Backtracking and Theorem Proving

1
Backtracking and Theorem Proving

• Lecture 23
• CS 312

2
Schedule

• Monday Hamiltonian cycles and extracting the
  solution.
• Wednesday review for midterm
• Midterm in testing center starting Wed at noon,
  going Thursday, Friday and Saturday. Ends Monday
  at noon.
• Friday no class.

3
Objectives

• Draw proofs as trees.
• Apply backtracking to automated theorem proving.

4
Parts of a Logic

• Axioms things you know are true without a
  proof.
• Inference rules how to build new true facts from
  old true facts
• forward proof
• Rewrite rules how to change one statement into
  another statement.
• backward proof

5
Kinds of Proofs

• Forward proof start from axioms and build facts
  using inference rules.

Axiom
Axiom
Fact
Fact
Fact
Fact
Fact
Fact
6
Kinds of Proofs

• Backward proof start from what you want to
  prove and rewrite back to axioms.

Axiom
Axiom
Fact
Fact
Fact
7
Backward proofs as trees.
New Theorem
Intermediate facts
Axioms
8
A Logic
Rewrite Rules
Axioms 1. a a 2. a 0 a 3. 0 a a 4.
(ab)c a(bc) 5. a -a 0 6. a implies a
1
5
2
6
3
7
4
9
A Backward Proof
ab ac implies b c
Rule 1 with x/(ab) and y/-a
(ab)-a (ac)-a implies b c
Rule 5 with x/a, y/b and z/-a
a(b-a) (ac)-a implies b c
Rule 5 with x/a, y/c and z/-a
a(b-a) a(c-a) implies b c
10
A Backward Proof
ab ac implies b c
Rule 1 with x/(ab) and y/-a
(ab)-a (ac)-a implies b c
Rule 5 with x/a, y/b and z/c
Backtrack
a(b-a) (ac)-a implies b c
Rule 5 with x/a, y/b and z/c
Backtrack
a(b-a) a(c-a) implies b c
11
A Backward Proof
ab ac implies b c
Rule 1 with x/(ab) and y/-a
(ab)-a (ac)-a implies b c
Rule 5 with x/a, y/c and z/-a
(ab)-a a(c-a) implies b c
Rule 5 with x/a, y/b and z/c
a(b-a) a(c-a) implies b c
12
A Backward Proof
ab ac implies b c
Rule 2 with x/(ab) and y/-a
-a(ab) -a(ac) implies b c
Rule 5 with x/-a, y/a and z/b
(-aa)b -a(ac) implies b c
Rule 7 with x/a.
0b (-aa)c) implies b c
13

• Apply the following algorithms to small examples
  dynamic programming knapsack problem, string
  editing, backtracking algorithm for graph
  coloring, backtracking algorithm for the knapsack
  problem and all pairs shortest paths.
• Define optimality and explain why optimality is
  important for dynamic programming.
• Compare and contrast dynamic programming and
  greedy algorithms.
• Identify components, biconnected components and
  articulation points in graphs

14

• Determine the order in which nodes are traversed
  in pre, post and inorder searches.
• Compare and contrast BFS, DFS and D-search (p325,
  4,5)
• Understand the role of bounding functions in
  backtracking problems and design bounding
  functions for simple problems
• Compare and contrast dynamic programming and
  backtracking solutions to problems.
• Compute worst-case bounds on backtracking
  algorithms (p368, 1)