Inference in first-order logic CE417: Introduction to Artificial - PowerPoint PPT Presentation

Inference in first-order logic CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2017 Soleymani “ Artificial Intelligence: A Modern Approach ” , 3 rd Edition, Chapter 9

Outline  Reducing first-order inference to propositional inference  Potentially great expense  Lifting inference (direct inference in FOL)  Unification  Generalized Modus Ponens  KB of Horn clauses  Forward chaining  Backward chaining  General KB  Resolution 2

FOL to PL  FOL to PL conversion  First order inference by converting the knowledge base to PL and using propositional inference.  How to remove universal and existential quantifiers?  Universel Instantiation  Existential Instantiation 3

Universal instantiation (UI)  Every instantiation of a universally quantified sentence is entailed by it: ∀𝑤 𝛽 𝑇𝑣𝑐𝑡𝑢( 𝑤 𝑕 , 𝛽)  𝑦 𝐿𝑗𝑜𝑕(𝑦) ∧ 𝐻𝑠𝑓𝑓𝑒𝑧(𝑦) ⇒ 𝐹𝑤𝑗𝑚(𝑦)  Example, 𝛽 :  𝑇𝑣𝑐𝑡𝑢 𝑦 𝐾𝑝ℎ𝑜 , 𝛽  𝐿𝑗𝑜𝑕(𝐾𝑝ℎ𝑜) ∧ 𝐻𝑠𝑓𝑓𝑒𝑧(𝐾𝑝ℎ𝑜) ⇒ 𝐹𝑤𝑗𝑚(𝐾𝑝ℎ𝑜)  𝑇𝑣𝑐𝑡𝑢( 𝑦 𝑆𝑗𝑑ℎ𝑏𝑠𝑒 , 𝛽)  𝐿𝑗𝑜𝑕(𝑆𝑗𝑑ℎ𝑏𝑠𝑒) ∧ 𝐻𝑠𝑓𝑓𝑒𝑧(𝑆𝑗𝑑ℎ𝑏𝑠𝑒) ⇒ 𝐹𝑤𝑗𝑚(𝑆𝑗𝑑ℎ𝑏𝑠𝑒)  𝑇𝑣𝑐𝑡𝑢( 𝑦 𝐺𝑏𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜) , 𝛽)  𝐿𝑗𝑜𝑕(𝐺𝑏𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜)) ∧ 𝐻𝑠𝑓𝑓𝑒𝑧(𝐺𝑏𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜)) ⇒ 𝐹𝑤𝑗𝑚(𝐺𝑏𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜)) 4

Existential instantiation (EI)  For any sentence 𝛽 , variable 𝑤 , and a new constant symbol 𝑙 : ∃𝑤 𝛽 𝑇𝑣𝑐𝑡𝑢( 𝑤 𝑙 , 𝛽)  Example, 𝛽 :  𝑦 𝐷𝑠𝑝𝑥𝑜(𝑦) ∧ 𝑃𝑜𝐼𝑓𝑏𝑒(𝑦, 𝐾𝑝ℎ𝑜)  𝑇𝑣𝑐𝑡𝑢 𝑦 𝐷 1 , 𝛽  𝐷𝑠𝑝𝑥𝑜(𝐷 1 ) ∧ 𝑃𝑜𝐼𝑓𝑏𝑒(𝐷 1 , 𝐾𝑝ℎ𝑜)  𝐷 1 is a new constant symbol, called a Skolem constant  EI can be applied one time (existentially quantified sentence can be discarded after it) 5

Reduction to propositional inference: Example  KB of FOL sentences:  𝑦 𝐿𝑗𝑜𝑕 𝑦  𝐻𝑠𝑓𝑓𝑒𝑧 𝑦  𝐹𝑤𝑗𝑚 𝑦  𝐿𝑗𝑜𝑕 𝐾𝑝ℎ𝑜  𝐻𝑠𝑓𝑓𝑒𝑧 𝐾𝑝ℎ𝑜  𝐶𝑠𝑝𝑢ℎ𝑓𝑠 𝑆𝑗𝑑ℎ𝑏𝑠𝑒, 𝐾𝑝ℎ𝑜   𝑦 𝐷𝑠𝑝𝑥𝑜(𝑦) ∧ 𝑃𝑜𝐼𝑓𝑏𝑒(𝑦, 𝐾𝑝ℎ𝑜)   A universally quantified sentence is replaced by all possible instantiations & an existentially quantified sentence by one instantiation: 𝐿𝑗𝑜𝑕(𝐾𝑝ℎ𝑜)  𝐻𝑠𝑓𝑓𝑒𝑧(𝐾𝑝ℎ𝑜) ⇒ 𝐹𝑤𝑗𝑚(𝐾𝑝ℎ𝑜)  𝐿𝑗𝑜𝑕(𝑆𝑗𝑑ℎ𝑏𝑠𝑒)  𝐻𝑠𝑓𝑓𝑒𝑧(𝑆𝑗𝑑ℎ𝑏𝑠𝑒) ⇒ 𝐹𝑤𝑗𝑚(𝑆𝑗𝑑ℎ𝑏𝑠𝑒)  𝐿𝑗𝑜𝑕(𝐾𝑝ℎ𝑜)  𝐻𝑠𝑓𝑓𝑒𝑧(𝐾𝑝ℎ𝑜)  𝐶𝑠𝑝𝑢ℎ𝑓𝑠(𝑆𝑗𝑑ℎ𝑏𝑠𝑒, 𝐾𝑝ℎ𝑜)  𝐷𝑠𝑝𝑥𝑜 𝐷 1  𝑃𝑜𝐼𝑓𝑏𝑒 𝐷 1 , 𝐾𝑝ℎ𝑜  6

