CS1502 Formal Methods in Computer Science

1. CS1502 Formal Methods in Computer Science Lecture Notes 1 Course Information Introduction to Logic Part 1

2. Content • 70% logic and proofs • Pervades computer science, e.g., hardware circuit design, artificial intelligence, knowledge representation, database systems, programming languages, software engineering (design, verification, specification), … • Will help you understand proofs in math, science, theoretical CS

3. Content • 30% Abstract models of computation • Needed for hardware design, compilers, computational complexity analysis, … • Interesting proofs; you will use what you learn in the first part of the course to understand them

4. Logistics • Prerequisites: • CS441 Discrete Structures for Computer Science • CS445 Data Structures • Materials on course website, reachable from www.cs.pitt.edu/~wiebe • Schedule of readings, homeworks, exams • Lecture notes, available before class • Support for lectures; filled in in class

5. Logistics • The TA will go over exercises and answer questions in recitation. • He will also cover logistics of submitting homeworks, and will post solutions to the homework.

6. Logistics • Workload is steady throughout the course • 3 exams • Weekly homeworks (no projects) • Many are graded electronically • You can submit solutions as many times as you like • Get feedback without having to wait for instructor grading • Purpose is to help you master the material for the exams • Exams 90% Homework 10%

7. Logistics • Exams • Challenging, but not devious • All of the questions will be related to an example worked out in the text, a homework exercise, or a problem we did in class.

8. Logistics • Work is steady throughout the course • If you keep up, you will likely do OK • The work gets harder! • Please ask questions (lectures, recitations, office hours) • My office hours are T,TH 1:30-2:30 and by appointment – just send mail to set up a time

9. Logistics • Do readings before lecture • Do the “you try it” exercises as you read the logic text

10. Software with LPL TextFitch • Named for Frederic Fitch • Construct formal proofs • Prove an argument is valid

11. Software with LPL TextBoole • Named for George Boole • Construct truth tables • Verify a sentence is a tautology • Verify two sentences are tautologically equivalent • Prove an argument is valid

12. Software with the TextTarski’s World • Named for Alfred Tarski • Construct sentences in first-order logic • Determine if a sentence is true wrt a world • Create a world that shows that an argument is invalid

13. Software with TextSubmit • Verify your solutions are correct (without instructor seeing) • Submit homework for final grading to TA (by due date)

14. More Notes on Homework • Many exercises will be submitted to the grade grinder • Please request that your grade assessments be sent to the TA (not to me!) • You may submit your solutions as many times as you like, but please send your TA only a single report per assignment • Now, let’s look at the webpage and syllabus…

15. Introduction to Logic

16. Examples circle(a) small(a) All x All y ((cube(x) ^ tet(y))  (leftof(x,y) ^ frontof(x,a))) feed(max,scruffy) All x All y ((pet(x,y) ^ hungry(y))  feed(x,y))

17. Logic • A simple grammar. Each sentence has a single interpretation (unlike English!) • Used to describe a world, which we define. • Once we define the world, we can say what things names refer to, and whether a logical sentence is true or false.

18. Names • Constants are used to name existing objects • a, b, c, d, e, f • max, claire, carl • No constant can name more than one object • An object can have more than one name or no name at all

19. Predicates • A property possessed by an object • Shape (e.g., Tet, Cube) • Size (e.g., Small, Large) • A relationship among objects • Shape relationship (e.g., SameShape) • Size relationship (e.g., Smaller) • Positional relationship (e.g., Between, LeftOf) • Equality =

20. Quick Example • Ackermann’s sentences and world in Tarski • Properties: • Cube, Tet (4 faces), Dodec (12 faces), Medium • Relations • Backof, Leftof • (Click verify, and you see that one of the sentences is false about the world)

21. Predicates • Each predicate has a fixed number of arguments or “arity”. This is the number of constants the predicate needs to form a sentence. • In English, OK, but not in logic: • Susanna is taller than Dimitri • Susanna is taller than Dimitri and Jerome

22. Predicates • Predicates must be “determinate” • Suppose p is an n-ary predicate. • For every n-tuple <o1,o2,…,oN> of objects, p(o1,o2,…,oN) is true or false (not kind of true). • What’s an n-tuple? • An n-tuple is a collection of n objects where order matters. Duplicates are allowed. In contrast, sets may not have duplicates, and the members of sets are not ordered.

23. Atomic Sentences (so far) • A sentence formed by a single predicate followed by one or more names • Max is tall Tall(max) • e is larger than b Larger(e,b) • e is identical to a e = a • A sentence expresses a claim that is either true or false

24. Atomic Sentences (so far) • Predicate(arg1, arg2,…, argn) • Predicates have names beginning with an uppercase letter or are represented by an operator symbol • The number of arguments is called the predicate’s arity • The order of the arguments is importantLarger(e,c) – e is larger than cLarger(c,e) – c is larger than eBetween(a,b,e) – a is between b and eBetween(b,a,e) – b is between a and e • =(a,b) • a and b are identical • Usually, written in infix form a = b

25. Function Symbols • A function is used to express complex names (a reference to an individual without using a name) • father(b) – b’s father • Used in a sentence: Tall(father(b)) • password(c) – c’s password • Used in a sentence: Long(password(c)) • A function may be nested • father(father(max)) • Used in a sentence: Short(father(father(max))) • Can’t nest predicates • Tall(Tall(max)) * not a legal sentence*

26. Functional Expressions • Function(arg1, arg2,…, argn) • Function names begin with a lowercase letter or are expressed with a symbol • father(max) Max’s father • father(mother(max)) Max’s mother’s father • youngestChild(max,ann) Max and Ann’s youngest child • *(5,+(2,4)) 30

27. Atomic Sentences (so far) • A sentence formed by a single predicate followed by one or more terms • A term is either a constant or a functional expression

28. Example Atomic Sentences (which are functions and which are predicates?) Predicate(term1,term2,…,term-n) Happy(bossof(sally)) Father(bill) Tall(fatherOf(motherOf(sally))) Happier(motherOf(bill),bossof(fatherOf(max))) Fake(santa(rossParkMall))) Real(santa(robinsonTownCenter)))

29. ConnectivesApply to sentences tocreate more complex sentences. • Not  • And, Or ,  • Material Conditional  • Biconditional 

30. Examples • Larger(e,c) • Cube(b)  Large(b) • SameRow(e,c)  BackOf(e,b) e is not larger than c b is a cube or b is large e and c are in the same row and e is in back of b

31. First Order Logic • Names • Predicates • Functions • Connectives Are there more?

32. Example FOL

33. Brando is Nancy’s favorite actor. brando = favoriteActor(nancy) Translation • Nancy’s favorite actor is better than Max’s favorite actor. • BetterActor(favoriteActor(nancy), favoriteActor(max)) • Sean is his own favorite actor. • sean = favoriteActor(sean) • Brando is someone’s favorite actor. • x(brando = favoriteActor(x))

34. Quantifiers and Variables • For every x x • x (man(x)  mortal(x)) • There exists y y • x(brando = favoriteActor(x))

35. First Order Logic • Names • Predicates • Functions • Connectives • Quantifiers and variables Revised List