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Logic Programming

Logic Programming. Formal Logics- Recap. Formulas w/out quantifiers Free Variables Bound Variables Assignments and satisfaction Validity and satisfiability. Formal Logics- Proofs. Satisfaction of a formula should be proved

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Logic Programming

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  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

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