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Logic: The Big Picture

- Propositional logic: atomic statements are facts
- Inference via resolution is sound and complete (though likely computationally intractable)

- First-order logic: adds variables, relations, and quantification
- Inference is essentially a generalization of propositional inference
- Resolution is still sound and complete, but not guaranteed to terminate on non-entailed sentences (semidecidable)
- Simple inference procedures (forward chaining and backward chanining) available for knowledge bases consisting of definite clauses

Logic programming: Prolog Prolog: Closed-world assumption: Inference by backward chaining, clauses are tried in the order in which they are listed in the program, and literals (predicates) are tried from left to right

- FOL:
King(x) Greedy(x) Evil(x)

Greedy(y)

King(John)

evil(X) :- king(X), greedy(X).

greedy(Y).

king(john).

- Every constant refers to a unique object
- Atomic sentences not in the database are assumed to be false

Prolog example

parent(abraham,ishmael).

parent(abraham,isaac).

parent(isaac,esau).

parent(isaac,jacob).

grandparent(X,Y) :- parent(X,Z), parent(Z,Y).

descendant(X,Y) :- parent(Y,X).

descendant(X,Y) :- parent(Z,X), descendant(Z,Y).

? parent(david,solomon).

? parent(abraham,X).

? grandparent(X,Y).

? descendant(X,abraham).

Prolog example

parent(abraham,ishmael).

parent(abraham,isaac).

parent(isaac,esau).

parent(isaac,jacob).

- What if we wrote the definition of descendant like this:
descendant(X,Y) :- descendant(Z,Y), parent(Z,X).

descendant(X,Y) :- parent(Y,X).

? descendant(W,abraham).

- Backward chaining would go into an infinite loop!
- Prolog inference is not complete, so the ordering of the clauses and the literals is really important

Graph coloring

colorable(Wa,Nt,Sa,Q,Nsw,V) :-

diff(Wa,Nt), diff(Wa,Sa), diff(Nt,Q), diff(Nt,Sa), diff(Q,Nsw), diff(Q,Sa), diff(Nsw,V), diff(Nsw,Sa), diff(V,Sa).

diff(red,blue). diff(red,green). diff(green,red).

diff(green,blue). diff(blue,red). diff(blue,green).

Prolog lists

- Appending two lists to produce a third:
append([],Y,Y).

append([X|L],Y,[X|Z]) :- append(L,Y,Z).

- query: append(A,B,[1,2])
- answers: A=[] B=[1,2]
A=[1] B=[2]

A=[1,2] B=[]

Logic: The Big Picture

- The original goal of formal logic was to axiomatize mathematics
- Hilbert’s program(1920’s): find a formalization of mathematics that is consistent, complete, and decidable

- Completeness theorem(Gödel, 1929):
- Deduction in FOL is consistent and complete
- Unfortunately, FOL is not strong enough to describe infinite structures such as natural or real numbers

- Incompleteness theorem(Gödel, 1931):
- Any consistent logic system strong enough to capture natural numbers and arithmetic will contain true sentences that cannot be proved

- Halting problem(Turing, 1936):
- There cannot be a general algorithm for deciding whether a given statement about natural numbers is true

- Profound implications for foundations of mathematics
- What about implications for AI?

Applications of logic

- Automated theorem proving in mathematics
- Robbins conjecture proved in 1996

- Software verification
- Software synthesis
- VLSI verification
- VLSI design
- Planning

http://www.cs.miami.edu/~tptp/OverviewOfATP.html

Planning

- What is planning?
- Finding a sequence of actions to achieve one’s goals

- How is planning different from regular search?
- States and action sequences typically have complex internal structure
- State space and branching factor are huge
- Multiple objectives, resource constraints

- Examples of planning applications
- Scheduling of tasks in space missions
- Logistics planning for the army
- Assembly lines, industrial processes

Propositional planning

- Start state, goal state are specified as conjunctions of predicates
- Start state: At(P1, RDU) Plane(P1) Airport(RDU) Airport(ORD)
- Goal state: At(P1, ORD)

- Actions are described in terms of their preconditions and effects:
- Fly(p, source, destination)
- Precond: At(p, source) Plane(p) Airport(source) Airport(destination)
- Effect:¬At(p, source) At(p, destination)

- Fly(p, source, destination)
- Search problem: starting with the start state, find all applicable actions (actions for which preconditions are satisfied), compute the successor state based on the effects, etc.

Complexity of planning

- Planning is PSPACE-complete
- Plans can be exponential in length!
- Example: tower of Hanoi

From propositional planning to real-world planning

- Incorporating the time dimension
- Resource constraints
- Contingencies: actions failing
- “Qualification problem”
- Hierarchical planning
- Uncertainty
- Observations
- Multiagent planning

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