Automated Reasoning in Propositional Logic Russell and Norvig: Chapters 6 and 9 Chapter 7, Sections 7.5—7.6 CS121 – Winter 2003

Automated Reasoning in Propositional Logic2 Problem Given: KB: a set of sentence  : a sentence Answer: KB  ?

Automated Reasoning in Propositional Logic3 KB  iff {KB,  } is unsatisfiable Deduction vs. Satisfiability Test Hence: Deciding whether a set of sentences entails another sentence, or not Testing whether a set of sentences is satisfiable, or not are closely related problems

Automated Reasoning in Propositional Logic4 Computational Approaches Enumeration of models Construction of a proof 1. Battery-OK  Bulbs-OK  Headlights-Work 2. Battery-OK  Starter-OK   Empty-Gas-Tank  Engine-Starts 3. Engine-Starts   Flat-Tire  Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9. Battery-OK  Starter-OK  (5+6) 10. Battery-OK  Starter-OK   Empty-Gas-Tank  (9+7) 11. Engine-Starts  (2+10) 12.  Engine-Starts  Flat-Tire  (3+8) 13. Flat-Tire  (11+12)

Automated Reasoning in Propositional Logic5 Enumeration of Models P : Set of propositional symbols in {KB,  } n: Size of P ENTAILS?( KB,  ) For each of the 2 n models on P do If it is a model of {KB,  } then return no Return yes

Automated Reasoning in Propositional Logic6 Satisfiability Test as CSP Each propositional symbol is a variable The domain of each variable is {True, False} Each sentence in {KB,  } is a constraint on the value(s) taken by one or several variables Recursive backtracking CSP techniques and heuristics are applicable

Automated Reasoning in Propositional Logic7 Construction of a Proof 1. Battery-OK  Bulbs-OK  Headlights-Work 2. Battery-OK  Starter-OK   Empty-Gas-Tank  Engine-Starts 3. Engine-Starts   Flat-Tire  Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9. Battery-OK  Starter-OK  (5+6) 10. Battery-OK  Starter-OK   Empty-Gas-Tank  (9+7) 11. Engine-Starts  (2+10) 12.  Engine-Starts  Flat-Tire  (3+8) 13. Flat-Tire  (11+12)

Automated Reasoning in Propositional Logic8 Construction of a Proof 1. Battery-OK  Bulbs-OK  Headlights-Work 2. Battery-OK  Starter-OK   Empty-Gas-Tank  Engine-Starts 3. Engine-Starts   Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9. Battery-OK  Starter-OK  (5+6) 10. Battery-OK  Starter-OK   Empty-Gas-Tank  (9+7) 11. Engine-Starts  (2+10) 12.  Engine-Starts  Flat-Tire  (3+8) 13. Flat-Tire  (11+12) What do we need? A complete set of sound inference rules A complete search algorithm to decide which rule to apply next and to which sentences

Automated Reasoning in Propositional Logic9 Complementary Literals A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P or  P Two literals are complementary if one is the negation of the other, e.g.: P and  P

Automated Reasoning in Propositional Logic10 Unit Resolution Rule Given two sentences: L 1  …  L p and M where L i,…, L p and M are all literals, and M and L i are complementary literals Infer: L 1  …  L i-1  L i+1  …  L p

Automated Reasoning in Propositional Logic11 Examples From:  Engine-Starts  Car-OK Engine-Starts Infer: Car-OK From:  Engine-Starts  Car-OK  Car-OK Infer:  Engine-Starts Modus ponens Modus tolens Engine-Starts  Car-OK

Automated Reasoning in Propositional Logic12 Another Example 1.  Engine-Starts  Flat-Tire  Car-OK 2. Engine-Starts 3.  Flat-Tire 4. Flat-Tire  Car-OK 5. Car-OK

Automated Reasoning in Propositional Logic13 Detection of Unsatisfiability 1.Car-OK 2.  Car-OK 3.False

Automated Reasoning in Propositional Logic14 Soundness of Unit Resolution Let m be a model of: L 1  …  L p and M where M and L i are complementary literals L i must be False in m, hence L 1  …  L i-1  L i+1  …  L p must be True

Automated Reasoning in Propositional Logic15 Shortcoming of Unit Resolution From:   Engine-Starts  Flat-Tire  Car-OK  Engine-Starts  Empty-Gas-Tank we can infer nothing!

