First Order Predicate Logic CPSC 386 Artificial Intelligence Ellen Walker Hiram College

First Order Logic (ch. 8) Has a greater expressive power than propositional logic –We no longer need a separate rule for each square to say which other squares are breezy/pits Allows for facts, objects, and relations –In programming terms, allows classes, functions and variables

Components of First Order Logic Term –Constant, e.g. Red –Function of constant, e.g. Color(Block1) Atomic Sentence –Predicate relating objects (no variable) Brother (John, Richard) Married (Mother(John), Father(John)) Complex Sentences –Atomic sentences + logical connectives Brother (John, Richard)  Brother (John, Father(John))

More Components… Quantifiers –Each quantifier defines a variable for the duration of the following expression, and indicates the truth of the expression… Universal quantifier “for all”  –The expression is true for every possible value of the variable Existential quantifer “there exists”  –The expression is true for at least one value of the variable

Quantification examples Everyone like chocolate –  x, likes(x, chocolate) Someone likes chocolate –  x, likes(x, chocolate) All children like chocolate –  x, child(x)  likes(x, chocolate) Everyone likes chocolate unless they are allergic to it –  x, likes(x, chocolate)  allergic(x, chocolate) –  x,  allergic (x, chocolate)  likes(x, chocolate)

Quantification and Negation Not everyone like chocolate  (  x, likes(x, chocolate))  x,  likes(x, chocolate) No one likes chocolate  (  x, likes(x, chocolate))  x,  likes(x, chocolate)

Nesting Variables Everyone likes some kind of food  y  x, food(x)  likes(y, x) There is a kind of food that everyone likes  x  y, food(x)  likes(y, x) Someone likes all kinds of food  y  x, food(x)  likes(y, x) Every food has someone who likes it  x  y, food(x)  likes(y, x)

Quantification with Classes To say “everyone likes chocolate”, the following is too broad! –  x, likes(x, chocolate) –Example: likes (chocolate, chocolate) We mean: Every one (who is a person) likes chocolate –  x, person(x)  likes(x, chocolate) Essentially, the left side of the rule declares the class of the variable x Constraints like this are often called “domain constraints”

Equality We allow the usual infix = operator –Father(John) = Henry –  x, sibling(x, y)   (x=y) Generally, we also allow mathematical operations when needed, e.g. –  x,y, NatNum(x)  NatNum(y)  x = (y+1)  x > y

Try some! Every successful person has a mentor. Every successful person is a mentor. Every mentor is a successful person. No person is successful without a mentor. There are no successful people who are mentors.

Using FOL to Model the World Tell the system assertions –Facts : Tell (KB, person (John) ) –Rules: Tell (KB,  x, person(x)  likes(x, chocolate)) Ask questions –Ask (KB, person(John)) –Ask (KB, likes(John, chocolate)) –Ask (KB, likes(x, chocolate))

Types of Answers Fact is in the KB –Yes. Fact is not in the KB –Yes (if it can be proven from the KB) –No (otherwise) Fact contains variables –List of bindings for which the fact can be proven, e.g. ((x John) (x Ellen) … )

Example Domains (pp ) Kinship domain –What is a second cousin once removed, anyway? Numbers, sets, and lists –This one is a classic. You should understand these, even if you don’t memorize them. The Wumpus World –Note how much simpler the description is in FOL! “Whatever your domain, if the axioms correctly and completely describe how the world works, any complete logical inference procedure will infer the strongest possible description of the world, given the available percepts” (AIMA, p. 260)

Knowledge Engineering Translating real-world knowledge to a usable form 1. Identify the task (PEAS description) 2. Assemble the relevant knowledge (knowledge acquisition) 3. Decide on the vocabulary: predicates, functions, constants (ontology) 4. Encode general knowledge 5. Encode a specific problem instance 6. Query and debug …

Types of Rules Diagnostic Rules lead from observed effects to hidden causes (like medical diagnosis) Causal Rules lead from (hidden) causes to observed effects (like simulation systems) Systems should reason either one way or the other, but not mix rules of both types (to avoid circular reasoning!)

