CS 357: Advanced Topics in Formal Methods Fall 2019 Lecture 7 Aleksandar Zelji´c (materials by Clark Barrett) Stanford University Logical Definitions SupposeΣ is a signature. AΣ -formula is a well-formed formula whose non-logical symbols are contained inΣ. LetΓ be a set ofΣ-formulas. We write j=M Γ[s] to signify that j=M φ[s] for every φ 2 Γ. IfΓ is a set ofΣ-formulas and φ is aΣ-formula, thenΓ logically implies φ, writtenΓ j= φ, iff for every model M ofΣ and every variable assignment s, if j=M Γ[s] then j=M φ[s]. We write j= φ as an abbreviation for f g j= φ. and φ are logically equivalent (written j= = j φ) iff j= φ and φ j= . AΣ-formula φ is valid, written j= φ iff ; j= φ (i.e. j=M φ[s] for every M and s). True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Examples Suppose that P is a unary predicate and Q a binary predicate. Which of the following are true? 1. 8 v1 Pv1 j= Pv2 True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Proof The proof is by induction on well-formed formulas φ. 1. If φ is an atomic formula, then all variables in φ occur free. Thus s1 and s2 agree on all variables in φ. It follows that s1(t) = s2(t) for each term t in φ (technically we should prove this by induction too). The result follows. 2. If φ is( :α) or( α ! β), the result is immediate from the inductive hypothesis. 3. Suppose φ = 8 x . The variables free in φ are the same as those free in except for x. Thus, for any d in dom(M), s1(xjd) and s2(xjd) agree at all variables free in . The result follows from the inductive hypothesis. As a corollary of this theorem, we have that for sentences, satisfaction is independent of the variable assignment. Invariance of Truth Values Theorem Suppose s1 and s2 are variable assignments over a model M which agree at all variables (if any) which occur free in the wff φ. Then j=M φ[s1] iff j=M φ[s2]. As a corollary of this theorem, we have that for sentences, satisfaction is independent of the variable assignment. Invariance of Truth Values Theorem Suppose s1 and s2 are variable assignments over a model M which agree at all variables (if any) which occur free in the wff φ. Then j=M φ[s1] iff j=M φ[s2]. Proof The proof is by induction on well-formed formulas φ.