PPT-Satisfiability Modulo Theories and DPLL(T)

Andrew Reynolds March 18 2015 Overview SAT Satisfiability for Propositional Logic A B C D B Does there exist truth values for A B C D that make this formula true. Lecturer: . Qinsi. Wang. May 2, 2012. Z3. high-performance theorem . prover. being developed at Microsoft Research.. mainly by Leonardo de . Moura. and . Nikolaj. . Bjørner. . . Free (online interface, APIs, …) . This paper presents a novel framework to extend the dynamic range of images called Unbounded High Dynamic Range (UHDR) photography with a modulo camera. A modulo camera could theoretically take unbounded radiance levels by keeping only the least significant bits. We show that with limited bit depth, very high radiance levels can be recovered from a single modulus image with our newly proposed unwrapping algorithm for natural images. We can also obtain an HDR image with details equally well preserved for all radiance levels by merging the least number of modulus images. Synthetic experiment and experiment with a real modulo camera show the effectiveness of the proposed approach. Spring 2011. Review. Overview. Course overview. Propositional Logic Example. CSP Example. Hints for Final. Course Review. AI introduction. Agents. Searching. Uninformed. Informed. Local. Adversarial search. Leonardo de Moura and Nikolaj . Bjørner. Microsoft Research. What. EPR . . . Deciding EPR using DPLL + Substitution sets. Why? EPR is the next SAT. SAT .  EPR. Deciding EPR using DPLL + Substitution sets. Terminology. Propositional variable: . boolean. variable (p). Literal: propositional variable or its negation. p . p. Clause: disjunction of literals q \/ . . p \/ . . r. . given by set of . LPAR 2008 . –. Doha, Qatar. Nikolaj . Bjørner. , . Leonardo de Moura. Microsoft Research. Bruno . Dutertre. SRI International. Satisfiability Modulo Theories (SMT). Accelerating lemma learning using joins. SAT/SMT Summer School 2014. Parallel SAT. Motivation. Technical. Clock frequency has hit the thermal wall. Multicore CPUs to cope with it. Algorithmic. Sequential SAT seems hard to improve . SAT applied to ever harder problems. Satisfiability. Modulo Theories . Frontiers . of . Computational Reasoning . 2009 . –. MSR Cambridge. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Rodrigo de Salvo Braz. Ciaran O’Reilly. Artificial Intelligence Center - SRI International. Vibhav Gogate. University of Texas at Dallas. Rina Dechter. University of California, Irvine. IJCAI-16. , . B. 50. 4. /. I. 538. :. . Introduction to. Cryptography. (2017—03—02). Tuesday’s lecture:. One-way permutations (OWPs). PRGs from OWPs. Today’s lecture:. Basic number theory. So far:. “secret key”. Our goal. Cover basic number theory quickly!. Cover the minimum needed for all the applications we will study. Some facts stated without proof. Can take entire classes devoted to this material. Abstracting some of the ideas makes things easier to understand. or . . For which of the following diagrams does . entail . . Select all that apply..  .  .  .  .  .  .  .  .  .  .  . A). B). C). D). E). Warm-up. The regions below visually enclose the set of models that satisfy the respective sentence . Modulo Theories . Manchester 2009. Leonardo de Moura. Microsoft Research. Symbolic Reasoning. Quantifiers in . Satisfiability. Modulo Theories. PSpace. -complete. (QBF). Undecidable. (First-order logic). Sriram Rajamani. (based on notes/slides by Matt Fredrickson, Andre . Platzer. , . Emina Torlak and Leonardo . De Moura). Modern SAT solvers. First convert a formula to CNF (Conjunctive Normal Form). Use variant of DPLL (Davis Putnam .