Propositionalization  Every FOL KB and query can be propositionalized  Algorithms for deciding PL entailment can be used  Problem: infinitely large set of sentences  Infinite set of possible ground-term substitution due to function symbols  e.g., 𝐺𝑏𝑢ℎ𝑓𝑠(… 𝐺𝑏𝑢ℎ𝑓𝑠(𝐺𝑏𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜)))  Solution:  Theorem (Herbrand, 1930): If a sentence 𝛽 is entailed by an FOL KB, it can be entailed by a finite subset of its propositionalized KB for 𝑜 = 0 to ∞ do Generate all instantiations with depth 𝑜 nested symbols if 𝛽 is entailed by this 𝐿𝐶 then return 𝑢𝑠𝑣𝑓 7

Propositionalization (Cont.)  Problem: When procedure go on and on, we will not know whether it is stuck in a loop or the proof is just about to pop out  Theorem (Turing-1936, Church-1936): Deciding entailment for FOL is semidecidable  Algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every non-entailed sentence. 8

Propositionalization is inefficient  Generates lots of irrelevant sentences  Example:  𝐿𝑗𝑜𝑕 𝑆𝑗𝑑ℎ𝑏𝑠𝑒  𝐻𝑠𝑓𝑓𝑒𝑧 𝑆𝑗𝑑ℎ𝑏𝑠𝑒  𝐹𝑤𝑗𝑚 𝑆𝑗𝑑ℎ𝑏𝑠𝑒 is irrelevant. KB Query  𝑦 𝐿𝑗𝑜𝑕 𝑦  𝐻𝑠𝑓𝑓𝑒𝑧 𝑦  𝐹𝑤𝑗𝑚 𝑦 𝐹𝑤𝑗𝑚(𝑦) 𝐿𝑗𝑜𝑕 𝐾𝑝ℎ𝑜 𝐻𝑠𝑓𝑓𝑒𝑧 𝐾𝑝ℎ𝑜 𝐶𝑠𝑝𝑢ℎ𝑓𝑠 𝑆𝑗𝑑ℎ𝑏𝑠𝑒, 𝐾𝑝ℎ𝑜  𝑞 𝑙 -ary predicates and 𝑜 constants  𝑞 × 𝑜 𝑙 instantiations 9

Lifting & Unification  Lifting: raising inference rules or algorithms from ground (variable-free) PL to FOL.  E.g., generalized Modus Ponens  Unification: Lifted inference rules require finding substitutions that make different expressions looks identical.  Instantiating variables only as far as necessary for the proof .  All variables are assumed universally quantified. 10

Unification: Simple example  Example: Infer immediately when finding a substitution θ such that 𝐿𝑗𝑜𝑕(𝑦) and 𝐻𝑠𝑓𝑓𝑒𝑧(𝑦) match 𝐿𝑗𝑜𝑕(𝐾𝑝ℎ𝑜) and 𝐻𝑠𝑓𝑓𝑒𝑧(𝑧) KB Query  𝑦 𝐿𝑗𝑜𝑕 𝑦  𝐻𝑠𝑓𝑓𝑒𝑧 𝑦  𝐹𝑤𝑗𝑚 𝑦 𝐹𝑤𝑗𝑚(𝑦) 𝐿𝑗𝑜𝑕 𝐾𝑝ℎ𝑜 ∀𝑧 𝐻𝑠𝑓𝑓𝑒𝑧 𝑧 𝐶𝑠𝑝𝑢ℎ𝑓𝑠 𝑆𝑗𝑑ℎ𝑏𝑠𝑒, 𝐾𝑝ℎ𝑜 11

Unification: examples  𝑉𝑂𝐽𝐺𝑍 𝑞, 𝑟 = 𝜄 where 𝑇𝑣𝑐𝑡𝑢 𝜄, 𝑞 = 𝑇𝑣𝑐𝑡𝑢(𝜄, 𝑟) 𝑞 𝑟 𝜄 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝐾𝑏𝑜𝑓) {𝑦/𝐾𝑏𝑜𝑓} { 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑧, 𝐶𝑗𝑚𝑚) 𝑦 𝐶𝑗𝑚𝑚, 𝑧/𝐾𝑝ℎ𝑜} 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑧, 𝑁𝑝𝑢ℎ𝑓𝑠(𝑧)) { 𝑦 𝑁𝑝𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜), 𝑧/𝐾𝑝ℎ𝑜} 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑦, 𝐹𝑚𝑗𝑨𝑏𝑐𝑓𝑢ℎ) 𝑔𝑏𝑗𝑚 12

