1. Logic Programming

2. Formal Logics- Recap Formulas – w/out quantifiers Free Variables Bound Variables Assignments and satisfaction Validity and satisfiability

3. Formal Logics- Proofs Satisfaction of a formula should be proved – Based on facts (assertions, assumptions, axioms) and inference rules We are led to a notion of Proof Trees – Why trees? Proofs are informative! – E.g. assignment to an existential variable

4. Logic and Programming Can we automate the process of proving logical statements? First we need a syntax for defining assertions, rules,.. Then we need an automated proof system

5. Logic and Programming Can we automate the process of proving logical statements? First we need a syntax for defining facts, rules, statements to-be-proved Then we need an automated proof system Note that in principle we may need to use a fact multiple times so no a-priori bound on the proof size

6. Applications Theory behind database query languages is based on logic – Querying a database = looking for a satisfying assignment to a query, based on the database facts Automated Verification – Describe (e.g. Java) program as logical rules, constraints on behavior as query Automated Theorem Proving

7. Is it even feasible? Hilbert’s first decision problem: given a first order logic formula, decide whether it is satisfiable Undecidable!

8. Tradeoff Classic tradeoff between expressive power of languages and complexity of evaluation There are decidable fragments of First Order Logic which are still quite expressive SQL, Datalog, Description Logic, Prolog,…

9. Sub-cases We start with Relational Logic Programming – Facts are relations between elements, relatively simple rules allow to infer new relations Then move on to Full Logic Programming – Allow function symbols etc.

10. Syntax We consider the Prolog syntax for describing facts and rules Very similar to the Datalog syntax Other LP languages use different syntax, but similar ideas

11. Facts % Signature: parent(Parent, Child)/2 % Purpose: Parent is a parent of Child parent(rina, moshe). parent(rina, rachel). parent(rachel, yossi). parent(reuven, moshe). % Signature: male(Person)/1 % Purpose: Person is a male. male(moshe). male(yossi). male(reuven). % Signature: female(Person)/1 % Purpose: Person is a female. female(rina). female(rachel).

12. Variables and constants Variables begin with an upper-case, constants with lower-case Variables in facts are universally quantified

13. Rules father(Dad, Child) :- parent(Dad, Child), male(Dad). ancestor (Anc, Child) :- parent(Anc, X), ancestor(X, Child).

14. Queries ?- father(D,C). – D = reuven, – C = moshe. ?- father(reuven,moshe). – true. ?- mother(M,C). – M = rina, – C = moshe ; – M = rina, – C = rachel ; – M = rachel, – C = yossi ; – fail.

15. Variables Variables in facts are universally quantified – In queries they are existentially quantified father(X,Ron). ?- father(Y,Ron). true

16. Proofs