Lecture 1, Thursday August 31

Class Overview

This class is an introduction to a collection of formal tools that we need to have in order to change "programming" into Computer Science. In particular, we study:

1. Logic. How to make concrete, specific statements about the world, or about, say,the behavior of our programs. How to change these statements into equivalent statements.
2. Proofs. How to demonstate if such concrete, specific statements are true.
3. Sets. How to define and manipulate collections of objects. These may be abstract objects like "possible inputs of a program", or "easily cracked passwords".
4. Counting. How to determine how many elements a set has. This is easy if the set is, say, "members of your dorm floor". But it is harder if it is "orders in which packets may be received in a network", or "number of possible poker hands".
5. Graphs and Relations. Beyond describing individual objects and collections of objects, modern science has benefited more and more by formal models of relationships between objects. Google, for example, was the first search engine to look not just at the set of words that was on a page, but instead at the "link to" structures of how pages linked to each other.

Logic and Computer Science

Why is logic important for Computer Science? This class used to be "Foundations of Computer Science", and it still is, the name just changed so that if you don't know anything about Computer Science, you'll still know what is to be taught. Logic plays a role at three specific places in Computer Science.

• First, languages like C, Java have logical operators such as && ("AND"), || ("OR"), and ! ("NOT"). In order to read and write these expressions correctly, we need to understand how to interpret and manipulate formulas like these (which make up "Propositional Logic").

• Second, logic is a key to the lower level of how computers and micro-processors work. This was not always the case, until the 1960's, the dominant form of electronic computation was analog computing. On these systems "writing a program" to simulate some natural phenomenon such as the interactions of springs and masses involves finding electrical components with analagous interactions such as capacitors, inductors and resistors (then simulate "speed = voltage" and "friction = resistance" and so on). Perhaps (?) surprisingly, most physical properties and forces do have analagous electrical components, and analog computers were used until the end of the Vietnam war for diverse tasks such as predictive bombsights (that take into account known winds, plane speed, altitiude, etc.), and systems to beat casinos by predicting roulette outcomes (this was also considered the first wearable computer).

  In the 70's, 80's and 90's, however, it became possible and more efficient to simulate all these physical interactions using much simpler electronic components, that could be made very small and very fast. Instead of using the amount of voltage in an electronic circuit to represent, say, the amount of force acting on an object, the system just uses the binary value "is there voltage on this wire or not", which we think of as ("0" or "1") or ("True" or "False"). Numbers, forces, and other kinds of data (such as web pages and MySpace friend comments) are represented as strings of 0's and 1's, and binary operations are used to manipulate this data.

• Beyond Propositional Calculus (the theory of logic formulas with AND, OR, NOT, and so no), Logic is the field of mathematics focussed on understand whether a set of axioms, and rules for manipulating those axioms are "sound" and "complete" --- starting with your axioms, is everything that you can prove true ("soundness"), and is everything that is true provable ("completeness"). In computer science we have similar concerns --- we would like to know if a language (say C) is enough to be able to write all the programs we need to (crikey, how can we define this set?), and can we prove that particular program is correct.

An Introduction to Logic

• Constructive Logic. Does not have the law of the excluded middle, any time you want to prove an "OR" statement you have to say which part is true.
• Temporal Logic. Formal system for statements whose truth value changes with time, (right now, "The cardinals win the 2006 world series" should have a logic value saying I-don't-know-yet, and the proposition "I am hungry" may change throughout the day).
• Inductive Logic. System that helps determine when it is *reasonable* to believe something based on accumulation of evidence. This is a different use of Induction that the proot technique we will talk about this semester.
• Quantum Logic. Craziness.

Propositions

Propositions are statements that have a truth value. "Washington University is in St. Louis" and "Online lecture notes are useful" are examples of propositions. English statements like "Clean your Room", "Do your homework", "Yikes" are not. In order to avoid several paradoxes, we do not allow a proposition to refer to its own truth value or the truth value of another proposition. That is we don't allow "This sentence is false", "This sentence is true", or "The next sentence is true", "The last sentence is false".

Using logical connectives (and, or, not), we can make propositional formulas out of propositions. For exampe the combined statement "Washington University is in St. Louis and Online lecture notes are useful" has a truth value (perhaps True?).

Since we are mostly interested in "how" we can think about propositional formulas, how we can combine and manipulate them (instead of being interested in listing facts about the world), we can shorten our formulas using variables. We will use letters P, Q, R to refer to propositions. They have a truth value, it is either true or false, but we may not know what it is.

We can make logical expression from propositional variables through the use of connectives. The most common of these connectives are:

• NOT, with symbol ~,
• AND, with symbol ^,
• OR, with symbol v,
• IMPLIES, with symbol -->.
• IF AND ONLY IF, sometimes shortened as IFF, with symbol <-->

• EXCLUSIVE OR, with symbol (plus with a circle around it).
• NAND
• NOR

We can define the formal meaning of these connectives through the use of a truth table.

1. Truth tables.

A logical expression, such as ((p AND q) or r), has a truth value that depends on the truth values of the propositions (the p's, q's, and r's) that make up the expression. A truth table is an organized way of writing down the truth value of an expression, by exhaustively considering every possible set of truth values for the propositions that make up the expression.

Complete truth tables

in a complete truth table, every new column is made as the "and", "or", "implies", "if and only if", or "not" of earlier columns.

How many rows are there in a complete truth table?

How many columns are there in a complete truth table?

Two expressions are called logically equivalent if their truth value is the same in EVERY condition, that is, if their truth value is the same in every row.

Now that we understand how to compute the logical meaning of connectives and propositional formulas, we can thing about trying to translate English sentences into logic *in a way that preserves their truth*. Some information on translating between logic and english can be found at the following fantastic web page, although note that they use a different notation than we do (a "dot" for and, and three horizontal lines for "<--->"). Some interesting questions for translating from english to logic and back:

• You cause a memory overflow error if you use pointers incorrectly
• You use pointers correctly, but you cause a memory overflow error.
• You cause a memory overflow error unless you use pointers incorrectly
• You cause a memory overflow error only if you use pointers incorrectly

Chapter 1.1 of the book does a very good job of talking about translating between english and logic. Please read that and Chapter 1.2 for the next class.