Adam Blank Spring 2017 CSE 311. Foundations of Computing I
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1 Adam Blank Spring 2017 CSE 311 Foundations of Computing I
2 CSE 311: Foundations of Computing I Lecture 1: Propositional Logic
3 Some Perspective Computer Science and Engineering Programming CSE 14x Theory Hardware CSE 311
4 About the Course We will study the theory needed for CSE: Logic:" How"can"we"describe"ideas"precisely? Formal Proofs: How"can"we"be"positive,we re"correct? Number2Theory: How"do"we"keep"data"secure? Relations/Relational2Algebra: How"do"we"store"information? Finite2State2Machines: How"do"we"design"hardware"and"software? Turing2Machines: Are"there"problems"computers"can t solve?
5 About the Course It s about perspective! Example: Sudoku Given one, solve it by hand Given most, solve them with a program Given any, solve it with computer science Tools for reasoning about difficult problems Tools for communicating ideas, methods, objectives Tools for automating difficult problems Fundamental structures for computer science This is NOT a programming course!
6 Administrivia Instructor: Adam Blank Teaching Assistants: Christopher Choi Michael Lee Halden Lin Elizabeth Moore Nicole Riley Logan Weber Jefferson Van Wagenen (Optional) Books: Rosen, Velleman, MIT Book Don t buy new copies! Aaron Johnston Andrew Li Karishma Mandyam Andrew Murray Forrest Timour Christine Wolf Homework: Due 11:30pm Write up individually Section: Thursdays Grading (roughly): 50% Homework 20% Midterm 30% Final Exam All Course cs.uw.edu/311
7 Logic: The Language of Reasoning Why not use English? Turn right here Does right mean the direction or now? Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo This means Bison from Buffalo, that bison from Buffalo bully, themselves bully bison from Buffalo. We saw her duck Does duck mean the animal or crouch down? Language of Reasoning like Java or English Words, sentences, paragraphs, arguments Today is about words and sentences
8 Why Learn A New Language? Logic, as the language of reasoning, will help us Be more precise Be more concise Figure out what a statement means more quickly
9 Propositions A proposition is a statement that has a truth value, and is well-formed
10 Are These Propositions? = 5 The home page renders correctly in IE. Turn in your homework on Wednesday. This statement is false. Akjsdf! Who are you? Every positive even integer can be written as the sum of two primes.
11 Are These Propositions? = 5 This is a proposition. It s okay for propositions to be false. The home page renders correctly in IE. This is a proposition. It s okay for propositions to be false. Turn in your homework on Wednesday. This is a command which means it doesn t have a truth value. This statement is false. This statement does not have a truth value! (If it s true, it s false, and vice versa.) Akjsdf! This is not a proposition because it s gibberish. Who are you? This is a question which means it doesn t have a truth value. Every positive even integer can be written as the sum of two primes. This is a proposition. We don t know if it s true or false, but we know it s one of them!
12 Propositions A proposition is a statement that has a truth value, and is well-formed We need a way of talking about arbitrary ideas Propositional Variables:!, #, $, %, Truth Values: T for true F for false
13 A Proposition Roger is an orange elephant who has toenails if he has tusks, and has toenails, tusks, or both. We d like to understand what this proposition means. This is where logic comes in. There are pieces that appear multiple times in the phrase (e.g., Roger has tusks ). These are called atomic propositions. Let s list them: RElephant: Roger is an orange elephant RTusks: Roger has tusks RToenails: Roger has toenails
14 Putting Them Together Roger is an orange elephant who has toenails if he has tusks, and has toenails, tusks, or both. RElephant: Roger is an orange elephant RTusks: RToenails: Roger has tusks Roger has toenails Now, we put these together to make the sentence: RElephant and (RToenails if RTusks) and (RToenails or RTusks or (RToenails and RTusks)) This is the general idea, but now, let s define our formal language.
15 Logical Connectives Negation (not)! Conjunction (and)!( (# Disjunction (or)!( (# Exclusive Or! # Implication! # Biconditional! # RElephant: Roger is an orange elephant RTusks: Roger has tusks RToenails: Roger has toenails Roger is an orange elephant who has toenails if he has tusks, and has toenails, tusks, or both. RElephant and (RToenails if RTusks) and (RToenails or RTusks or (RToenails and RTusks))
16 Logical Connectives Negation (not)! Conjunction (and)!( (# Disjunction (or)!( (# Exclusive Or! # Implication! # Biconditional! # RElephant: Roger is an orange elephant RTusks: Roger has tusks RToenails: Roger has toenails Roger is an orange elephant who has toenails if he has tusks, and has toenails, tusks, or both. RElephant and (RToenails if RTusks) and (RToenails or RTusks or (RToenails and RTusks)) RElephant (RToenails if RTusks ) (RToenails RTusks (RToenails RTusks))
17 Some Truth Tables p p p q p# q p q p# q p q p# q
18 Some Truth Tables p p T F F T p q p# q T T T T F F F T F F F F p q p# q T T T T F T F T T F F F p q p# q T T F T F T F T T F F F
19 Implication If it s raining, then I have my umbrella It s useful to think of implications as promises. That is Did I lie? p q p q T T T T F F F T T F F T I"have"my" umbrella I"do"not"have" my"umbrella It s"raining It s"not"raining
20 Implication If it s raining, then I have my umbrella It s useful to think of implications as promises. That is Did I lie? p q p q T T T T F F F T T F F T I"have"my" umbrella I"do"not"have" my"umbrella It s"raining No Yes It s"not"raining No No The only lie is when: (a) It s raining AND (b) I don t have my umbrella
21 Implication If it s raining, then I have my umbrella Are these true? = 4 earth is a planet p q p q T T T T F F F T T F F T = 5 bears are commonly found near seals
22 !( (# (1) I have collected all 151 Pokémon if I am a Pokémon master (2) I have collected all 151 Pokémon only if I am a Pokémon master These sentences are duals of each other: (1) (2)
23 !( (# Implication: p implies q whenever p is true q must be true if p then q q if p p is sufficient for q p only if q p q p q T T T T F F F T T F F T
24 Biconditional:!( (# p iff q p is equivalent to q p implies q and q implies p p q p( q T T T T F F F T F F F T
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Numbers that are divisible by 2 are even. The above statement could also be written in other logically equivalent ways, such as:
3.4 THE CONDITIONAL & BICONDITIONAL Definition. Any statement that can be put in the form If p, then q, where p and q are basic statements, is called a conditional statement and is written symbolically
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Chapter 2 Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic. Lewis Carroll You will have encountered several languages
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A statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial