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Advanced Proof Methods in Logical Agents: Inference Strategies and Algorithms

This chapter delves into proof methods within logical agents, categorizing them into two fundamental types: the application of inference rules and the legitimate generation of new sentences from existing ones. Key techniques include model checking, truth table enumeration, and resolution, all utilizing conjunctive normal form (CNF). It discusses reasoning patterns in propositional logic, emphasizing rules like Modus Ponens, along with algorithms for efficient searching. Additionally, it provides insights into forward and backward chaining, focusing on Horn forms and their applications.

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Advanced Proof Methods in Logical Agents: Inference Strategies and Algorithms

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  1. Logical Agentsعاملهاي منطقي Chapter 7 (part 2) Modified by Vali Derhami

  2. Proof methods • Proof methods divide into (roughly) two kinds: • Application of inference rules • Legitimate (sound) generation of new sentences from old • Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm • Typically require transformation of sentences into a normal form • Model checking • truth table enumeration (always exponential in n) improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL) • heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms

  3. Reasoning Patterns in Propositional Logic • Inference rules: The patterns of inference: 1-Modus Ponens (قياس استثنايي) 2-And –Elimination • تمامي هم ارزهاي منطقي در اسلايد 28از قسمت اول مي توانند بعنوان قاعده استنتاج بكار روند • اثبات : دنباله اي از كاربردهاي قواعد اسنتاج • يكنواختي:مجموعه جملات ايجاب شده فقط زماني مي تواند افزايش يابند كه اطلاعاتي به پايگاه دانش افزوده شود.

  4. Resolution تحليل: تحليل: يك روش استنتاج ساده در صورت همراه شدن با هر الگوريتم جستجوي كامل، يك الگوريتم استنتاج كامل را ايجاد ميكند • In Wumpus world, consider the agent returns from [2,1] to [1,1] and thengoes to [1,2], where it perceives a stench, but no breeze. We add the following facts to theknowledge base:

  5. تحليل (ادامه) we can now derive the absence of pits in [2,2] and [1,3],([1,1] is already known( Respect to R11 and R12: Known from past: : Respect to R15 and Rr13: Respect to R10 (R10 : P11), and R16:

  6. تحليل(ادامه) Conjunctive Normal Form (CNF)(فرم نرمال عطفي) conjunction of disjunctions of literals (clauses) تركيب AND از ليترال كه با هم ORشده اند. Literal: a atomic sentence. Ex; P , Q , or , A Clause: a disjunction of literals. E.g., (A B)  (B C D) • Unit resolution inference rule: Where li and m are complementary literals

  7. تحليل(ادامه) • Full Resolution inference rule (for CNF): l1… lk, m1 … mn l1 … li-1 li+1  … lkm1 … mj-1 mj+1... mn where li and mj are complementary literals. E.g., P1,3P2,2, P2,2 P1,3 • Using any complete search, • Resolution is sound and complete for propositional logic توجه، تحليل هميشه مي تواند براي اثبات ياتكذيب يك جمله استفاده شود

  8. Conversion to CNFتبديل به فرم نرمال عطفي B1,1 (P1,2 P2,1) • Eliminate , replacing α  β with (α  β)(β  α). (B1,1 (P1,2 P2,1))  ((P1,2 P2,1)  B1,1) 2. Eliminate , replacing α  β with α β. (B1,1 (P1,2 P2,1))  ((P1,2 P2,1)  B1,1) 3. Move  inwards using de Morgan's rules and double-negation: (B1,1  P1,2 P2,1)  ((P1,2 P2,1)  B1,1) 4. Apply distributivity law ( over ) and flatten: (B1,1 P1,2 P2,1)  (P1,2  B1,1)  (P2,1 B1,1)

  9. Resolution algorithm • Proof by contradiction, i.e., to Prove KB a, show KBα unsatisfiable

  10. Steps in Resolution algorithm • (KB a) is converted into CNF. • The resolution rule is applied to the resulting clauses. Each pair that contains complementary literals is resolved to produce a new clause, which is added to the set if it is not already present. • The process continues until one of two things happens: 1- two clauses resolve to yield the empty clause, in which case KB entails a. 2- there are no new clauses that can be added, in which case KB does not entail a.

