Propositional Logic
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...
... A formula is in prenex form if it is of the form Q1 x1 .Q2 x2 . . . . Qn xn .ψ (possibly with n = 0) where each Qi is a quantifier (either ∀ or ∃) and ψ is a quantifier-free formula . Proposition For any formula of first-order logic, there exists an equivalent formula in prenex form. Proof. Such a p ...
Chapter 5 – Simplifying Formulas and Solving Equations Section 5A
... part, so they are like terms. Think of it like 5 apples plus 3 apples. We would have 8 apples, not 8 apples squared or 8 double apples. So 5a + 3a = 8a. We can combine the like terms and keep the letter part the same. Look at the example of 7a + 2b. Are these like terms? Since they have different le ...
... part, so they are like terms. Think of it like 5 apples plus 3 apples. We would have 8 apples, not 8 apples squared or 8 double apples. So 5a + 3a = 8a. We can combine the like terms and keep the letter part the same. Look at the example of 7a + 2b. Are these like terms? Since they have different le ...
Simplifying Formulas and Like Terms
... Look at the example of 7a + 2b. Are these like terms? Since they have different letter parts, they are not like terms. Hence we cannot add them. Think of it like 7 apples plus 2 bananas. That will not equal 9 apple/bananas. It is just 7 apples and 2 bananas. That is a good way of thinking about add ...
... Look at the example of 7a + 2b. Are these like terms? Since they have different letter parts, they are not like terms. Hence we cannot add them. Think of it like 7 apples plus 2 bananas. That will not equal 9 apple/bananas. It is just 7 apples and 2 bananas. That is a good way of thinking about add ...
Creativity and Artificial Intelligence
... is contained in at most one connection while in the former this linearity restriction in its original form [Bibel, 1986a]) is not satisfied because the literal is contained in more than one, namely in two connections. To cover the general case considered in the present paper this linearity restricti ...
... is contained in at most one connection while in the former this linearity restriction in its original form [Bibel, 1986a]) is not satisfied because the literal is contained in more than one, namely in two connections. To cover the general case considered in the present paper this linearity restricti ...
Predicate Logic Review
... The strategy for defining truth in a model that we have just outlined is due in its essentials to Alfred Tarski. It was one of his great achievements, because it showed logicians how to use semantic notions rigorously. Once we’ve defined truth in a model, of course, we can define logical consequence ...
... The strategy for defining truth in a model that we have just outlined is due in its essentials to Alfred Tarski. It was one of his great achievements, because it showed logicians how to use semantic notions rigorously. Once we’ve defined truth in a model, of course, we can define logical consequence ...
Handout 14
... where the Ai can be either atomic formulas or their negations. It is thus a conjuction of clauses, i.e. disjunctions of literals (variables or their negations). CNF is used in machine proving of theorems. Resolution in Prolog is also based on a special form of CNF. Definition 5.10 (Disjunctive norma ...
... where the Ai can be either atomic formulas or their negations. It is thus a conjuction of clauses, i.e. disjunctions of literals (variables or their negations). CNF is used in machine proving of theorems. Resolution in Prolog is also based on a special form of CNF. Definition 5.10 (Disjunctive norma ...
POSSIBLE WORLDS AND MANY TRUTH VALUES
... than β. Finitely many replacements of this sort lead to a formula α00 having only standard connectives and , in which τa ’s apply only to variables; and α00 is valid on the same frames as α0 (in fact, α0 ⇔ α00 is valid on every frame). In α00 , replace every variable p not within the scope of any τ ...
... than β. Finitely many replacements of this sort lead to a formula α00 having only standard connectives and , in which τa ’s apply only to variables; and α00 is valid on the same frames as α0 (in fact, α0 ⇔ α00 is valid on every frame). In α00 , replace every variable p not within the scope of any τ ...
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 ...
L11
... Logic is a study of principles used to − distinguish correct from incorrect reasoning. Formally it deals with − the notion of truth in an abstract sense and is concerned with the principles of valid inferencing. A proposition in logic is a declarative statements ...