Necessary algorithms (Ch. 9) We already know enough to implement TELL (although maybe not efficiently) But how do we implement ASK? Recall 3 cases –Direct matching –Finding a proof (inference) –Finding a set of bindings (unification)

Inference with Quantifiers Universal Instantiation: –Given  x, person(x)  likes(x, chocolate) –Infer person(john)  likes(john, chocolate) Existential Instantiation: –Given  x, likes(x, chocolate) –Infer  likes(S1, chocolate) –S1 is a “Skolem Constant” that is not found anywhere else in the KB and refers to (one of) the indviduals that likes chocolate.

Reduction to Propositional Inference Use instantiation rules to create relevant (!) propositional facts in the KB, then use propositional reasoning Problems: –May generate infinite sentences when it cannot prove –Will generate many irrelevant sentences along the way!

Generalized Modus Ponens This is a general inference rule for FOL that does not require instantiation Given: – p1’, p2’ … pn’ (p1  … pn)  q –Subst(theta, pi’) = subst(theta, pi) for all i Conclude: –Subst(theta, q)

Simple Example of GMP Rule: king(x)  rich(x) –p1 = king(x), q = rich(x) Fact: king(John) –p1’ = king(John) Theta is x = John –Subst(x=John, king(x)) = king(John) –Subst(x=John, rich(x)) = rich(John) We can conclude rich(John).

GMP in “CS terms” Given a rule containing variables If there is a consistent set of bindings for all of the variables of the left side of the rule (before the arrow) Then you can derive the result of substituting all of the same variable bindings into the right side of the rule

GMP Example  x, parent(x,y)  parent(y,z)  grandparent(x,z) Parent(james, john), parent(james, richard), parent(harry, james), parent(charles,hal) We can derive: –Grandparent(harry, john), bindings: ((x harry) (y james) (z john)) –Grandparent(harry, richard), bindings: ((x harry) (y james) (z richard))

Unification Process of finding all legal substitutions Given 2 expressions Find a set of unifiers (variable bindings) that causes the expressions to “match” This is a beautifully recursive algorithm.

Base Cases for Unification If two expressions are identical, the result is (NIL) (succeed with empty unifier set) If two expressions are different constants, the result is NIL (fail) If one expression is a variable and is not contained in the other, the result is ((variable other-exp))

Recursive Case If both expressions are lists, –Combine the results of unifying the CAR with unifying the CDR –(append (unify (car list1) (car list2)) (unify (cdr list1) (cdr list2))

A few more details… Don’t reuse variable names –Before actually unifying, give each rule a separate set of variables. –The lisp function gentemp creates uniquely numbered variables Keep track of bindings –If a variable is already bound to something, it must retain the same value throughout the computation. –This requires substituting each successful binding in the remainder of the expression.

Storage and retrieval Most systems don’t use variables on predicates Therefore, hash statements by predicate for quick retrieval (predicate indexing) Subsumption lattice for efficiency (see p. 279)

Forward Chaining Given a new fact, generate all consequences Assumes all rules are of the form –C1 and c2 and c3 and…. --> result Each rule & binding generates a new fact This new fact will “trigger” other rules Keep going until the desired fact is generated (Semi-decidable as is FOL in general)

Efficient Forward Chaining Order conjuncts appropriately –E.g. most constrained variable Don’t generate redundant facts; each new fact should depend on at least one newly generated fact. –Production systems –RETE matching –CLIPS

Backward Chaining Consider the item to be proven a goal Find a rule whose head is the goal (and bindings) Apply bindings to the body, and prove these (subgoals) in turn If you prove all the subgoals, increasing the binding set as you go, you will prove the item. Logic Programming (SWI Prolog)