Automated Reasoning in Propositional Logic16 Full Resolution Rule Given two sentences: L 1  …  L p and M 1  …  M q where L 1,…, L p, M 1,…, M q are all literals, and L i and M j are complementary literals Infer: L 1  …  L i-1  L i+1  …  L k  M 1  …  M j-1  M j+1  …  M k in which only one copy of each literal is retained (factoring)

Automated Reasoning in Propositional Logic17 Example From:  Engine-Starts  Flat-Tire  Car-OK Engine-Starts  Empty-Gas-Tank Infer: Empty-Gas-Tank  Flat-Tire  Car-OK

Automated Reasoning in Propositional Logic18 Example From:  P  Q (  P  Q)  Q  R (  Q  R) Infer:  P  R (  P  R)

Automated Reasoning in Propositional Logic19 Not All Inferences are Useful! From:  Engine-Starts  Flat-Tire  Car-OK Engine-Starts   Flat-Tire Infer:  Flat-Tire  Flat-Tire  Car-OK

Automated Reasoning in Propositional Logic20 Not All Inferences are Useful! From:  Engine-Starts  Flat-Tire  Car-OK Engine-Starts   Flat-Tire Infer:  Flat-Tire  Flat-Tire  Car-OK tautology

Automated Reasoning in Propositional Logic21 Not All Inferences are Useful! From:  Engine-Starts  Flat-Tire  Car-OK Engine-Starts   Flat-Tire Infer:  Flat-Tire  Flat-Tire  Car-OK  True tautology

Automated Reasoning in Propositional Logic22 Soundness of Full Resolution Left as an exercise

Automated Reasoning in Propositional Logic23 Full Resolution Rule Given two sentences: L 1  …  L p and M 1  …  M q Infer: L 1  …  L i-1  L i+1  …  L k  M 1  …  M j-1  M j+1  …  M k clauses

Automated Reasoning in Propositional Logic24 Sentence  Clause Form Example: (A   B)  (C  D)

Automated Reasoning in Propositional Logic25 Sentence  Clause Form Example: (A   B)  (C  D) 1. Eliminate   (A   B)  (C  D)

Automated Reasoning in Propositional Logic26 Sentence  Clause Form Example: (A   B)  (C  D) 1. Eliminate   (A   B)  (C  D) 2. Reduce scope of  (  A  B)  (C  D)

Automated Reasoning in Propositional Logic27 Sentence  Clause Form Example: (A   B)  (C  D) 1. Eliminate   (A   B)  (C  D) 2. Reduce scope of  (  A  B)  (C  D) 3. Distribute  over  (  A  (C  D))  (B  (C  D)) (  A  C)  (  A  D)  (B  C)  (B  D)

Automated Reasoning in Propositional Logic28 Sentence  Clause Form Example: (A   B)  (C  D) 1. Eliminate   (A   B)  (C  D) 2. Reduce scope of  (  A  B)  (C  D) 3. Distribute  over  (  A  (C  D))  (B  (C  D)) (  A  C)  (  A  D)  (B  C)  (B  D) Set of clauses: {  A  C,  A  D, B  C, B  D}

Automated Reasoning in Propositional Logic29 Resolution Refutation Algorithm RESOLUTION-REFUTATION(KB  ) clauses  set of clauses obtained from KB and  new  {} Repeat: For each C, C’ in clauses do res  RESOLVE(C,C’) If res contains the empty clause then return yes new  new U res If new  clauses then return no clauses  clauses U new

Automated Reasoning in Propositional Logic30 Completeness of Resolution Refutation Left as an exercise

