1 Quantifier Complexity and Bounded Quantifiers

... So far we have used ordinary quantiﬁers ∀ and ∃. In order to study quantiﬁer complexity, we now introduce bounded versions, deﬁned here: (∀y ≤ t)A(y) ↔ (∀y)(y ≤ t → A(y)) (∃y ≤ t)A(y) ↔ (∃y)(y ≤ t ∧ A(y)) where t is a term not involving y. Deﬁne a formula to be ∆0 if all of its quantiﬁers are bounde ...

... So far we have used ordinary quantiﬁers ∀ and ∃. In order to study quantiﬁer complexity, we now introduce bounded versions, deﬁned here: (∀y ≤ t)A(y) ↔ (∀y)(y ≤ t → A(y)) (∃y ≤ t)A(y) ↔ (∃y)(y ≤ t ∧ A(y)) where t is a term not involving y. Deﬁne a formula to be ∆0 if all of its quantiﬁers are bounde ...

Logic: Propositional Tree Rules (Handout)

... Using these nine simple “algorithmic” rules you can: (i) prove any valid sequent; (ii) construct a counter-example for any invalid sequent. ...

... Using these nine simple “algorithmic” rules you can: (i) prove any valid sequent; (ii) construct a counter-example for any invalid sequent. ...

INF3170 Logikk Spring 2011 Homework #8 Problems 2–6

... 12. The language of sets has a single binary relation symbol ∈, where x ∈ y is meant to denote the fact that x is an element of y. In the intended interpretation, everything is a set; that is, every object is a set, whose elements are sets, and so on. In this language, formalize the following state ...

... 12. The language of sets has a single binary relation symbol ∈, where x ∈ y is meant to denote the fact that x is an element of y. In the intended interpretation, everything is a set; that is, every object is a set, whose elements are sets, and so on. In this language, formalize the following state ...

Normal Forms

... Input: a formula F Output: an equisatisfiable, rectified, closed formula in Skolem form ∀y1 . . . ∀yk G where G is quantifier-free 1. Rectify F by systematic renaming of bound variables. The result is a formula F1 equivalent to F . 2. Let y1 , y2 , . . . , yn be the variables occurring free in F1 . ...

... Input: a formula F Output: an equisatisfiable, rectified, closed formula in Skolem form ∀y1 . . . ∀yk G where G is quantifier-free 1. Rectify F by systematic renaming of bound variables. The result is a formula F1 equivalent to F . 2. Let y1 , y2 , . . . , yn be the variables occurring free in F1 . ...

.pdf

... Last time, we defined the notion of a formula ϕ being true under an interpretation v0 , written v0 |= ϕ. Consider some examples: • p ∧ q: If v0 is the interpretation v0 (p) = t, v0 (q) = t, then v0 |= p ∧ q. • (p∧q) ⇒ p: This is the formula that everyone, two lectures asgo, agree was true, despite n ...

... Last time, we defined the notion of a formula ϕ being true under an interpretation v0 , written v0 |= ϕ. Consider some examples: • p ∧ q: If v0 is the interpretation v0 (p) = t, v0 (q) = t, then v0 |= p ∧ q. • (p∧q) ⇒ p: This is the formula that everyone, two lectures asgo, agree was true, despite n ...

IUMA Máster MTT, Métodos, 2015-2016 Examen 22 febrero 2016

... 3. Comentar: A1, . . . ,An ⊨ B means that formula B is valid: it is true in all situations in which A1, . . . ,An are true. It implies the use of semantics. However a theorem is a formula that can be established (‘proved’) by a given proof system. We write A is a theorem as ⊢ A, A is a theorem. A pr ...

... 3. Comentar: A1, . . . ,An ⊨ B means that formula B is valid: it is true in all situations in which A1, . . . ,An are true. It implies the use of semantics. However a theorem is a formula that can be established (‘proved’) by a given proof system. We write A is a theorem as ⊢ A, A is a theorem. A pr ...

Ch 11 Patterns (WP)

... (b) Write a formula using symbols. ( P = ... x T + ... ) (c) Use the formula to find how many people can sit at 11 tables. this is chapter 11 ...

... (b) Write a formula using symbols. ( P = ... x T + ... ) (c) Use the formula to find how many people can sit at 11 tables. this is chapter 11 ...

Lecture 2a: First-order Logic over words Formal Semantics for FO(<)

... u, [x 7→ i] |= ¬ψ iff u, [x 7→ i] 6|= ψ iff v, [x 7→ j] 6|= ψ iff v, [x 7→ j] |= ¬ψ Case 3: ϕ = ϕ1 ∨ ϕ2 u, [x 7→ i] |= ϕ1 ∨ ϕ2 iff u, [x 7→ i] |= ϕ1 or u, [x 7→ i] |= ϕ2 and by induction hypothesis, u, [x 7→ i] |= ϕk iff v, [x 7→ j] |= ϕk for k ∈ 1, 2. The result follows. Case 4: ϕ = ∃x. ψ If u, [x ...

