setup has its origins in Buss’ celebrated thesis Bounded arithmetic, Bibliopolis, 1986, for complexity classes beyond PH, and following Zambella Notes on polynomially bounded ari ...
Document
... • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false ...
... • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false ...
3.1.3 Subformulas
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
Slides
... A generic framework for reducing decidable logics to propositional logic (beyond NP). ...
... A generic framework for reducing decidable logics to propositional logic (beyond NP). ...
.pdf
... list of metavariables. The notation v denotes a copy of the formula denoted by in which all occurrences (even those within the scope of 2 ) of the variables of v are replaced by the formulas denoted by the corresponding variables of . This method for eliminating axiom schemes does not work in ...
... list of metavariables. The notation v denotes a copy of the formula denoted by in which all occurrences (even those within the scope of 2 ) of the variables of v are replaced by the formulas denoted by the corresponding variables of . This method for eliminating axiom schemes does not work in ...
Adding the Everywhere Operator to Propositional Logic (pdf file)
... as follows. First, extend language C to a language C . The formulas of C will include those of C; the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C —using a finite number of axioms. Finally, we show that the theorems of C that are concrete are preci ...
... as follows. First, extend language C to a language C . The formulas of C will include those of C; the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C —using a finite number of axioms. Finally, we show that the theorems of C that are concrete are preci ...
MUltseq: a Generic Prover for Sequents and Equations*
... error-prone and complex computations by hand. We hope that its simplicity, and the fact that no previous knowledge (except the truth tables) of the logic is needed to experiment, make the system useful for all those researches interested in these logics. In addition, since equations and quasi-equati ...
... error-prone and complex computations by hand. We hope that its simplicity, and the fact that no previous knowledge (except the truth tables) of the logic is needed to experiment, make the system useful for all those researches interested in these logics. In addition, since equations and quasi-equati ...
Exam #2 Wednesday, April 6
... There are no further clauses to be obtained from these by resolution. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and then Q to get no clauses, we also see that the original formula is not valid. 3. (P -> Q) -> ( (P -> R ) -> (Q -> R)) The negation of the formula in CNF is: ...
... There are no further clauses to be obtained from these by resolution. If we use the Davis-Putnam Procedure, first eliminating P to get {Q} and then Q to get no clauses, we also see that the original formula is not valid. 3. (P -> Q) -> ( (P -> R ) -> (Q -> R)) The negation of the formula in CNF is: ...
Lesson 12
... In addition to the above rules of inference one also requires a set of equivalences of propositional logic like “A /\ B” is equivalent to “B /\ A”. A number of such equivalences were presented in the discussion on propositional logic. ...
... In addition to the above rules of inference one also requires a set of equivalences of propositional logic like “A /\ B” is equivalent to “B /\ A”. A number of such equivalences were presented in the discussion on propositional logic. ...
Propositional Logic: Why? soning Starts with George Boole around 1850
... ∀x∃y(P (x, y) → (¬∃z∃yR(z, y) ∧ ¬∀xS(x))) into a (set of) formula(s) in prenex conjunctive normal form ...
... ∀x∃y(P (x, y) → (¬∃z∃yR(z, y) ∧ ¬∀xS(x))) into a (set of) formula(s) in prenex conjunctive normal form ...
ppt - Purdue College of Engineering
... Examples where propositional logic fails Every positive number is greater than zero. Five is a positive number. Therefore, five is greater than zero. Minimal statements? A = Every positive number is greater than zero. B = Five is a positive number. C = Five is greater than zero. Hypotheses: A, B. C ...
... Examples where propositional logic fails Every positive number is greater than zero. Five is a positive number. Therefore, five is greater than zero. Minimal statements? A = Every positive number is greater than zero. B = Five is a positive number. C = Five is greater than zero. Hypotheses: A, B. C ...
Sequent calculus - Wikipedia, the free encyclopedia
... The Sequent calculus LK was introduced by Gerhard Gentzen as a tool for studying natural deduction. It has turned out to be a very useful calculus for constructing logical derivations. The name itself is derived from the German term Logischer Kalkül, meaning "logical calculus." Sequent calculi are t ...
... The Sequent calculus LK was introduced by Gerhard Gentzen as a tool for studying natural deduction. It has turned out to be a very useful calculus for constructing logical derivations. The name itself is derived from the German term Logischer Kalkül, meaning "logical calculus." Sequent calculi are t ...
Intro to First
... Is this true? You might say, yes, it is true, but its truth value depends on what x can be, i.e. the meaning of the symbol x. If x can be a negative number, this statement is not true. In this sense, mathematicians are rather sloppy: there are often unwritten assumptions in the statements they make. ...
... Is this true? You might say, yes, it is true, but its truth value depends on what x can be, i.e. the meaning of the symbol x. If x can be a negative number, this statement is not true. In this sense, mathematicians are rather sloppy: there are often unwritten assumptions in the statements they make. ...
Normal Forms
... The Skolem form of a formula F in RPF is the result of applying the following algorithm to F : while F contains an existential quantifier do Let F = ∀y1 ∀y2 . . . ∀yn ∃z G (the block of universal quantifiers may be empty) Let f be a fresh function symbol of arity n that does not occur in F . F := ∀y ...
... The Skolem form of a formula F in RPF is the result of applying the following algorithm to F : while F contains an existential quantifier do Let F = ∀y1 ∀y2 . . . ∀yn ∃z G (the block of universal quantifiers may be empty) Let f be a fresh function symbol of arity n that does not occur in F . F := ∀y ...
