Propositional Reasoning

1
Propositional Reasoning

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University
• Based on
• Russell Norvig, Edn 1, Ch 6
• Lecture Notes by Ng Hwee Tou (Singapore)

2
Outline

• The Resolution Method
• Conjunctive Normal Form
• Resolution and the Empty Clause
• Horn Clauses and Modus Ponens
• Backward and Forward Chaining
• Satisfiability and NP Completeness
• Efficient Heuristics
• Phase Changes
• Limitations of Propositional Calculus

3
References

• Russell, S Norvig, P Artificial Intelligence
  A Modern Approach, Prentice Hall
• The library has edition 1, call number 006.3
  R917a
• To buy, edition 2 (1995), ISBN 0137903952
• Either is fine
• Nilsson, NJ Artificial Intelligence A New
  Synthesis, Morgan Kaufmann, 1998, ISBN 1 55860
  535 5
• Library call no 006.3 N599a
• More detail than youll ever need
• Leitsch, A The Resolution Calculus, Springer
  1997, ISBN 3 540 61882 1
• Library call number 511.3 L537r

4
Proof methods

• Proof methods divide (roughly) into two kinds
• Application of inference rules
• Legitimate (sound) generation of new sentences
  from old
  
• Like our method from last lecture
• Proof a sequence of inference rule applications
• We could use inference rules as steps in a
  standard search algorithm
  
• Effective computer searches usually transform
  sentences into a normal form which is simpler for
  automated searching
• Model checking
• truth table enumeration (always exponential in n)
  
• improved backtracking, e.g., Davis--Putnam-Logeman
  n-Loveland (DPLL)
  
• heuristic search in model space (sound but
  incomplete)
• e.g., min-conflicts-like hill-climbing
  algorithms
  

5
Proof methods for automated search

• Our proof method from last lecture isnt very
  good for automated search
• In essence, guessing the right combination of
  rules to use is a hard problem (even if humans do
  it well)
• Its even harder for the predicate calculus (next
  lecture)
• Well look at an alternative, resolution, which
  works well for automated search in both the
  propositional and predicate calculus
• It requires sentences to be converted to a normal
  form
• Conjunctive Normal Form (CNF)

6
Conjunctive Normal Form (CNF)

• An atom means one of the variables of our
  language
• (A, B, C,.)
• A literal means either an atom or its negation
• (A, B, C,.)
• A clause means a disjunction (or) of literals
• (A ? B ? C,.)
• A CNF sentence means a conjunction (and) of
  clauses
• (A ? ?B) ? (B ? ?C ? ?D)

7
Converting to CNF

• Every propositional sentence can be
  (systematically) converted to CNF
• First, replace all equivalences using
  biconditional elimination
• (A ? B) (A ? B) ? (B ? A)
• Next, replace all implications using implication
  elimination
• (A ? B) (?A ? B)
• Next, move all negations inwards towards atoms
  using de Morgans laws
• ?(A ? B) (?A ? ?B)
• ?(A ? B) (?A ? ?B)
• And remove all multiple negations
• ? ?A A
• Now, always move conjunctions to the right using
  commutativity
• (A ? B) (B ? A)
• But at the same time, move disjunctions inside
  conjunctions when possible, using distributivity
• (A ? (B ? C)) (A ? B) ? (A ? C)

8
Converting to CNF

• The algorithm for converting to CNF finishes in a
  finite amount of time
• If you wish, you can calculate the maximum time
  it will take to convert a formula of n symbols
• The sentence it generates is logically equivalent
  to the original sentence
• Ie it has the same models
• In particular
• If one of the sentences is valid, so is the other
• If one of the sentences is unsatisfiable, so is
  the other

9
Representing in Clausal Form

• A sentence in CNF consists of a conjunction of
  disjunctions of literals
• Remember that we dont care about the order of
  conjunctions or disjunctions, or about
  repetitions
• So we can just represent CNF as a set of sets
• A, ?B,B, ?C, ?D
• Instead of
• (A ? ?B) ? (B ? ?C ? ?D)
• We just have to remember that the outer set means
  conjunction, the inner one disjunction
• This is known as clausal form
• Its easier to explain the resolution method in
  clausal form

10
The Resolution Method

• Resolution is the basic process used in the
  resolution method
• There are a number of different (equivalent) ways
  of explaining it
• Well use a different (slightly more general) one
  from the book
• Suppose we want to prove A is valid
• One way of doing this is to prove that ?A is
  unsatisfiable
• So we start by converting ?A into clausal form
• Lets take last lectures
• (A ? (A ? B)) ? B
• As an example

