Class 5: Conjunctive Normal Form, Computing, Qualifiers

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cs2102: Discrete Mathematics Class 5: Conjunctive Normal Form, Computing, and Quantifiers David Evans University of Virginia

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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.

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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?

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Conjunctive Normal Form

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Conjunctive Normal Form clause literal variable or its negation

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Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B

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Conjunctive Normal Form Can we write any logical formula in CNF? A ∧ B A ⇒ B

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Logical Formula to Truth Table Can we convert any logical formula to a truth table?

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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

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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

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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)

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Truth Table to Conjunctive Form De Morgan’s Law

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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

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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)

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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).

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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:

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(A ∨ B ∨ C ∨ D ∨ E ∨ …) Convert clause with any number of literals to 3CNF:

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Any logical formula → truth table → CNF → 3CNF Universality of 3CNF

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Computing Machines and Logic

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First Transistor John Bardeen, Walter Brattain, William Shockley Bell Labs, 1947 Nobel Prize (Physics), 1956 24

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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?

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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

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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.

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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

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Are NANDs efficient? 2-bit adder b a s c

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Are NANDs the most efficient? 2-bit adder b a s c

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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

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4 transistors XOR design Efficient XORs 12 T Hashes/s, each hash is ~40,000 XORs

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Logic Synthesis Are all the transistors in a chip used to make NAND gates?

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Car theft rate (by model year) Source: hldi.org

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Inside the Mifare Chip 0.001 mm (1000 nm) 0.01 mm (10000 nm) Pictures by Karsten Nohl

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Interconnection Layers Logic Layer Pictures by Karsten Nohl

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Zooming in on the Logic… rotated rotated + mirrored 4 NAND: Y = !(A & B & C & D) match match Pictures by Karsten Nohl

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This room has roughly 100 Trillion NAND gates in it…what are most of the doing?

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b … …

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Quantifiers

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Logical Quantifiers ∀ ∈ . () ∃ ∈ . ()

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Nesting Quantifiers ∀ ∈ ℕ. ∃ ∈ ℕ. = + 1

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Nesting Quantifiers ∀ ∈ ℕ − {0}. ∃ ∈ ℕ. = + 1

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Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element. Can we express this using a quantified logical formula?

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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