... Logic is a study of principles used to − distinguish correct from incorrect reasoning. Formally it deals with − the notion of truth in an abstract sense and is concerned with the principles of valid inferencing. A proposition in logic is a declarative statements ...
1.3.4 Word Grammars
... Proof. By contradiction. Assume an infinite sequence u1 , u2 , . . . such that for any two words uk , uk+l they are not embedded, i.e., uk 6v uk+l . Furthermore, I assume that the sequence is minimal at any word with respect to length, i.e., considering any uk , there is no infinite sequence with th ...
... Proof. By contradiction. Assume an infinite sequence u1 , u2 , . . . such that for any two words uk , uk+l they are not embedded, i.e., uk 6v uk+l . Furthermore, I assume that the sequence is minimal at any word with respect to length, i.e., considering any uk , there is no infinite sequence with th ...
1. What is propositional logic? With respect to AI, what is it good for
... to AI, it is good for a nice, simple framework for reasoning. Propositional logic also with respect to AI can help take existing knowledge and derive new knowledge or answers. 2. Using BNF, define the language of all “fully parenthesized” propositional logic formulas, where a propositional logical f ...
... to AI, it is good for a nice, simple framework for reasoning. Propositional logic also with respect to AI can help take existing knowledge and derive new knowledge or answers. 2. Using BNF, define the language of all “fully parenthesized” propositional logic formulas, where a propositional logical f ...
SECTION 8-4 Binomial Formula
... We are now ready to try to discover a formula for the expansion of (a b)n using ordinary induction; that is, we will look at a few special cases and postulate a general formula from them. We will then try to prove that the formula holds for all natural numbers, using mathematical induction. To sta ...
... We are now ready to try to discover a formula for the expansion of (a b)n using ordinary induction; that is, we will look at a few special cases and postulate a general formula from them. We will then try to prove that the formula holds for all natural numbers, using mathematical induction. To sta ...
1.1 Recursively defined sequences PP
... Recursive rule- defines the nth term in relation to the previous term. Usually we start with the 0th term or the 1st term, but you can really start at any term you like. Then just say how to get from one term to the next after that. This is what is meant by the rule. ...
... Recursive rule- defines the nth term in relation to the previous term. Usually we start with the 0th term or the 1st term, but you can really start at any term you like. Then just say how to get from one term to the next after that. This is what is meant by the rule. ...
Exam 2 study guide
... then strengthen it as much as you can. In most cases, your answer will be a single countermodel, in the strongest system in which the formula is invalid. But in one case, you will need two countermodels: if the formula is valid in S5 but not valid in any weaker systems, you will need both a B and an ...
... then strengthen it as much as you can. In most cases, your answer will be a single countermodel, in the strongest system in which the formula is invalid. But in one case, you will need two countermodels: if the formula is valid in S5 but not valid in any weaker systems, you will need both a B and an ...
11-1 Mathematical Patterns
... Ex. 3 Using a Recursive Formula a. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162, . . . . Write a recursive formula for the sequence. Multiply a term by 3 to find the next term. A recursive formula is an = an – 1 • 3, where a1 = 2. b. Find the sixth and ...
... Ex. 3 Using a Recursive Formula a. Describe the pattern that allows you to find the next term in the sequence 2, 6, 18, 54, 162, . . . . Write a recursive formula for the sequence. Multiply a term by 3 to find the next term. A recursive formula is an = an – 1 • 3, where a1 = 2. b. Find the sixth and ...
INTLOGS16 Test 2
... that might make sense for one or more of these new quantifiers (we still only have rules for ∀ and ordinary ∃). Suggest in detail two new valid inference rules that you think makes sense for one or more of the new quantifiers. Q3 Let A be some finite alphabet of symbols {s1 , s2 , . . . , sn }. Prof ...
