PDF-Decidable fragments of FIrst order logic

RRamanujamTheInstituteofMathematicalSciencesChennaiIndiajamimscresin SummaryI Modallogicshavedecentalgorithmicpropertiesusefulforspecicationandverication I Vardi1996Whyaremodallogicssorobu. Please do not alter or modify contents All rights reserved 1FQMFXIFFMMZVDDFGVMJNQMFNFUJHUIJLJMM hy does my child always have an attitude Shes often disruptive disrespectful or picking on other children Shes always the one with a chip on her shoulder Probability Conditional Probability Mean Median Mo de and Standard Deviation Random Variables Distributions uniform normal exponential Poisson Binomial Set Theory Algebra Sets Relations Functions Groups Partial Orders Lattice Boolean Algebra Comb A is a decidable language q0 0111 q0 q1q2q3 01 q0q2q3 0111 Decidable Problems q0 A B is an NFA B accepts w NFA Th A is decidable NFA q1q2q3 01 q0 q0 q1 brPage 2br Decidable Problems DFA E B is a DFA LB is nonempty Th E is decidable DFA L a context decidable Sipser. 4.1 (pages 165-173). Hierarchy of languages. All languages. Turing-recognizable. Turing-decidable. Context-free languages. Regular languages. 0. n. 1. n. a. n. b. n. c. n. D. 0. *. 1. *. Describing Turing machine input. Onramps . Table of Contents. W. Welcome Aboard! Who’s Here?. Destination: Where . A. re . W. e . G. oing?. GPS Navigation: How Do We Engage our City Contextually?. 4. Major Boulevards: What . P. 1. . Theory of Computation Peer Instruction Lecture Slides by . Dr. Cynthia Lee, UCSD.  are licensed under a . Creative Commons Attribution-. NonCommercial. -. ShareAlike. 3.0 . Unported. License. bn wwR Our main goal is to exhibit a language L that Use . WalkSAT. Use min-Conflict heuristic. Similar to hill climbing and simulated annealing. Pick unsatisfied clause then pick a symbol to flip to satisfy the clause by. Use Min Conflict. Or Random. Downside. Assertions;. t/f. Epistemological. commitment. Ontological. commitment. t/f/u. Deg. belief. facts. Facts. Objects. relations. Prop. logic. Prob. prop. logic. FOPC. Prob. FOPC. Atomic. PropositionalRelationalFirst order. Computability and Logic. Central Question. Given any statement about mathematics, is it true or is it false?. Example: . Goldbach. Conjecture: Every even number greater than 2 is the sum of two prime numbers.. Properties of Propositional Logic. Pros. Compositional. Declarative. Cons. Limited expressive power. Represents facts. First Order Logic. Used to represent. Objects – Martin the cat. Relations – Martin and Moses are brothers. Turing Machine: Languages. Recall that a collection of strings that a TM M accepts is called. the language of M or language recognized by M, denoted L(M).. Definition. A . language is Turing-recognizable (or recursively enumerable) if . Xiaokang Qiu. x=1;. y=1;. while (*) {. x=x 2;. y=y 1;. }. Q: is . x y. >=2 always true?.  . Q: Are . these formulae valid in arithmetic?. First-Order Theories. Q: Which statements are true in arithmetic/set-theory/groups/fields?.