  11. Resolution example • KB = (B1,1 (P1,2 P2,1))  B1,1 ,α = P1,2

  12. Forward and backward chaining • Horn Form (restricted): فرم محدود شده اي از فرم نرمال عطفي است • تركيب OR از ليترال ها كه حداكثر يكي از آنها مثبت است. • مزاياي آن: 1- تبديل به فرم شرطي P 1,1  B1,1  P1,2 (P 1,1   P1,2)  B1,1 (P 1,1   P1,2)  B1,1 KB = conjunction of Horn clauses • Horn clause = • proposition symbol; or • (conjunction of symbols)  symbol • E.g., C  (B  A)  (C  D  B) • بندهاي هورني كه دقيقا يك ليترال مثبت دارند را بند معين (Definite clause) مي نامند

  13. Forward and backward chaining • 2-اجازه استناج با استفاده از قياس استثنايي: • Modus Ponens (for Horn Form): complete for Horn KBs α1, … ,αn, α1 …  αnβ β • Can be used with forward chaining or backward chaining. • 3- زمان تصميم گيري در مورد ايجاب در بندهاي هورن مي تواند بر حسب اندازه پايگاه دانش بصورت خطي باشد: • These algorithms are very natural and run in linear time

  14. Forward and backward chaining A Horn clause with no positive literals can be written as an implication whose conclusion is the literal False. Ex: —wumpus cannot be in both [1,1] and [1,2]—is equivalent to . Such sentences are called integrity(کامل( constraints in the database world,

  15. Forward chaining • Idea: fire any rule whose premises are satisfied in the KB, • add its conclusion to the KB, until query is found

  16. Forward chaining algorithm • Forward chaining is sound and complete for Horn KB

  17. Forward chaining example

  18. Forward chaining example

  19. Forward chaining example

  20. Forward chaining example

  21. Forward chaining example

  22. Forward chaining example

  23. Forward chaining example

  24. Forward chaining example

  25. Proof of completeness • FC derives every atomic sentence that is entailed by KB • FC reaches a fixed point where no new atomic sentences are derived • Consider the final state as a model m, assigning true/false to symbols • Every clause in the original KB is true in m a1  … ak  b • Hence m is a model of KB • If KB╞ q, q is true in every model of KB, including m

  26. Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new subgoal is already on the goal stack Avoid repeated work: check if new subgoal has already been proved true, or has already failed

  27. Backward chaining example

  28. Backward chaining example

  29. Backward chaining example

  30. Backward chaining example

  31. Backward chaining example

  32. Backward chaining example

  33. Backward chaining example

  34. Backward chaining example

  35. Backward chaining example

  36. Backward chaining example

  37. Forward vs. backward chaining • FC is data-driven, automatic, unconscious processing (پروسس بي هدف), appropritate for Design, Control. • e.g., object recognition, routine decisions • May do lots of work that is irrelevant to the goal • BC is goal-driven, appropriate for problem-solving, Diagnosis, • e.g., Where are my keys? How do I get into a PhD program? • Complexity of BC can be much less than linear in size of KB

  38. Efficient propositional inference Two families of efficient algorithms for propositional inferenceon Model checking: • Complete backtracking search algorithms • DPLL algorithm (Davis, Putnam, Logemann, Loveland) • Incomplete local search algorithms • WalkSAT algorithm

  39. The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. استفاده از روش عمق اول براي يافتن مدل. Improvements over truth table enumeration: • Early termination A clause is true if any literal is true. A sentence is false if any clause is false. • Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C  A), A and B are pure, C is impure. Make a pure symbol literal true. آگر جمله مدلي داشته باشد لذا مقدار سمبل هاي خالص مي تواند به گونه اي مقدار دهی شودند كه بند مربوطه اشان را صحيح كند. • Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true. در واقع مقدار دهی تمام بندهای واحد قبل از انشعاب درخت

  40. The DPLL algorithm

  41. The WalkSAT algorithm • Incomplete, local search algorithm • هدف يافتن انتسابي است كه تمام بندها را ارضا كند.تابع ارزيابي كه تعداد بندهاي ارضا نشده را بشمارد از عهده اين كار بر مي آيد. • On every iteration, the algorithm picks an unsatisfied clause and picks a symbol in the clause to flip. It chooses randomly between two ways to pick which symbol to flip: A) a "min-conflicts" step thatminimizes the number of unsatisfied clauses in the new state, and B) a random walk" step that picks the symbol randomly. • Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses • Balance between greediness and randomness

  42. The WalkSAT algorithm

  43. Hard satisfiability problems • Consider random 3-CNF sentences. e.g., (D B  C)  (B A C)  (C B  E)  (E D  B)  (B  E C) m = number of clauses n = number of symbols • Hard problems seem to cluster near m/n = 4.3 (critical point)

  44. Hard satisfiability problems

  45. Hard satisfiability problems • Median runtime for 100 satisfiable random 3-CNF sentences, n = 50

  46. تکالیف : 5، 8 (الف- تا د) و 9 و سوالات تست

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