21: Virtual Substitution & Real Arithmetic Logical Foundations of Cyber-Physical Systems André Platzer Logical Foundations of Cyber-Physical Systems André Platzer André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 1 / 22
Outline Learning Objectives 1 Real Arithmetic 2 Recap: Quadratic Equations Quadratic Weak Inequalities Infinity ∞ Virtual Substitution Expedition: Infinities Quadratic Strict Inequalities Infinitesimal ε Virtual Substitution Quantifier Elimination by Virtual Substitution of Quadratics 3 Summary 4 André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 2 / 22
Outline Learning Objectives 1 Real Arithmetic 2 Recap: Quadratic Equations Quadratic Weak Inequalities Infinity ∞ Virtual Substitution Expedition: Infinities Quadratic Strict Inequalities Infinitesimal ε Virtual Substitution Quantifier Elimination by Virtual Substitution of Quadratics 3 Summary 4 André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 2 / 22
Learning Objectives Virtual Substitution & Real Equations rigorous arithmetical reasoning miracle of quantifier elimination logical trinity for reals switch between syntax & semantics at will virtual substitution lemma bridge gap between semantics and inexpressibles infinities & infinitesimals CT M&C CPS analytic complexity verifying CPS at scale modeling tradeoffs André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 3 / 22
Outline Learning Objectives 1 Real Arithmetic 2 Recap: Quadratic Equations Quadratic Weak Inequalities Infinity ∞ Virtual Substitution Expedition: Infinities Quadratic Strict Inequalities Infinitesimal ε Virtual Substitution Quantifier Elimination by Virtual Substitution of Quadratics 3 Summary 4 André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 3 / 22
Quadratic Virtual Substitution Theorem (Virtual Substitution: Quadratic Equation x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c = 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ �� ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) ¯ ¯ x x Lemma (Virtual Substitution Lemma for √· ) F ( a + b √ ≡ F ( a + b √ Extended logic c ) / d c ) / d FOL R x ¯ x [ F ( a + b √ � c ) / d ω r x ∈ [ [ F ] ] iff ω ∈ [ ] ] where r = ( ω [ [ a ] ]+ ω [ [ b ] ] ω [ [ c ] ]) / ( ω [ [ d ] ]) ∈ R ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 4 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ �� ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) ¯ ¯ x x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ F small ... ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ F − ∞ ... ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ F − ∞ ... ¯ x − ∞ the rubber band number that’s smaller on any comparison André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ ( ax 2 + bx + c ≤ 0 ) − ∞ ∧ F − ∞ ... ¯ x ¯ x − ∞ needs to satisfy the quadratic inequality (obvious for roots, not − ∞ ) André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ ( ax 2 + bx + c ≤ 0 ) − ∞ ∧ F − ∞ ... ¯ x ¯ x Lemma (Virtual Substitution Lemma for − ∞ ) F − ∞ ≡ F − ∞ x ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ ( ax 2 + bx + c ≤ 0 ) − ∞ ∧ F − ∞ ... ¯ x ¯ x Lemma (Virtual Substitution Lemma for − ∞ ) Extended logic FOL R ∪{− ∞ , ∞ } FOL R F − ∞ ≡ F − ∞ x ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) a � = 0 ∨ b � = 0 ∨ c � = 0 → � ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ a = 0 ∧ b � = 0 ∧ F − c / b ¯ x √ √ ∨ a � = 0 ∧ b 2 − 4 ac ≥ 0 ∧ � F ( − b + b 2 − 4 ac ) / ( 2 a ) ∨ F ( − b − b 2 − 4 ac ) / ( 2 a ) � ¯ ¯ x x � ∨ ( ax 2 + bx + c ≤ 0 ) − ∞ ∧ F − ∞ ... ¯ x ¯ x Lemma (Virtual Substitution Lemma for − ∞ ) Extended logic FOL R ∪{− ∞ , ∞ } FOL R F − ∞ ≡ F − ∞ x ¯ x ω r [ F − ∞ x ∈ [ [ F ] ] iff ω ∈ [ ] ] where r → − ∞ ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 5 / 22
Virtual Substitution of Infinities p = ∑ n i = 0 a i x i Virtual Substitution of − ∞ into Comparisons ( p = 0 ) − ∞ ≡ ¯ x ( p ≤ 0 ) − ∞ ≡ ¯ x ( p < 0 ) − ∞ ≡ ¯ x ( p � = 0 ) − ∞ ≡ ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 6 / 22
Virtual Substitution of Infinities p = ∑ n i = 0 a i x i Virtual Substitution of − ∞ into Comparisons n � ( p = 0 ) − ∞ ≡ a i = 0 ¯ x i = 0 ( p ≤ 0 ) − ∞ ≡ ¯ x ( p < 0 ) − ∞ ≡ ¯ x ( p � = 0 ) − ∞ ≡ ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 6 / 22
Virtual Substitution of Infinities p = ∑ n i = 0 a i x i Virtual Substitution of − ∞ into Comparisons n � ( p = 0 ) − ∞ ≡ a i = 0 ¯ x i = 0 ( p ≤ 0 ) − ∞ ≡ ( p < 0 ) − ∞ ∨ ( p = 0 ) − ∞ ¯ ¯ ¯ x x x ( p < 0 ) − ∞ ≡ ¯ x ( p � = 0 ) − ∞ ≡ ¯ x André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 6 / 22
Virtual Substitution of Infinities p = ∑ n i = 0 a i x i Virtual Substitution of − ∞ into Comparisons n � ( p = 0 ) − ∞ ≡ a i = 0 ¯ x i = 0 ( p ≤ 0 ) − ∞ ≡ ( p < 0 ) − ∞ ∨ ( p = 0 ) − ∞ ¯ ¯ ¯ x x x ( p < 0 ) − ∞ ≡ p ( - ∞ ) < 0 ¯ x ( p � = 0 ) − ∞ ≡ ¯ x Ultimately negative at − ∞ lim x →− ∞ p ( x ) < 0 � if def p ( - ∞ ) < 0 ≡ if André Platzer (CMU) LFCPS/21: Virtual Substitution & Real Arithmetic LFCPS/21 6 / 22
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