21: Virtual Substitution & Real Arithmetic 15-424: Foundations - PowerPoint PPT Presentation

21: Virtual Substitution & Real Arithmetic 15-424: Foundations of Cyber-Physical Systems Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 1 / 21

Outline Learning Objectives 1 Recap: Quadratic Equations Real Arithmetic 2 Quadratic Weak Inequalities Virtual Substitution of Infinities Expedition: Infinities Quadratic Strict Inequalities Infinitesimals Virtual Substitution of Infinitesimals Quantifier Elimination by Virtual Substitution 3 Summary 4 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 2 / 21

Outline Learning Objectives 1 Recap: Quadratic Equations Real Arithmetic 2 Quadratic Weak Inequalities Virtual Substitution of Infinities Expedition: Infinities Quadratic Strict Inequalities Infinitesimals Virtual Substitution of Infinitesimals Quantifier Elimination by Virtual Substitution 3 Summary 4 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 2 / 21

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 CT M&C CPS analytic complexity verifying CPS at scale modeling tradeoffs Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 3 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) F ( − b + ∨ F ( − b − ¯ ¯ x x Lemma (Virtual Substitution Lemma for √· ) F ( a + b √ c ) / d ≡ F ( a + b √ c ) / d Extended logic FOL R x x ¯ [ F ( a + b √ c ) / d � ω r x ∈ [ [ F ] ] iff ω ∈ [ ] ] where r = ([ [ a ] ] ω + [ [ b ] ] ω [ [ c ] ] ω ) / [ [ d ] ] ω ∈ R x ¯ Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 4 / 21

Outline Learning Objectives 1 Recap: Quadratic Equations Real Arithmetic 2 Quadratic Weak Inequalities Virtual Substitution of Infinities Expedition: Infinities Quadratic Strict Inequalities Infinitesimals Virtual Substitution of Infinitesimals Quantifier Elimination by Virtual Substitution 3 Summary 4 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 4 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) F ( − b + ∨ F ( − b − ¯ ¯ x x Lemma (Virtual Substitution Lemma for √· ) F ( a + b √ c ) / d ≡ F ( a + b √ c ) / d Extended logic FOL R x x ¯ [ F ( a + b √ c ) / d � ω r x ∈ [ [ F ] ] iff ω ∈ [ ] ] where r = ([ [ a ] ] ω + [ [ b ] ] ω [ [ c ] ] ω ) / [ [ d ] ] ω ∈ R x ¯ Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 5 / 21

Quadratic Inequality Virtual Substitution Theorem (Virtual Substitution: Quadratic Inequality x �∈ a , b , c ) ∃ x ( ax 2 + bx + c ≤ 0 ∧ F ) ↔ Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) F ( − b + ∨ F ( − b − ¯ ¯ x x Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − ¯ ¯ x x � ∨ F small ¯ x Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − ¯ ¯ x x � ∨ F −∞ ¯ x Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − x ¯ ¯ x � ∨ F −∞ ¯ x −∞ the rubber band number that’s smaller on any comparison Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − x ¯ ¯ x � ∨ ( ax 2 + bx + c ≤ 0) −∞ ∧ F −∞ ¯ x ¯ x −∞ needs to satisfy the quadratic inequality (obvious for roots) Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − ¯ ¯ x x � ∨ ( ax 2 + bx + c ≤ 0) −∞ ∧ F −∞ ¯ ¯ x x Lemma (Virtual Substitution Lemma for −∞ ) F −∞ ≡ F −∞ x ¯ x Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − ¯ ¯ 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 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 ∧ � b 2 − 4 ac ) / (2 a ) b 2 − 4 ac ) / (2 a ) � F ( − b + ∨ F ( − b − ¯ ¯ 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 [ F −∞ ω r x ∈ [ [ F ] ] iff ω ∈ [ ] ] where r → −∞ x ¯ Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 6 / 21

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 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 7 / 21

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 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 7 / 21

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 Andr´ e Platzer (CMU) FCPS / 21: Virtual Substitution & Real Arithmetic 7 / 21