Note that quantification is over “objects”. This is what makes it “First-order” If you can quantify over predicate symbols, then it will be “Second-order” E.g. Can’t say “A symmetric predicate is one which holds even if the arguments are reversed” in a single sentence E.g2. Can’t write Mathematical Induction schema as a single FOPC sentence (Goedel’s incompleteness theorem won’t hold for FOPC ) Why roughly? Because if you have a function symbol, then the conjunction (or disjunction) can be infinitely large
Apt-pet An apartment pet is a pet that is small Dog is a pet Cat is a pet Elephant is a pet Dogs and cats are small. Some dogs are cute Each dog hates some cat Fido is a dog
Caveat: Decide whether a symbol is predicate, constant or function… Make sure you decide what are your constants, what are your predicates and what are your functions Once you decide something is a predicate, you cannot use it in a place where a predicate is not expected! In the previous example, you cannot say
Family Values: Falwell vs. Mahabharata According to a recent CTC study, “….90% of the men surveyed said they will marry the same woman..” “…Jessica Alba.”
Caveat: Order of quantifiers matters “either Fido loves both Fido and Tweety; or Tweety loves both Fido and Tweety” “ Fido or Tweety loves Fido; and Fido or Tweety loves Tweety” Loves(x,y) means x loves y
More on writing sentences Forall usually goes with implications (rarely with conjunctive sentences) There-exists usually goes with conjunctions—rarely with implications Everyone at ASU is smart Someone at UA is smart Will hold if there exists a single person who doesn’t go to UA (in which case At(x,UA) will be false, and so the implication will be true, making the entire disjunction true..
Notes on encoding English statements to FOPC You get to decide what your predicates, functions, constants etc. are. All you are required to do it be consistent in their usage. When you write an English sentence into FOPC sentence, you can “double check” by asking yourself if there are worlds where FOPC sentence doesn’t hold and the English one holds and vice versa Since you are allowed to make your own predicate and function names, it is quite possible that two people FOPCizing the same KB may wind up writing two syntactically different KBs If each of the DBs is used in isolation, there is no problem. However, if the knowledge written in one DB is supposed to be used in conjunction with that in another DB, you will need “Mapping axioms” which relate the “vocabulary” in one DB to the vocabulary in the other DB. This problem is PRETTY important in the context of Semantic Web The “Semantic Web” Connection
Two different Tarskian Interpretations This is the same as the one on The left except we have green guy for Richard Problem: There are too darned many Tarskian interpretations. Given one, you can change it by just substituting new real-world objects Substitution-equivalent Tarskian interpretations give same valuations to the FOPC statements (and thus do not change entailment) Think in terms of equivalent classes of Tarskian Interpretations (Herbrand Interpretations) We had this in prop logic too—The real World assertion corresponding to a proposition
10/31 Midterm returned Make-up class on Friday 11/9 (morning—usual class time)
Herbrand Interpretations Herbrand Universe –All constants Rao,Pat –All “ground” functional terms Son-of(Rao);Son-of(Pat); Son-of(Son-of(…(Rao)))…. Herbrand Base –All ground atomic sentences made with terms in Herbrand universe Friend(Rao,Pat);Friend(Pat,Rao);Friend(P at,Pat);Friend(Rao,Rao) Friend(Rao,Son-of(Rao)); Friend(son-of(son-of(Rao),son-of(son- of(son-of(Pat)) –We can think of elements of HB as propositions; interpretations give T/F values to these. Given the interpretation, we can compute the value of the FOPC database sentences If there are n constants; and p k-ary predicates, then --Size of HU = n --Size of HB = p*n k But if there is even one function, then |HU| is infinity and so is |HB|. --So, when there are no function symbols, FOPC is really just syntactic sugaring for a (possibly much larger) propositional database Let us think of interpretations for FOPC that are more like interpretations for prop logic
But what about Godel? Godel’s incompleteness theorem holds only in a system that includes “mathematical induction”—which is an axiom schema that requires infinitely many FOPC statements –If a property P is true for 0, and whenever it is true for number n, it is also true for number n+1, then the property P is true for all natural numbers –You can’t write this in first order logic without writing it once for each P (so, you will have to write infinite number of FOPC statements) So, a finite FOPC database is still semi-decidable in that we can prove all provably true theorems
Proof-theoretic Inference in first order logic For “ground” sentences (i.e., sentences without any quantification), all the old rules work directly (think of ground atomic sentences as propositions) –P(a,b)=> Q(a); P(a,b) |= Q(a) –~P(a,b) V Q(a) resolved with P(a,b) gives Q(a) What about quantified sentences? –May be infer ground sentences from them…. –Universal Instantiation (a universally quantified statement entails every instantiation of it) –Existential instantiation (an existentially quantified statement holds for some term (not currently appearing in the KB). Can we combine these (so we can avoid unnecessary instantiations?) Yes. Generalized modus ponens Needs UNIFICATION
UI can be applied several times to add new sentences --The resulting KB is equivalent to the old one EI can only applied once --The resulting DB is not equivalent to the old one BUT will be satisfiable only when the old one is
How about knows(x,f(x)) knows(u,u)? x/u; u/f(u) leads to infinite regress (“occurs check”)
GMP can be used in the “forward” (aka “bottom-up”) fashion where we start from antecedents, and assert the consequent or in the “backward” (aka “top-down”) fashion where we start from consequent, and subgoal on proving the antecedents.
