PPT-Quantifiers in

Satisfiability Modulo Theories Frontiers of Computational Reasoning 2009 MSR Cambridge Leonardo de Moura Microsoft Research Symbolic Reasoning Quantifiers in Satisfiability. Nested Quantifiers. Needed to express statements with multiple variables . Example 1. : “. x+y. = . y+x. for all real numbers” . . xy. (. x+y. = . y+x. ) . where the domains of . x. and . Goals. : . Explain . how to work with nested . quantifiers. S. how that . the order . of quantification . matters. . Work . with . logical . expressions involving multiple . quantifiers.. Copyright © . Logic and Proof. Fall . 2011. Sukumar Ghosh. Predicate Logic. Propositional logic has limitations. Consider this:. Is . x. . > 3. a proposition? No, it is a . predicate. . Call it . P(x. ). . The pictures describe an incident. Write sentences based on the pictures and words given. You must add your own words to make the story interesting. Brainstorm the verbs for each of the picture. taxi driver – passenger – airport – Noir Hotel. Propositional Logic Not Enough. Given the statements: . “All men are mortal.”. “Socrates is a man.”. It follows that “Socrates is mortal.”. This can’t be represented in propositional logic. . Chapter 1, Part II: Predicate Logic. With Question/Answer Animations. Summary. Predicate Logic (First-Order Logic (FOL), Predicate Calculus). The Language of Quantifiers. Logical Equivalences. Nested Quantifiers. A. n. v. e. s. h. . K. o. murave. l. l. i. work done at Carnegie Mellon University. Joint work with . Nikolaj. . Bjørner. , . Arie. . Gurfinkel. , and Kenneth McMillan. In essence…. 1. Efficiently under-approximating projections,. What . is your quantifier quotient (QQ)?. Barbara . Zurer. . Pearson. bpearson@research.umass.edu. University of Massachusetts . Amherst. Language Acquisition . Colloquium. Background. This work is part of the preliminary studies for an . Goals. : . Explain . how to work with nested . quantifiers. S. how that . the order . of quantification . matters. . Work . with . logical . expressions involving multiple . quantifiers.. Copyright © . Logic and Proof. Fall 2014. Sukumar Ghosh. Predicate Logic. Propositional logic has limitations. Consider this:. Is . x. . > 3. a proposition? No, it is a . predicate. . Call it . P(x. ). . P(4) . Structures. Logic and Proof. Spring 2014. Sukumar Ghosh. Predicate Logic. Propositional logic has limitations. Consider this:. Is . x. . > 3. a proposition? No, it is a . predicate. . Call it . Section 1.4. Section Summary. Predicates . Variables. Quantifiers. Universal Quantifier. Existential Quantifier. Negating Quantifiers. De Morgan’s Laws for Quantifiers. Translating English to Logic. Lecture 5. Predicate Logic. Spring 2013. 1. Announcements. Reading assignments. Predicates and Quantifiers. 1.4, 1.5 7. th. Edition. 1.3, 1.4 6. th. . Edition. Hand in Homework 1 now. Homework 2 is available on the website. Modulo Theories . Manchester 2009. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Modulo Theories. PSpace. -complete. (QBF). Undecidable. (First-order logic).