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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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. ...
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 ...
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: ...
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 ...
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 ...
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 ...
.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 ...
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 . ...
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 ...
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. ...
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 ...
