1 Inference in First Order Logic CS 171/271 (Chapter 9) Some text and images in these slides were drawn from Russel & Norvig’s published material

2 Inference Algorithms Reduction to Propositional Inference Lifting and Unification Chaining Resolution

3 Propositionalization Strategy: convert KB to propositional logic and then use PL inference Ground atomic sentences become propositional symbols What about the quantifiers?

4 Example KB in FOL:  x King(x)  Greedy(x)  Evil(x) King(John) Greedy(John) Brother(Richard,John) The last 3 sentences can be symbols in PL Apply Universal Instantiation to the first sentence

5 Universal Instantiation UI says that from a universally quantified sentence, we can infer any sentence obtained by substituting a ground term for the variable Back to Example From:  x King(x)  Greedy(x)  Evil(x) To: King(John)  Greedy(John)  Evil(John) King(Richard)  Greedy(Richard)  Evil(Richard) …

6 Issue with UI Ground terms: all symbols that refer to objects as well as function applications (recall that function applications return objects) For example, suppose Father is a function: Father(John) and Father(Richard) are also objects/ground terms But so are Father(Father(John)) and Father(Father(Father(John))) Infinitely many ground terms/instantiations

7 Existential Instantiation Whenever there is a sentence,  x P, introduce a new object symbol called the skolem constant and then add the unquantified sentence P, substituting the variable with that constant Example: From:  x Crown(x)  OnHead(x, John) To: Crown(C new )  OnHead(C new, John)

8 Substitution UI and EI apply substitutions A substitution is represented by a variable v and a ground term g; {v/g} Can have sets of these pairs if there are more variables involved Let  be a sentence (possibly containing v) SUBST( {v/g},  ) stands for the sentence that applies the substitution to 

9 UI and EI Defined UI:  v α ___ for any ground term g SUBST({v/g}, α) EI:  v α ___ for some constant symbol k not SUBST({v/k}, α) yet in the knowledge base

10 Back to Propositionalization Given a KB in FOL, convert KB to PL by 1. applying UI and EI to quantified sentences 2. converting atomic sentences to symbols If there are no functions (Datalog KB), UI application does not result in infinitely many sentences Regular PL Inference can now be carried out without problems What if there are functions?

11 Dealing with Infinitely Many Ground Terms Can set a depth-limit for ground terms Depth specifies levels of function nesting allowed Carry out reduction and inference process for depth 1, then 2, then 3, … Stop when entailment can be concluded This works if there is such a proof, but goes into an endless loop if there is not The strategy is complete The entailment problem in this sense is semidecidable

12 Inefficiencies in Propositionalization An inordinate number of irrelevant sentences may be generated, resulting from UI This motivates generating only those sentences that are important in entailment

13 Example Suppose KB contains:  x King(x)  Greedy(x)  Evil(x)  y Greedy(y) King(John) Suppose we want to conclude Evil(John) Because of the existence of objects other than John (such as Richard) and the existence of functions, UI will generate many sentences

14 Example, continued It is sufficient to generate: King(John)  Greedy(John)  Evil(John) Greedy(John) Which is just: SUBST( {x/John}, King(x)  Greedy(x)  Evil(x) ) SUBST( {y/John}, Greedy(y) ) Applying the substitution matches the Premises: King(x)  Greedy(x) With other sentences in the KB: Greedy(y), King(John)

15 Lifted Modus Ponens Lifting: Raising propositional inference rules to first order logic Example: Generalized Modus Ponens If there is a substitution θ, such that SUBST(θ, p i ) = SUBST(θ, p i ’) for all i, then p 1 ', p 2 ', …, p n ’, ( p 1  p 2  …  p n  q) _______________________________________________________________________________ SUBST(θ,q) In our example,  = {x/John, y/John}

16 Unification Process that makes logical expressions identical Goal: match the premises of implications so that conclusions can be derived UNIFY algorithm takes two sentences and returns a unifier (substitution) if it exists

17 Unification Algorithm

18 Unification Algorithm

19 About UNIFY UNIFY returns a Most General Unifier (MGU) There are efficiency issues with OCCUR-CHECK function May need to standardize apart: rename variables to avoid name clashes Unification is a key component of all first-order algorithms

20 What’s Next? Forward and backward chaining algorithms for FOL that use unification Resolution-based theorem proving systems