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First Order Logic Logic is a mathematical attempt to formalize the way we think. First-order predicate calculus was created in an attempt to mechanize.

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Presentation on theme: "First Order Logic Logic is a mathematical attempt to formalize the way we think. First-order predicate calculus was created in an attempt to mechanize."— Presentation transcript:

1 First Order Logic Logic is a mathematical attempt to formalize the way we think. First-order predicate calculus was created in an attempt to mechanize reason.

2 First Order Logical Representations All zombies eat brains. This assumption can be represented in propositional logic by the propositional variable p. In first order logic, the expression can be broken down X (zombie(X) eatsbrains(X)) X (zombie(X) eats(X, brains)) X Y (zombie(X) eats(X, Y) ^ brains(Y)) Some people with no heartbeats are zombies. X (person(X) ^ zombie(X) ^ ¬ heartbeat(X))

3 Natural Deductive Rules Natural deductive rules are logical rules accepted as being intuitively true. Universal Instantiation: If a variable is assigned a value, it will always have that value. If X = joe, X (zombie(X) eatsbrains(X)) is the same as (zombie(joe) eatsbrains(joe)) Modus Ponens: For some pair of expressions p and q, p ^ (p q) q. If zombie(X), (zombie(X) eatsbrains(X)) becomes (eatsbrains(X))

4 Truth Tables Truth tables can be constructed from first order logic by transforming expressions into atomic formulas (something with value true or false). Atomic formulas can be constructed from first order logic by either quantifying a formula or mapping it to a value (analogous to valuation). Interpretations are analogous to truth tables.

5 Truth Tables zombie(joe) zombie(X) eatsbrains(joe) eatsbrains(X)... 0?0?1?0?0?0?1?0? zombie(joe) zombie(fred) eatsbrains(joe) eatsbrains(fred) 0000100100001001 By mapping (valuation) of X to fred

6 Brief Intro: Herbrand Base The Herbrand base is the set of ground atomic formulas. In the previous example, this is the set {zombie(joe), zombie(fred), eatsbrains(joe), eatsbrains(fred)}. A model is satisfiable if and only if it has a Herbrand Interpretation (a valid truth assignment given its Herbrand base).

7 Function Symbols Function symbols map an individual in the domain to another in the domain. Everyone loves their mother. In the expression X loves(X, mother(X)), mother(X) maps X to the element in the domain. Peano's Axioms for Arithmetic In the expression X (number(X) number(S(X)), the occurance of number in number(X) is a function symbol. In the second occurrence, number is a predicate and S(X) is a function symbol.

8 Relational Databases Assume there is a relational database with a patient table, visit table, and a prescription table. The patient table has fields ID, Date of Birth, and Gender. The visit table has fields ID, Visit Date, and Diagnosis Code. The prescription table has fields ID, Date, and Drug.

9 Relational Databases Patient Visit Prescription

10 Relational Databases In a relational database, the table can be represented by an atomic formula of a predicate objects composed of constants and/or function symbols. One such formula exists for each patient. Patient table: patient(id, date of birth, gender) Visit table: visit(id, date, diagnosis code) Prescription table: prescription(id, date, drug)

11 Relational Databases Any query can be represented as a formula. To find all male patients with diagnosis code 111, use the query formula P (interest(P)patient(P, B, male) ^ visit(P, D, 111) The query result can define a new table containing the desired data. Extensional: Write out entries in a table. Intensional: Write out tables via formula

12 First Order Logical Inference-Datalog Datalog is a syntactic variation on relational database queries. It forms an intensional representation to the query equivalent to those formed by relational algebra. All quanitifiers are assumed universal and out front of the formula. All formulas are atomic or implications. No function symbols are used. (interest(P)patient(P, B, male), visit(P, D, 111)

13 Queries in Datalog Datalog queries can be called with a goal. male-flu(P, D)(patient(P, B, male), visit(P, D, 111) Goals can be satisfied by finding all bindings on the goal variables and using pattern matching. The process of matching all atomic formulas by binding the variables to force matching is known as Unification.

14 Unification is most frequently executed by running a backtracking search for proofs satisfying all subgoals. Running unification on the male-flu example, if there exists patient(1, 1-1-01, male) and visit(1, 3-3-03, 111), there will be a new row binding in the resulting male-flu table with P=1, D=3-3-03. Queries in Datalog

15 Prolog All formulas have the format: atomic formula atomic formula [, atomic formula(s)] written in Prolog as: atomic formula :- atomic formula [, atomic formula(s)]. Prolog uses only universally quantified variables. Some arithmetic functions are allowed, such as W is X*Y Specification of correct output is allowed.

16 Prolog Lists cons(H, T) constructs the list with H as the head and T as the tail. This is equivalent to [H|T]. [1,2,3]=[1|[2, 3]]=[1|2|3|nil]]]=cons(1, cons(2, cons(3, nil))) where nil is the empty list. All functions are evaluated based on unification. Conventionally, arguments to a Prolog expression are given with inputs first, then outputs. Prolog is Turing-complete so long as function symbols are allowed.

17 Defining Prolog Functions Member function: member(H, [H|T]). member(H, [S|T]) member(H,T). Insertion Sort: sort([ ], [ ]). sort([H|T], L) sort(T, T2), insert(H, T2, L). insert(H, [A|T], [H,A|T] H = A, insert(H, T, L).

18 Running Prolog On CSL machines, Prolog is run via the yap interpreter. To load a file into the interpreter: ?- consult('filename'). Once a file has been loaded, definitions contained in that file may be executed. The code will then be run top down until a solution is found and then bound to the output. ?- sort([5, 1, 10, 3, 9], X). %Bind sorted list to X

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