Automated Reasoning in Propositional Logic31 Example 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire 10.  Starter-OK  Empty-Gas-Tank  Engine-Starts 11.  Battery-OK  Empty-Gas-Tank  Engine-Starts 12.  Battery-OK   Starter-OK  Engine-Starts 13.  Engine-Starts  Flat-Tire 14.  Engine-Starts  Car-OK

Automated Reasoning in Propositional Logic32 Example 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire 10.  Starter-OK  Empty-Gas-Tank  Engine-Starts 11.  Battery-OK  Empty-Gas-Tank  Engine-Starts 12.  Battery-OK   Starter-OK  Engine-Starts 13.  Engine-Starts  Flat-Tire 14.  Engine-Starts  Car-OK

Automated Reasoning in Propositional Logic33 Example … Battery-OK Starter-OK  Empty-Gas-Tank  Car-OK  Flat-Tire …  Battery-OK   Starter-OK  Engine-Starts  Engine-Starts  Flat-Tire …

Automated Reasoning in Propositional Logic34 Example … Battery-OK Starter-OK  Empty-Gas-Tank  Car-OK  Flat-Tire …  Battery-OK   Starter-OK  Engine-Starts  Engine-Starts  Flat-Tire …  Battery-OK   Starter-OK  Flat-Tire …  Starter-OK  Flat-Tire … Flat-Tire … False (empty clause)

Automated Reasoning in Propositional Logic35 Resolution Heuristics Set-of-support heuristics: At least one ancestor of every inferred clause comes from  Shortest-clause heuristics: Generate a clause with the fewest literals first Simplifications heuristics: Remove any clause containing two complementary literals (tautology) Remove any clause C that contains all the literals of another clause C’ If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

Automated Reasoning in Propositional Logic36 Example (Set-of-Support) 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire

Automated Reasoning in Propositional Logic37 Example (Set-of-Support) 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire 10.  Engine-Starts  Car-OK 11.  Engine-Starts 12.  Battery-OK   Starter-OK  Empty-Gas-Tank 13.  Starter-OK  Empty-Gas-Tank 14. Empty-Gas-Tank 15. False Note the goal-directed flavor

Automated Reasoning in Propositional Logic38 Resolution Heuristics Set-of-support heuristics: At least one ancestor of every inferred clause comes from  Shortest-clause heuristics: Generate a clause with the fewest literals first Simplifications heuristics: Remove any clause containing two complementary literals (tautology) Remove any clause C that contains all the literals of another clause C’ If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

Automated Reasoning in Propositional Logic39 Example (Shortest-Clause) 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire

Automated Reasoning in Propositional Logic40 Example (Shortest-Clause) 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlight-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire 10.  Engine-Starts  Car-OK 11.  Engine-Starts 12.  Bulbs-OK  Headlights-Work 13.  Battery-OK   Starter-OK  Empty-Gas-Tank 14.  Starter-OK  Empty-Gas-Tank 15. Empty-Gas-Tank 16. False

Automated Reasoning in Propositional Logic41 Resolution Heuristics Set-of-support heuristics: At least one ancestor of every inferred clause comes from  Shortest-clause heuristics: Generate a clause with the fewest literals first Simplifications heuristics: Remove any clause containing two complementary literals (tautology) Remove any clause C that contains all the literals of another clause C’ If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

Automated Reasoning in Propositional Logic42 Example (Pure Literal) 1.  Battery-OK   Bulbs-OK  Headlights-Work 2.  Battery-OK   Starter-OK  Empty-Gas-Tank  Engine-Starts 3.  Engine-Starts  Flat-Tire  Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7.  Empty-Gas-Tank 8.  Car-OK 9.  Flat-Tire

Automated Reasoning in Propositional Logic43 When to Use Logic? When the knowledge base is large and can be made explicit When formulating a problem as a CSP problem would yield complex constraints When the agent’s environment is conveniently described by true or false sentences

Automated Reasoning in Propositional Logic44 Summary Entailment problem Resolution rule Clause form of a set of sentences Resolution refutation algorithm Resolution heuristics