PPT-Quantifiers, Arithmetic and Fixed-points

Quantifier Elimination Procedures in Z3 Support for Nonlinear arithmetic Fixedpoints features and a preview Quantifier Elimination O ption ELIMQUANTIFIERStrue LRA Linear real arithmetic. Nikolaj . Bjørner. Microsoft Research. Bit-Precise Constraints: . Applications and . Decision Procedures. Tutorial Contents. Bit-vector decision procedures by categories. Bit-wise operations . Vector Segments. 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 © . 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. . The Divisor function . The . divisor function . . counts the number of divisors of an integer . . . Dirichlet. divisor problem:. Determine the asymptotic behaviour as . of the sum. . This is a count of lattice points under the hyperbola . CSE 2541. Rong. Shi. Pointer definition. A variable whose value . refers directly to (or "points to") another value stored elsewhere in the computer memory using its . address. Memory addresses. Z. +. Goals. : . Explain . how to work with nested . quantifiers. S. how that . the order . of quantification . matters. . Work . with . logical . expressions involving multiple . quantifiers.. Copyright © . IIIS, Tsinghua University. Logic Conference, Tsinghua Oct. 2013. From . Classical Proof Theory . to. . P . vs.. NP. Complexity Theory. P = PTIME. : . Efficiently computable problems;. . . Algorithms of polynomial run-time. Satisfiability. Modulo Theories . Frontiers . of . Computational Reasoning . 2009 . –. MSR Cambridge. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Section 1.4. Section Summary. Predicates . Variables. Quantifiers. Universal Quantifier. Existential Quantifier. Negating Quantifiers. De Morgan’s Laws for Quantifiers. Translating English to Logic. Fixed point (can overflow). Floating point (can overflow, underflow). (Boolean / Character). Note: most embedded systems deal with physical input and output (medical data, mechanical quantities, temperature, concentration of a material in a mixture, etc.). FIXED POINTS AND FREEZING SETS IN DIGITAL TOPOLOGY L aurence B oxer 1 2 I ntro to digital topology In computer memory, a digital image is not a set of “continuous bodies.” Rather, a digital image is a set of discrete For example : 5, 10, 15, 20, 25…... In this each term is obtained by adding 5 to the preceding term except first term. The general form of an . Arithmetic . Progression is . a , a +d , a + 2d , a + 3d ………………, a + (n-1)d. Modulo Theories . Manchester 2009. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Modulo Theories. PSpace. -complete. (QBF). Undecidable. (First-order logic).