Logic the big picture
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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

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Logic: The Big Picture

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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

  • FOL:

    King(x)  Greedy(x)  Evil(x)



  • Prolog:

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



  • Closed-world assumption:

    • Every constant refers to a unique object

    • Atomic sentences not in the database are assumed to be false

  • 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

  • Prolog example





    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





    • 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

    Backward chaining algorithm

    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([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



    • 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)

    • 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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