Apt-pet An apartment pet is a pet that is small Dog is a pet Cat is a pet Elephant is a pet Dogs, cats and skunks are small. Fido is a dog Louie is a skunk Garfield is a cat Clyde is an elephant Is there an apartment pet?
11/7 Are there irrational numbers p and q such that p q is a rational number? Hint: Suppose p=q= Rational Irrational Why is the set that is the set of all sets jumping up and down excitedly?..it couldn’t contain itself
Efficiency can be improved by re-ordering subgoals adaptively e.g., try to prove Pet before Small in Lilliput Island; and Small before Pet in pet-store.
Forward (bottom-up) vs. Backward (top-down) chaining Forward chaining fires rules starting from facts –Using P, derive Q –Using Q & R, derive S – Using S, derive Z – Using Z, Q, derive W –Using Q, derive J –No more inferences. Check if J holds. It does. So proved Backward chaining starts from the theorem to be proved –We want to prove J. –Using Q=>J, we can subgoal on Q –Using P=>Q, we can subgoal on P –P holds. We are done. Suppose we have P => Q Q & R =>S S => Z Z & Q => W Q => J P R We want to prove J Forward chaining allows parallel derivation of many facts together; but it may derive facts that are not relevant for the theorem. Backward chaining concentrates on proving subgoals that are relevant to the theorem. However, it proves theorems one at a time. Some similarity with progression vs. regression…
Datalog and Deductive Databases A deductive database is a generalization of relational database, where in addition to the relational store, we also have a set of “rules”. –The rules are in definite clause form (universally quantified implications, with one non-negated head, and a conjunction of non-negated tails) When a query is asked, the answers are retrieved both from the relational store, and by deriving new facts using the rules. The inference in deductive databases thus involves using GMP rule. Since deductive databases have to derived all answers for a query, top-down evaluation winds up being too inefficient. So, bottom-up (forward chaining) evaluation is used (which tends to derive non-relevant facts A neat idea called magic-sets allows us to temporarily change the rules (given a specific query), such that forward chaining on the modified rules will avoid deriving some of the irrelevant facts. Base facts P(a,b),Q(b) R(c).. Rules P(x,y),Q(y)=>R(y) ?R(z) RDBMS R(c); R(b).. Connection to Progression becoming goal directed w.r.t. P.G. reachability heuristics
Similar to “Integer Programming” or “Constraint Programming”
Generate compilable matchers for each pattern, and use them
Example of FOPC Resolution.. Everyone is loved by someone If x loves y, x will give a valentine card to y Will anyone give Rao a valentine card? y/z;x/Rao ~loves(z,Rao) z/SK(rao);x’/rao Anyone who gets a valentines card will not go on a rampage on 2/14 Prove: Rao won’t go on rampage..
Finding where you left your key.. Atkey(Home) V Atkey(Office) 1 Where is the key? Ex Atkey(x) Negate Forall x ~Atkey(x) CNF ~Atkey(x) 2 Resolve 2 and 1 with x/home You get Atkey(office) 3 Resolve 3 and 2 with x/office You get empty clause So resolution refutation “found” that there does exist a place where the key is… Where is it? what is x bound to? x is bound to office once and home once. so x is either home or office
Existential proofs.. Are there irrational numbers p and q such that p q is rational? Rational Irrational This and the previous examples show that resolution refutation is powerful enough to model existential proofs. In contrast, generalized modus ponens is only able to model constructive proofs..