1
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
In Class Slides
... whether something in your proof has been assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown use “We must show that” ...
... whether something in your proof has been assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown use “We must show that” ...
curry
... Note on lambda syntax • (lambda X (foo X)) is a way to define a lambda expression that takes any number of arguments • In this case X is bound to the list of the argument values, e.g.: > (define f (lambda x (print x))) ...
... Note on lambda syntax • (lambda X (foo X)) is a way to define a lambda expression that takes any number of arguments • In this case X is bound to the list of the argument values, e.g.: > (define f (lambda x (print x))) ...
323-670 ปัญญาประดิษฐ์ (Artificial Intelligence)
... TodayIsThursday) JerryGivingLecture • It is equivalent to a single long sentence: the conjunction of all sentences (JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture ...
... TodayIsThursday) JerryGivingLecture • It is equivalent to a single long sentence: the conjunction of all sentences (JerryGivingLecture (TodayIsTuesday TodayIsThursday)) JerryGivingLecture ...
I am an L&S CS major. Why do I have to take this class?
... The number 0 (called logic 0) is represented with a voltage near 0 V. The number 1 (called logic 1) is represented with a voltage between 2 and 5 V, depending on the technology. Circuits perform computation by taking voltage inputs and allowing current to flow, creating voltage outputs. ...
... The number 0 (called logic 0) is represented with a voltage near 0 V. The number 1 (called logic 1) is represented with a voltage between 2 and 5 V, depending on the technology. Circuits perform computation by taking voltage inputs and allowing current to flow, creating voltage outputs. ...
PDF
... The completeness theorem of propositional logic is the statement that a wff is tautology iff it is a theorem. The if part of the statement is the soundness theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. ...
... The completeness theorem of propositional logic is the statement that a wff is tautology iff it is a theorem. The if part of the statement is the soundness theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. ...
coppin chapter 07e
... that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? ...
... that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? ...
Programming and Problem Solving with Java: Chapter 14
... that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? ...
... that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? ...
powerpoint - IDA.LiU.se
... Rewrite (or p (or q r)) as (or p q r), with arbitrary number of arguments, and similarly for and The result is an expression on conjunctive normal form Consider the arguments of and as separate formulas, obtaining a set of or-expressions with literals as their arguments Consider these or-expressions ...
... Rewrite (or p (or q r)) as (or p q r), with arbitrary number of arguments, and similarly for and The result is an expression on conjunctive normal form Consider the arguments of and as separate formulas, obtaining a set of or-expressions with literals as their arguments Consider these or-expressions ...
PDF
... such that each Bj (where j ≤ m) is either an axiom, or a formula in ∆0 . Then certainly this is also a deduction with assumptions in ∆ and conclusion A → B. Therefore, ∆ ` A → B. The deduction theorem holds in most of the widely studied logical systems, such as classical propositional logic and pre ...
... such that each Bj (where j ≤ m) is either an axiom, or a formula in ∆0 . Then certainly this is also a deduction with assumptions in ∆ and conclusion A → B. Therefore, ∆ ` A → B. The deduction theorem holds in most of the widely studied logical systems, such as classical propositional logic and pre ...
Schematics 201 - Ivy Tech -
... engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits ...
... engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits ...
(draft)
... A ∗ B → C ≡ A → B → C which is a statement about currying and uncurrying. Also, deMorgan’s law, A ∧ B ⇔ ¬(¬A ∨ ¬B) suggests that there is a way to encode sums using products and vice versa. Lastly, double negation translates to τ ≡ (τ → 0) → 0 is continuation passing under the isomorphism: ...
... A ∗ B → C ≡ A → B → C which is a statement about currying and uncurrying. Also, deMorgan’s law, A ∧ B ⇔ ¬(¬A ∨ ¬B) suggests that there is a way to encode sums using products and vice versa. Lastly, double negation translates to τ ≡ (τ → 0) → 0 is continuation passing under the isomorphism: ...
Assignment 6
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.