Formal Languages, Part Two

1. Formal Languages, Part Two SIE 550 Lecture Matt Dube Doctoral Student – Spatial

2. Where we left off on Friday • Formal Languages • Terminal Symbols – base-level instances • “symbols that we can’t break down further”-Jake • Non-terminal Symbols – rules to extract specific sequences of terminal symbols • “symbols that can be broken down” • Well Formed Formulas (WFF) • Valid outputs of a formal language • Semantics DO NOT matter! • Logic is not involved NON- TERMINAL SYMBOLS atom::=proton{proton}{electron}{neutron} TERMINAL SYMBOLS Carbon 14 = 6 protons, 6 electrons, 8 neutrons Helium = 2 protons, 5 electrons, 2 neutrons

3. Homework from Friday • You are a math teacher and want to take a nap during class. • Create a formal language that will generate addition and subtraction problems involving arbitrary terms over positive integers. 48 + 97 –9+ 17 – 4–1 … =?

4. One possible solution • start ::= problem • problem ::= integer sign {term} integer end • term ::= integer sign • integer ::= digit/0 {digit} • sign ::= “+” | “-” • end ::= “=?” • digit/0 ::= “1” | “2” | … | “9” • digit ::= “0” | “1” | “2” | … | “9”

5. undo Another Example to Discuss Language for Computer Drawing? Just like text, we can interpret this as a formal language and use the same concepts and terminology! Terminal Symbols? Non-terminal Symbols?

6. Questions, Comments, Concerns? • Please voice before we move forward

7. Today’s Class • Apply formal languages to a class of English sentences • Propositional logic • First Order Languages • Predicates

8. Peter Naur John Backus Backus-Naur Form • Recall the operators: | ::= [] {} “” () • These operators are referred to as the Backus-Naur Form. • “Language of the language” – meta-language • Standard syntax

9. Example of BNF • start ::= transitive_sentence • transitive_sentence ::= [article] noun verb [article] noun end • article ::= “a” | “an” | “the” • noun ::= “tricycle” | “cat” | “veterinarian” | “rubber” | “ferrari” • verb ::= “rides” | “fixes” | “eats” | “burns” | “jumps” • end ::= “.” the cat fixes the veterinarian. an ferrari eats a cat. ferrari jumps a veterinarian The tricycle burns rubber. the cat rode a tricycle. a tricycle burns the ferrari.

10. What are the goals? • We want a computer to retrieve correct information. • We want a computer to tell us correct information. • We want a computer to test the correctness of information. • We want a computer to infer correct information. Query Language Print Commands Formal Language ??? How should we go about that?

11. Well… • How do we do it minus a computer? • Propositional Logic • Related statements and then moving information between them • Example: • Jake lives in Hampden • Hampden is in Maine • Therefore Jake lives in Maine. • Very effective system if… You Can Think!

12. You expect me to think? I am only a machine! My aren’t they temperamental sometimes!

13. How do we rectify the situation? • We need a logic system • “Propositional logic”-like, but… • We know a computer can: • Handle a formal language • Cross-reference and replace terms • Pass information through code • So what’s missing? • We need functions to pass through

14. First Order Logic • First order logic = functional propositional logic • Example: • man(X) • X=socrates • X=plato • man(socrates) -> socrates is a man. • man(plato) -> plato is a man. • Could we use this? YES • What if we established “man(socrates)” as a terminal symbol and put something above it in the code saying mortal(X)::=man(X)? • Do we find out anything else about socrates?

15. Note • Things in a computer program starting with a capital letter are variables. • Things in a computer program starting with a lowercase letter are constants. • IMPLICATION: The program sees what satisfies the capital letter and then passes that information through the rest of the program.

16. A New Language • It is now time to define a new type of formal language: a first order language. • WFFs in a first order language • a predicate • (WFF or WFF) • (WFF and WFF) • (WFF implies WFF) • (WFF = WFF) WHAT IS A PREDICATE? animal( )

17. Predicates • Gateways • man( ) • Whatever we put inside the parentheses is a man. • father( , ) • Whatever we put first is the father; second is the child. • User defined linkages • Predicates are non-terminal symbols • Variables are non-terminal symbols • Constants are terminal symbols • Number of terms in the predicate = arity

18. Use formal language now • Treat predicate information based on constants as if they were terminal symbols • man(mike), man(henry), etc. • Treat predicate information based on variables as if they were non-terminal symbols • mortal(X)::=man(X) • If X is a man, than X is also a mortal. • We can now build in as many crazy combinations as we choose.

19. Example PREDICATE CALCULUS • parent(X,A) ::= father(X,A) | mother(X,A) • child(X,A) ::= father(A,X) | mother(A,X) • father(john,suzy) • father(jack,daniel) • mother(almice,jack) • mother(almice,tim) TERMINAL SYMBOLS Suzy John AXIOMS Jack Almice What can we say about: Tim Daniel

20. Other Example Commands • We used an or statement • Mother or father makes you a parent • father(X,A)=child(A,X) • If X is A’s father, then A is a child of X IMPLY • parents(X,Y,A)=parent(X,A) parent(Y,A) • X and Y are A’s parents if X is A’s parent and Y is also A’s parent AND