11
Clausal Form Conversion

• ?((A ? (A ? B)) ? B)
• ?(?(A ? (? A ? B)) ? B)
• (? ?(A ? (? A ? B)) ? ? B)
• (A ? (? A ? B)) ? ? B
• A, ?A, B, ?B

12
The Resolution Method

• The resolution method uses an operator called
  resolution
• Resolution takes two clauses which have
  complementary literals
• The same literal with opposite signs
• X1, X2, Xn, Y and Z1, Z2, Zm, ?Y
• And creates their resolvent by joining them
  together, deleting the complementary literals
• X1, X2, Xn, Z1, Z2, Zm
• You can add the resolvent to the original clause
  set without changing its validity

13
The Resolution Method (Cont)

• Sometimes, the resolvent is bigger than the
  original clauses
• But sometimes, its also smaller
• Notice that as clauses get smaller, they are more
  difficult to satisfy
• A is harder to satisfy than (A ? B ? C)
• In the limit, it is consistent to regard the
  empty clause as unsatisfiable ( false)
• (this requires proof, but its not hard)
• So a clause set which contains the empty clause
  is a conjunction of clauses, one of which is
  unsatisfiable
• So the clause set is also unsatisfiable
• The resolution method uses repeated resolutions
  to try to derive the empty set

14
Resolution Example

• Take our example
• ?((A ? (A ? B)) ? B)
• Which we converted to
• A, ?A, B, ?B
• We can resolve the first two clauses
• A resolved with ?A, B gives B
• Adding this to our clause set gives
• A, ?A, B, ?B, B
• And resolving the last two clauses
• ?B and B gives

15
Resolution Example (continued)

• So our new clause set is
• A, ?A, B, ?B, B
• Since is unsatisfiable, so is the whole clause
  set
• So ?((A ? (A ? B)) ? B) is unsatisfiable
• Which means ((A ? (A ? B)) ? B) is valid
• Notice that there were fewer choices to make than
  in our previous proof
• Theres only one rule to choose (resolution)
• Our only choice is which (of possibly many)
  resolutions to choose

16
Properties of the Resolution Method

• For the Propositional calculus, the resolution
  method is
• Sound
• It will never give incorrect answers
• Complete
• For any unsatisfiable clause set, the empty
  clause can be derived in finite time
• In fact, in exponential time
• Algorithmic
• There is an algorithm to guarantee this
• Relatively efficient
• Proofs can be quite effective
• Highly efficient for some sorts of formulas

17
Horn Clauses

• A Horn Clause is a clause which has at most one
  positive literal
• There are three cases
• 1) No positive literal (generally not used much)
• ? X1, ? X2, ? Xn (? X1 ? ? X2 ? ? ? Xn)
• 2) No negative literal
• X (X)
• A fact
• 3) General Case
• ? X1, ? X2, ? Xn, Y (? X1 ? ? X2 ? ? ?
  Xn ? Y)
• (X1 ? X2 ? ? Xn) ? Y
• A rule

18
Horn Clauses and Resolution

• Lets look at the sub-cases
• 1) and 1) - Cant resolve (no positive literals)
• 1) and 2) - only one way to do it (not very
  useful)
• (? X1 ? ? X2 ? ? ? Xn), X1
• _________________________
• (? X2 ? ? ? Xn)
• 1) and 3) - again only one choice (not very
  useful)
• (? X1 ? ? X2 ? ? ? Xn), (Y1 ? Y2 ? ? Yn) ?
  X1
• _________________________
• (? X2 ? ? ? Xn ? ? Y1 ? ? Y2 ? ? ? Yn)

19
Horn Clauses and Resolution

• 2) and 2) - Cant resolve (no negative literals)
• 2) and 3) - only one way to do it
• (Generalised Modus Ponens)
• X1, (X1 ? X2 ? ? Xn) ? Y
• _________________________
• (X2 ? ? Xn) ? Y
• 3) and 3) - Now there are two choices, lets take
  one
• (X1 ? X2 ? ? Xn) ? Y1, (Y1 ? Y2 ? ? Yn) ?Z
• _________________________
• (X1 ? X2 ? ? Xn ? Y2 ? ? Yn) ?Z
• In fact, we only need generalised modus ponens
  for Horn clause reasoning
• Runs in Linear Time

20
Forward and Backward Chaining

• Modus ponens gives us very efficient (short)
  proofs for Horn Clauses
• But how do we find an efficient proof
  efficiently?
• There are two main approaches
• Start from what we know, and work forwards
• Forward Chaining
• Start from what we want to prove, and work
  backwards
• Backward Chaining
• Of course, its also possible to combine the two
• Mixed Chaining

21
Forward chaining

• Idea fire any rule whose premises are satisfied
  in the KB,
• add its conclusion to the KB, until query is
  found
• Sound and complete for Horn clauses

22
Forward chaining example
23
Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
30
Sketch of completeness proof

• To prove forward chaining derives every atomic
  sentence that is entailed by KB
  
• Forward chaining must reach a final state, a
  fixed point where no new atomic sentences can be
  derived
• Because there are only finitely many sentences
• 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
  

31
Backward chaining

• Idea work backwards from the query q, keeping a
  stack of open queries
• to prove q by Backward Chaining, add q to the
  goal stack, then recursively
• Pop the top of the goal stack, q
• If q is already known, do nothing, its proven
• Otherwise, find a rule (q1 ? q2 ? ? qn) ? q
• Push q1, q2, qn onto the goal stack
• Until the goal stack is empty (the goal is
  proven!)

