PPT-A New Characterisation of Propositional Proofs

Iddo Tzameret Royal Holloway University of London Joint work with Fu Li Tsinghua and Zhengyu Wang Harvard Sketch 2 Sketch a major open problem in proof complexity stems from seemingly weak results. 1. Tautologies, Contradictions, and Contingencies. A . tautology. is a proposition that is always . true. .. Example: . p. . ∨¬. p. . A . contradiction. is a proposition that is always . false. Chapter 1, Part III: Proofs. Summary. Proof Methods. Proof Strategies. Introduction to Proofs. Section 1.7. Section Summary. Mathematical Proofs. Forms of Theorems. Direct Proofs. Indirect Proofs. Proof of the . 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. Learner Objective: I will calculate midpoints of segments and complete proofs 
 requiring that more than one pair of triangles be shown congruent.. Advanced Geometry. Learner Objective: I will calculate midpoints of segments and complete proofs 
 requiring that more than one pair of triangles be shown congruent.. Chapter 1, Part III: Proofs. With Question/Answer Animations. Summary. Valid Arguments and Rules of Inference. Proof Methods. Proof Strategies. Rules of Inference. Section 1.6. Section Summary. Valid Arguments. 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. 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. 1. NP-Completeness . Proofs. Presentation for use with the textbook, . Algorithm Design and Applications. , by M. T. Goodrich and R. Tamassia, Wiley, 2015. © 2015 Goodrich and Tamassia . NP-Completeness Proofs. Nat 4/5. What is a characterisation techniques?. Characterisation. . is the process of developing a . role. . into a character. .. When creating a character, thought should be given to developing appropriate and distinctive . 1.1 Propositional Logic. 1.2 Propositional Equivalences. 1.3 Predicates and Quantifiers. 1.4 Nested Quantifiers. 1.5 Rules of Inference. 1.6 Introduction to Proofs. 1.7 Proof Methods and Strategy. We wish to establish the truth of. 1.1 Propositional Logic. 1.2 Propositional Equivalences. 1.3 Predicates and Quantifiers. 1.4 Nested Quantifiers. 1.5 Rules of Inference. 1.6 Introduction to Proofs. 1.7 Proof Methods and Strategy. To prove an argument is valid or the conclusion follows . Chapter 1, Part III: Proofs. Summary. Proof Methods. Proof Strategies. Introduction to Proofs. Section 1.7. Section Summary. Mathematical Proofs. Forms of Theorems. Direct Proofs. Indirect Proofs. Proof of the . Chapter . 1. , Part I: Propositional Logic. With Question/Answer Animations. Chapter Summary. Propositional Logic. The Language of Propositions. Applications. Logical Equivalences. Predicate Logic. The Language of Quantifiers. Alexander A. . Razborov. University of . Chicago. Steklov. . Mathematical Institute . IAS, . Avi. is 60 conference, October 5, 2016. Subtitle: on my . (and others’) largely unsuccessful . attempts to make .