COMP 2600: Formal Methods for Software Engineeing

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1 COMP 2600: Formal Methods for Software Engineeing Dirk Pattinson Semester 2, 2013 What do we mean by FORMAL? Oxford Dictionary in accordance with convention or etiquette or denoting a style of writing or public speaking characterized by more elaborate grammatical structures and more conservative and technical vocabulary. officially sanctioned or recognized of or concerned with outward form or appearance as distinct from content The validity of logical arguments depends on form, rather than content. 1. Abstracting content allows us to study the mechanics of reasoning 2. We have more confidence in formalised arguments 3. Automation requires formalisation. Topics in COMP 2600 Logic and Natural Deduction Proving Properties of Functional Programs (Induction) Proving Properties of Imperative Programs Formal Specification of Systems Automata, Languages and Parsing Turing Machines and Computability 1

2 Assessment Assignments: 4 9% One for each major topic, released in weeks 5, 7, 9 and 12 Tutorials: 4% demonstrate a reasonable attempt at the tutorial questions Mid-Semester Quiz: 10% redeemable: replaced by exam mark if better, covers weeks 2 6 Final Exam: 50% or 60% if quiz not attempted or quiz score < exam score Final Mark capped at Exam * (100/60) + 10% Boring Administrative Stuff Tutorials Register for a tutorial in Streams. Tutorials start in week 3. Use the Course Web Page Textbooks Tentative Schedule Forum Lecture Notes Course Web Page: 2

3 Aristotle ( BC) Leibniz ( ) Boole ( ) Russel ( ) History of Logic, the Science of Reasoning Aristotle: syllogistic logic logic. Leibniz: symbolic logic Boole: algebraic logic (boolean algebras) algebra. Russel: logic as foundation of mathematics logic. Later: Church, Turing, Curry, Goedel, Scott, Milner etc.: Formal models of computational systems Why study logic? Coolness: Logic is cool trust me. Hardware: Binary logic is logical. Software: Programming languages have logical constructs. Semantics: The language of logic is unambiguous and can be used to give meaning to programs. Proof: Arguments should be logical. Every day: Clearer thinking in every day situations. More effective communication. Propositions: Basic Building Blocks Definition 1. A statement is a sentence for which it makes sense to ask whether it is true or false. A proposition is a statement that does not depend on any variables. Example 2. John had tea for breakfast. x2 > 12 3

4 Every integer > 2 is the sum of two primes (Goldbach s conjecture) We often use variables p, q, r to denote propositions ( atomic propositions or propositional variables ). Logical Connectives Operators on Statements conjunction, and disjunction, or implication, if... then... negation, not, equivalence, if and only if, true, false Definition 3. Propositional Formulae are built from a set of atomic propositions using the logical connectives. Example 4. If p, q and John had toast for breakfast and John is hungry are atomic propositions, then p ( q (p )) (p q) r John had toast for breakfast John is hungry are propositional formulae. Precedences and Meaning Precedences Operator priorities ( strength of binding ) to minimise parentheses where stands for binds more strongly or has higher precedence. Example 5. p q r s (( p) q) (r s) p q r p (q r) Meaning of Connectives: Truth Tables p q p q (and similarly for other connectives) 4

5 Tautologies Definition 6. A propositional formula is said to be a tautology if it is true for all possible assignments of truth values to its atomic propositions. Example 7. p p p p In general, to prove a tautology, one can construct a truth table for the proposition and checks that its column is all s. Contradictions and Contingencies Definition 8. A contradiction is a compound proposition which evaluates to for all values of its elementary propositions.[1ex] A contingency is a compound proposition which may evaluate to or for different values of its elementary propositions. Example John had toast for breakfast is a contingency. 2. John had toast for breakfast John had toast for breakfast is a contradiction 3. p ( q (p )) (p q) r can be complicated! (Algebraic) Laws of Propositional Calculus Associative laws p (q r) (p q) r p (q r) (p q) r Distributive laws p (q r) (p q) (p r) p (q r) (p q) (p r) (These are just a representative few. tautologies.) Replacing by they are all 5