... that might make sense for one or more of these new quantifiers (we still only have rules for ∀ and ordinary ∃). Suggest in detail two new valid inference rules that you think makes sense for one or more of the new quantifiers. Q3 Let A be some finite alphabet of symbols {s1 , s2 , . . . , sn }. Prof ...
Exercises for CS3511 Week 31 (first week of practical)
... functionally complete set of connectives. Making use of this result, can you prove that {NAND} is also functionally complete? Answer: the reasoning is similar to that in 4a, but it may be a bit trickier to find the right formulas: p can be expressed as p p|p. (It helps to do negation before conj ...
... functionally complete set of connectives. Making use of this result, can you prove that {NAND} is also functionally complete? Answer: the reasoning is similar to that in 4a, but it may be a bit trickier to find the right formulas: p can be expressed as p p|p. (It helps to do negation before conj ...
Modal Logic
... (i) Every tautology of propositional logic is valid. (ii) ❏(❏❏ (iii) Suppose that is valid. Then, ❏must also be valid. ...
... (i) Every tautology of propositional logic is valid. (ii) ❏(❏❏ (iii) Suppose that is valid. Then, ❏must also be valid. ...
P 1
... • Logic is a study of methods and principles used to distinguish correct from incorrect reasoning. It is supposed to explicate the laws of thought. Formally it deals with the notion of truth in an abstract sense and is concerned with the principles of valid inferencing. A proposition in logic ...
... • Logic is a study of methods and principles used to distinguish correct from incorrect reasoning. It is supposed to explicate the laws of thought. Formally it deals with the notion of truth in an abstract sense and is concerned with the principles of valid inferencing. A proposition in logic ...
Geometric-Sequences-and-Series
... Since we would plug in the integers 1 through 18, n must be 18. ...
... Since we would plug in the integers 1 through 18, n must be 18. ...
PDF
... 1. given any premise, every formula in it is also a formula in the conclusion, or 2. every formula in the conclusion is also a formula in some premise. In the former case, if there is a formula B in the conclusion not in any of the premises, then B is said to be introduced by the rule. In the later ...
... 1. given any premise, every formula in it is also a formula in the conclusion, or 2. every formula in the conclusion is also a formula in some premise. In the former case, if there is a formula B in the conclusion not in any of the premises, then B is said to be introduced by the rule. In the later ...
Warm-up
... The sequence above is called an _______________ sequence because it goes on forever (notice the …). If the sequence ends (ex: 2, 4, 6, 8), then it is called a _______________ sequence. ...
... The sequence above is called an _______________ sequence because it goes on forever (notice the …). If the sequence ends (ex: 2, 4, 6, 8), then it is called a _______________ sequence. ...
ARITHMETIC SERIES
... let t1 be 1 since the first term is 1. let n be 100 since there are 100 terms. let tn be 100 since the nth term is 100. ...
... let t1 be 1 since the first term is 1. let n be 100 since there are 100 terms. let tn be 100 since the nth term is 100. ...
Book Question Set #1: Ertel, Chapter 2: Propositional Logic
... A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional formulas to be logically equivalent? If two propositional formulas are logically equivalent, they must evaluate to the same truth values for all interpretations. 7.) What does it mean for a logical for ...
... A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional formulas to be logically equivalent? If two propositional formulas are logically equivalent, they must evaluate to the same truth values for all interpretations. 7.) What does it mean for a logical for ...
Lecture Notes 2
... where Fi is a conjunctive term for any i. Theorem For each Boolean formula G with variables X1 , X2 , . . . , Xn there exists a Boolean formula F in DNF, with variables from {X1 , X2 , . . . , Xn }, such that G ≡ F . We don’t prove this theorem in this course. Examples (1) Let G be the formula X → Y ...
... where Fi is a conjunctive term for any i. Theorem For each Boolean formula G with variables X1 , X2 , . . . , Xn there exists a Boolean formula F in DNF, with variables from {X1 , X2 , . . . , Xn }, such that G ≡ F . We don’t prove this theorem in this course. Examples (1) Let G be the formula X → Y ...