Class 5: Conjunctive Normal Form, Computing, Qualifiers

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Class 5: Conjunctive Normal Form, Computing, and Quantifiers David Evans University of Virginia
2. Plan Review: Logic Formulas, Valid, Satisfiable Conjunctive Normal Form, 3CNF Computing Machines Quantifiers Add date is today! If you aren’t enrolled in the class yet, need to get a course action form signed and delivered today. PS1: grading almost finished, everyone should receive grade by tonight. Comments will be posted soon.
3. A formula is valid if there is no way to make it false. A formula is satisfiable if there is some way to make it true. If NOT(X) is valid, what does that mean about X?
4. Conjunctive Normal Form
5. Conjunctive Normal Form clause literal variable or its negation
6. Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B
7. Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B A ⇒ B
8. Logical Formula to Truth Table Can we convert any logical formula to a truth table?
9. Logical Formula to Truth Table How big is the table? a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F
10. Truth Table to Disjunctive Normal Form Disjunctive normal form: OR of clauses, each clause is AND of literals. a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F
11. Truth Table to Disjunctive Normal Form a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F clauses = Set() # empty is False for each row in table: if result(row): clauses.add(row as conjunct clause)
12. Truth Table to Conjunctive Form De Morgan’s Law
13. Truth Table to Conjunctive Normal Form a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F
14. Truth Table to Conjunctive Normal Form a1 a2 a3 … an Result T T T … T T T T … F … … … … … … … … F F F … T F F F … F clauses = Set() for each row in table: if not result(row): clauses.add(row as disjunctive clause)
15. 3CNF Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF).
16. 3CNF Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B ∨ ¬C) ∧ (A ∨ B ∨ C) ∧ (¬A ∨ ¬B ∨ C)
17. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B) Convert clause with two literals to 3CNF:
18. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A) Convert clause with one literal to 3CNF:
19. Definition. A logical formula that is written as a conjunction of clauses, where each clause is a disjunction of exactly three literals, and each literal is either a variable or a negation of a variable, is in three- conjunctive normal form (3CNF). (A ∨ B ∨ C ∨ D) Convert clause with four literals to 3CNF:
20. (A ∨ B ∨ C ∨ D ∨ E ∨ …) Convert clause with any number of literals to 3CNF:
21. Any logical formula → truth table → CNF → 3CNF Universality of 3CNF
22. Computing Machines and Logic
24. First Transistor John Bardeen, Walter Brattain, William Shockley Bell Labs, 1947 Nobel Prize (Physics), 1956 24
25. Physics of Computation #25 control source output If the control is “on”, the source flows to the output. Otherwise, the output is “off”. A transistor is a device used to amplify or switch electronic signals (like a faucet). What logical operation is this?
26. Logic from Transistors With transistors it is possible to make two switches: control source output S C O 1 1 1 1 0 0 0 1 0 0 0 0 normal control
27. Making Transistors Today • A transistor is made of a solid piece of semiconductor material. • A semiconductor is a material that has electrical conductivity that varies dynamically between that of a conductor (on) or an insulator (off). • Silicon is a semiconductor.
28. Transistors in your Smartphone 28 Qualcomm Snapdragon 821 ∼2 Billion transistors Cost: $18 (> 1 Million transistors/cent) 14 nanometers process (about 25M transistors per mm2) Snapdragon 835, mid-2017: 3B transistors, 10nm process
29. Are NANDs efficient? 2-bit adder b a s c
30. Are NANDs the most efficient? 2-bit adder b a s c
31. Are NANDs the most efficient? 2-bit adder b a s c a b s c 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 Check Check
32. 4 transistors XOR design Efficient XORs 12 T Hashes/s, each hash is ~40,000 XORs
33. Logic Synthesis Are all the transistors in a chip used to make NAND gates?
34. Car theft rate (by model year) Source: hldi.org
35. Inside the Mifare Chip 0.001 mm (1000 nm) 0.01 mm (10000 nm) Pictures by Karsten Nohl
36. Interconnection Layers Logic Layer Pictures by Karsten Nohl
37. Zooming in on the Logic… rotated rotated + mirrored 4 NAND: Y = !(A & B & C & D) match match Pictures by Karsten Nohl
38. This room has roughly 100 Trillion NAND gates in it…what are most of the doing?
40. b … …
41. Quantifiers
42. Logical Quantifiers ∀ ∈ . () ∃ ∈ . ()
43. Nesting Quantifiers ∀ ∈ ℕ. ∃ ∈ ℕ. = + 1
44. Nesting Quantifiers ∀ ∈ ℕ − {0}. ∃ ∈ ℕ. = + 1
45. Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. Can we express this using a quantified logical formula?
46. Charge • Add date is today: make sure you are enrolled (or get a course action form signed now!) • Computers are all built on logic – In theory, NAND is all we need – In practice, efficiency matters Due Friday (6:29pm): PS2