Unification: examples  𝑉𝑂𝐽𝐺𝑍 𝑞, 𝑟 = 𝜄 where 𝑇𝑣𝑐𝑡𝑢 𝜄, 𝑞 = 𝑇𝑣𝑐𝑡𝑢(𝜄, 𝑟) 𝑞 𝑟 𝜄 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝐾𝑏𝑜𝑓) {𝑦/𝐾𝑏𝑜𝑓} { 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑧, 𝐶𝑗𝑚𝑚) 𝑦 𝐶𝑗𝑚𝑚, 𝑧/𝐾𝑝ℎ𝑜} 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑧, 𝑁𝑝𝑢ℎ𝑓𝑠(𝑧)) { 𝑦 𝑁𝑝𝑢ℎ𝑓𝑠(𝐾𝑝ℎ𝑜), 𝑧/𝐾𝑝ℎ𝑜} 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑦, 𝐹𝑚𝑗𝑨𝑏𝑐𝑓𝑢ℎ) 𝑔𝑏𝑗𝑚 13

Unification: complications  Complications  T wo sentences using the same variable name  Standardizing apart: renaming the variables causing name clashes in one of sentences  More than one unifier  Most General Unifier (MGU) 𝑞 𝑟 𝜄 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑦, 𝐹𝑚𝑗𝑨𝑏𝑐𝑓𝑢ℎ) 𝑔𝑏𝑗𝑚 { 𝐿𝑜𝑝𝑥𝑡(𝐾𝑝ℎ𝑜, 𝑦) 𝐿𝑜𝑝𝑥𝑡(𝑦 2 , 𝐹𝑚𝑗𝑨𝑏𝑐𝑓𝑢ℎ) 𝑦 𝐹𝑚𝑗𝑨𝑏𝑐𝑓𝑢ℎ, 𝑦 2 /𝐾𝑝ℎ𝑜} 14

Unification (MGU)  𝑉𝑂𝐽𝐺𝑍( 𝐿𝑜𝑝𝑥𝑡 𝐾𝑝ℎ𝑜, 𝑦 , 𝐿𝑜𝑝𝑥𝑡 𝑧, 𝑨 )  𝜄 1 = {𝑧/𝐾𝑝ℎ𝑜, 𝑦/𝑨 }  𝜄 2 = {𝑧/𝐾𝑝ℎ𝑜, 𝑦/𝐾𝑝ℎ𝑜, 𝑨/𝐾𝑝ℎ𝑜}  𝜄 1 is more general  There is a single most general unifier (MGU) that is unique up to renaming of variables. 15

Generalized Modus Ponens (GMP) ′ = 𝑇𝑣𝑐𝑡𝑢 𝜄, 𝑞 𝑗 ′ , 𝑟 & 𝑇𝑣𝑐𝑡𝑢 𝜄, 𝑞 𝑗  For atomic sentences 𝑞 𝑗 , 𝑞 𝑗 for all 𝑗 ′ , … , 𝑞 𝑜 ′ , ′ , 𝑞 2 𝑞 1 (𝑞 1 ∧ 𝑞 2 ∧ … ∧ 𝑞 𝑜 ⇒ 𝑟) 𝑇𝑣𝑐𝑡𝑢(𝜄, 𝑟) KB ′ : 𝐿𝑗𝑜𝑕(𝐾𝑝ℎ𝑜) 𝑞 1 𝑞 1 : 𝐿𝑗𝑜𝑕(𝑦) 𝐿𝑗𝑜𝑕 𝑦  𝐻𝑠𝑓𝑓𝑒𝑧 𝑦  𝐹𝑤𝑗𝑚 𝑦 ′ : 𝐻𝑠𝑓𝑓𝑒𝑧(𝑧) 𝑞 2 𝑞 2 : 𝐻𝑠𝑓𝑓𝑒𝑧(𝑦) 𝐿𝑗𝑜𝑕 𝐾𝑝ℎ𝑜 𝑟 : 𝐹𝑤𝑗𝑚(𝑦) 𝐻𝑠𝑓𝑓𝑒𝑧 𝑧 … 𝜄 = {𝑦/𝐾𝑝ℎ𝑜, 𝑧/𝐾𝑝ℎ𝑜} Query 𝑇𝑣𝑐𝑡𝑢(𝜄, 𝑟) is 𝐹𝑤𝑗𝑚(𝐾𝑝ℎ𝑜) 𝐹𝑤𝑗𝑚(𝑦) 16