32
Backward chaining

• Efficiency considerations
• Avoid loops
• check if new subgoal is already on the goal
  stack
  
• Avoid repeated work check if new subgoal
• has already been proved true
• has already failed

33
Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
43
Forward vs. backward chaining

• Forward Chaining is data-driven
• Suitable for automatic, unconscious processing,
• object recognition
• routine decisions
• May do lots of work that is irrelevant to the
  goal
• Particularly frustrating for human users
• Backward Chaining is goal-driven
• appropriate for problem-solving,
• Where are my keys?
• How do I get into a PhD program?
• Complexity can be much less than linear in size
  of KB
• Appears more purposeful to human users

44
Efficient propositional inference

• Recall that we discussed resolution in terms of
  unsatisfiability
• Is this sentence unsatisfiable?
• We can turn this question on its head
• Is this sentence satisfiable?
• How hard is this problem?
• Actually, a very deep question

45
How hard is it to compute satisfiability

• It can be solved in polynomial (linear) time by
  guessing
• A problem which guessing can solve in polynomial
  time is known as an NP problem
• Satisfiability can be shown to be as hard as any
  other NP problem
• A problem which can be solved in polynomial time
  without guess is known as a P problem
• It is almost universally believed that P ? NP
• (in fact, that satisfiability requires
  exponential time in the worst case)
• But it has never been proven

46
Efficient propositional inference

• There are two main families of efficient
  algorithms for propositional inference
  
• Complete backtracking search algorithms
• DPLL algorithm (Davis, Putnam, Logemann,
  Loveland)
  
• Incomplete local search algorithms
• WalkSAT algorithm
• Both are efficient for most cases, but can be
  very slow on the worst cases

47
The DPLL algorithm

• Uses heuristics to remove alternatives in 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
• A pure symbol is a symbol that always appears
  with the same "sign" in all clauses.
• In the clauses (A ? ?B), (?B ? ?C), (C ? A), A
  and B are pure, C is impure
• Make any pure symbol literal true
• Unit clause heuristic
• A unit clause is a clause with only one literal
• Make the only literal in a unit clause true

48
The DPLL algorithm
49
The WalkSAT algorithm

• An incomplete, local search algorithm
• Evaluation function
• Minimise the number of unsatisfied clauses
• Balance between greediness and randomness

50
The WalkSAT algorithm
51
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
• If there are many clauses per symbol, it is
  highly unlikely that some combination of truth
  assignments will satisfy all clauses
• If there are only a few clauses per symbol, its
  easy to find truth assignments satisfying them
  all
• The hardest problems cluster near m/n 4.3
  (critical point)
  

52
Are most sentences satisfiable?
53
Hard satisfiability problems

• Median runtime for 100 satisfiable random 3-CNF
  sentences, n 50
  

54
Propositional Logic and the wumpus world

• A wumpus-world description using propositional
  logic
  
• ?P1,1
• ?W1,1
• Bx,y ? (Px,y1 ? Px,y-1 ? Px1,y ? Px-1,y)
• Sx,y ? (Wx,y1 ? Wx,y-1 ? Wx1,y ? Wx-1,y)
• W1,1 ? W1,2 ? ? W4,4
• ?W1,1 ? ?W1,2
• ?W1,1 ? ?W1,3
• ? 64 distinct proposition symbols, 155 sentences

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Expressiveness limitation of propositional logic

• The KB has to contain "physics" sentences for
  every single square
  
• But even worse, some propositions change with
  time
• Those which describe where the agent is
• So this means we need time-stamped variables for
  every possible time and location for the agent
• So every statement involving the agent has to be
  repeated once for each possible time
• For every time t and every location x,y,
• Lx,y ? FacingRightt ? Forwardt ? Lx1,y
• In general, we probably dont even know how many
  times we might need
• Though we can probably make a decent guess for
  wumpus

t
t
57
Summary

• Logical systems apply inference to a knowledge
  base to derive new information and make decisions
  
• Basic concepts of logic
• syntax formal structure of sentences
• semantics truth of sentences wrt models
• entailment necessary truth of one sentence given
  another
  
• inference deriving sentences from other
  sentences
  
• soundness derivations produce only entailed
  sentences
  
• completeness derivations can produce all
  entailed sentences
  
• Wumpus world requires the ability to represent
  partial and negated information, reason by cases,
  etc.
  
• Resolution is complete for propositional
  logicForward, backward chaining are linear-time,
  complete for Horn clauses
  
• Propositional logic lacks expressive power
• Propositional logic probably requires exponential
  time in the worst case