... u, [x 7→ i] |= ¬ψ iff u, [x 7→ i] 6|= ψ iff v, [x 7→ j] 6|= ψ iff v, [x 7→ j] |= ¬ψ Case 3: ϕ = ϕ1 ∨ ϕ2 u, [x 7→ i] |= ϕ1 ∨ ϕ2 iff u, [x 7→ i] |= ϕ1 or u, [x 7→ i] |= ϕ2 and by induction hypothesis, u, [x 7→ i] |= ϕk iff v, [x 7→ j] |= ϕk for k ∈ 1, 2. The result follows. Case 4: ϕ = ∃x. ψ If u, [x ...

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: ...

EECS 203-1 – Winter 2002 Definitions review sheet

... it is true for all possible assignments of truth values to its variables. A contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional ...

... it is true for all possible assignments of truth values to its variables. A contradictory expression is false for all assignments of truth values to its variables. A satisfiable formula is an expression which is true for at least one assignment. • Logical equivalence and implication in propositional ...

Arithmetic Sequences

... given and there is a method of determining the nth tem by using the terms that precede it. ...

... given and there is a method of determining the nth tem by using the terms that precede it. ...

Agenda 1/8 & 1/9

... 2) Determine what is happening to your pattern. Find the constant difference, d. 3) Pick a term in the sequence. You will get 2 pieces of information from this term: tn is the actual value and n is the term number. 4) Sub d, n and tn into tn =dn+ ___ and find out what number needs to fill in the bla ...

... 2) Determine what is happening to your pattern. Find the constant difference, d. 3) Pick a term in the sequence. You will get 2 pieces of information from this term: tn is the actual value and n is the term number. 4) Sub d, n and tn into tn =dn+ ___ and find out what number needs to fill in the bla ...

as a PDF

... value). iii) If Ae #and A = 0, then A* = 0. We will use the notation φ w for the extension Henle matrix containing 2n elements.3 If we try to find a formula containing more than one variable, which, if added to S5, would give a proper extension of S5, we can delimit ourselves to formulas in MNCF of ...

... value). iii) If Ae #and A = 0, then A* = 0. We will use the notation φ w for the extension Henle matrix containing 2n elements.3 If we try to find a formula containing more than one variable, which, if added to S5, would give a proper extension of S5, we can delimit ourselves to formulas in MNCF of ...

Patterns and Sequences

... Note: To divide by a number is the same as multiplying by its reciprocal. The pattern for a geometric sequence is represented as a multiplication pattern. For example: to divide by 2 is represented as the pattern multiply by ½. ...

... Note: To divide by a number is the same as multiplying by its reciprocal. The pattern for a geometric sequence is represented as a multiplication pattern. For example: to divide by 2 is represented as the pattern multiply by ½. ...

this material

... very compressed. It has not given any meaning to the symbols, but its structure is clearly modeled on that of (a non-metaphorical!) natural language. The eight logical constants correspond to the words “not,” “or”, “and”, “if … then” and “if and only if.” Names and other subject-expressions from the ...

... very compressed. It has not given any meaning to the symbols, but its structure is clearly modeled on that of (a non-metaphorical!) natural language. The eight logical constants correspond to the words “not,” “or”, “and”, “if … then” and “if and only if.” Names and other subject-expressions from the ...

CSE596, Fall 2015 Problem Set 1 Due Wed. Sept. 16

... course webpage’s handout on the Myhill-Nerode Theorem, which fills the same purposes as the (in)famous “Pumping Lemma” but is better in my opinion (and also a UB-Cornell joint product in 1958). Also read problem (A) below. The first problem is “for discussion” but is not assigned for credit. Sometim ...

... course webpage’s handout on the Myhill-Nerode Theorem, which fills the same purposes as the (in)famous “Pumping Lemma” but is better in my opinion (and also a UB-Cornell joint product in 1958). Also read problem (A) below. The first problem is “for discussion” but is not assigned for credit. Sometim ...

The dnf simplification procedure

... simplest possible dnf equivalent of a formula. It turns out that they do not, that if one is after the shortest possible equivalent sometimes one must expand a formula before dropping some of its clauses so that other of its clauses can be dropped in addition or instead. This process will be describ ...

... simplest possible dnf equivalent of a formula. It turns out that they do not, that if one is after the shortest possible equivalent sometimes one must expand a formula before dropping some of its clauses so that other of its clauses can be dropped in addition or instead. This process will be describ ...

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 ...

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 ...

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. ...

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. ...

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 ...

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. ...