6 Logical Arguments, or: Formal Proofs Example: Arguments in English If the professor is naked then the class is amused. If the class is amused and the material is organized then the class is happy. The lecture is good if both the material is organized and the class is happy. Therefore, the lecture is good if the professor is naked. Typical Structure the assumptions appear above the horizontal line the conclusions appear below. (Is this a valid argument?) Exercise: Reconstruct the Argument in Logic identify the atomic propositions, or variables translate assumptions and conclusions into logical form For Example Propositions n naked, a amused, o organised, h happy, g good Translation If the professor is naked then the class is amused: n a If the class is amused and the material is organised, then the class is happy: a o h... In Logical Form n a a o h o h g n g 6 As a single formula: (n a) (a o h) (o h g) (n g)

7 Disjunctive Syllogism An inference rule Definition 10. An inference rule is a blueprint for a valid argument. Its propositions are variables, and applying an inference rule amounts to providing formulae for the variables. Disjunctive Syllogism One of Aristotle patterns of valid deduction: Example 11. p q p q The student was happy The student was awake The student was happy The student was awake Disjunctive Syllogism Proof in the Algebraic Style (p q) p q p (p q) q ( p p) ( p q) q ( p q) q p q q ( p q) q (p q) q p ( q q) p (commutativity) (distribution) (contradiction) (or-simplification) (implication) (De Morgan) (associativity) (excluded middle) (or-simplification) A non-example Or in logic In natural language, or is almost always exclusive. Do you want to have chips or peas with your dinner? The car was grey or green 7

8 In logic, or is always inclusive see the truth table. Non-Example The student was happy The student was awake The student was happy The student was awake This reasoning is INVALID. Modus ponens Inference rule Modus Ponens: the most important rule p q p q Example 12. Non-Example The student worked hard The student passed The student worked hard The student passed The student worked hard The student passed The student passed The student worked hard This reasoning is INVALID. Limitations of propositional logic Is this argument valid? useful? Natural language All COMP2600 students are happy. Lisa is a COMP2600 student. Therefore, Lisa is happy. Propositional logic p q r Maybe but not in propositional logic! Not a valid in terms of propositional logic, since p q r is not a tautology. (No relationship between the propositions.) Problem The identity of Lisa is not maintained through the propositions. In other words, the propositions don t take arguments. 8

9 Predicate Calculus: Propositions on Steroids Formalising the Argument What are the propositions? p(x) x is a COMP2600 student. q(x) x is happy. Logical Form of the argument: x.p(x) q(x) p(lisa) q(lisa)... which we might be able to deal with... Predicates Introduction Definition 13. A statement that depends on (zero, one or more) variables is called a predicate. They are usually written as expressions with (unbound) variables, e.g. x > 5. How to think about predicates As a a mapping from some domain to a Boolean value, such as p N Bool where p(x) x > 5. The domain of a predicate is some set of appropriate values for the variable(x) in the predicate expression. In the above example the domain of p is N. If a predicate contains more than one variable, then the elements of the domain are tuples. That is x+y = 5 is the function p N N Bool where p(x, y) (x + y = 5). Instantiating the variables in a predicate with values yields a proposition, e.g. p(12, 17) = 5. Quantifiers What they mean and how they are written Definition 14. If P (x) is a predicate that depends on a variable x, then x.p (x) means that P (x) is true for all values of x (chosen from a domain of discourse), and x.p (x) means that P (x) is true for some choice of x (from a suitable domain of discourse). In both cases 9

10 (resp. ) is the quantifier x is the quantified (or bound) variable. Formulae of predicate logic are built from atomic predicates, propositional connectives and the quantifiers. Abbreviations and Simple Laws Abbreviations x, y.p (x, y) x. y.p (x, y) x.( y.p (x, y)) i.e. we can remove (some) parentheses and group quantifiers. Quantifiers of the same type commute x. y.p (x, y) y. x.p (x, y) and x. y.p (x, y) y. x.p (x, y) Quantifiers of different types don t commute Predicates. C(x, y) car x has colour y Translation. x y.c(x, y) every car has a colour. y. x.c(x, y) every car has the same colour. 10