Using Definite Knowledge Notes for Ch 3 of

Using Definite Knowledge Notes for Ch. 3 of Poole et al. CSCE 580 Marco Valtorta UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Databases and Recursion • Datalog can be used to define relational algebra • Datalog is more powerful than relational algebra: – It has variables – It allows recursive definitions of relations – Therefore, e. g. , datalog allows the definition of the transitive closure of a relation. • Datalog has no function symbols: it is a subset of definite clause logic. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Relational DB Operations • A relational db is a kb of ground facts • datalog rules can define relational algebra database operations • The examples refer to the database in course. pl UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Selection • Selection: – cs_course(X) <- department(X, comp_science). – math_course(X) <- department(X, math). UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Union • Union: multiple rules with the same head – cs_or_math_course(X) <- cs_course(X). – cs_or_math_course(X) <- math_course(X). • In the example, the cs_or_math_course relation is the union of the two relations defined by the rules above. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Join • Join: the join is on the shared variables, e. g. : – ? enrolled(S, C) & department(C, D). – One must find instances of the relations such that the values assigned to the same variables unify • in a DB, unification simply means that the same variables have the same value! UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Projection • When there are variables in the body of a clause that don’t appear in the head, you say that the relation is projected onto the variables in the head, e. g. : – in_dept(S, D) <- enrolled(S, C) & department(C, D). • In the example, the relation in_dept is the projection of the join of the enrolled and department relations. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Recursion • Define a predicate in terms of simpler instances of itself – Simpler means: easier to prove • Examples: – west in west. pl – live in elect. pl • “Recursion is a way to view mathematical induction top-down. ” UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Well-founded Ordering • Each relation is defined in terms of instances that are lower in a well-founded ordering, a one-to-one correspondence between the relation instances and the non-negative integers. • Examples: – west: induction on the number of doors to the west---imm_west is the base case, with n=1. – live: number of steps away from the outside--live(outside) is the base case. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Verification of Logic Programs • Verifiability of logic programs is the prime motivation behind using semantics! • If g is false in the intended interpretation and g is proved from the KB, – Find the clause used to prove g – If some atom in the body of the clause is false in the intended interpretation, then debug it – Else return the clause as the buggy clause instance UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Verifiability II • Also need to show that all cases are covered: if an instance of a predicate is true in the intended interpretation, then one of the clauses is applicable to prove the predicate. • Also need to show termination---this is in general impossible, due to semidicidability results, but it is possible in many practical situations. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Limitations • No notion of complete knowledge! – Cannot conclude that something is false. – Cannot conclude something from lack of knowledge. Example: • The relation empty_course(X) with the obvious intended interpretation cannot be defined from enrolled(S, C) relation. • The Closed World Assumption (CWA) allows reasoning from lack of knowledge. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering

Case Study: University Rules • • univ. pl DB of student records DB of relations about the university Rules about satisfying degree requirements. UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering