Logic Programming Part 3: Control Flow

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Logic Programming Part 3 Control Flow

• James Cheney
• CS 411

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Declarative Programming

• Ideal Write down logical formulas that define
  programs clearly concisely
• In logic, clause order doesnt matter
• Control flow, efficiency invisible
• Reality Must know how programs are run
• Efficiency (backtracking, clause order)
• Correctness (determinism, termination)

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Clause order matters

• p(X) - p(X).
• p(a).
• Has answer X ? p(a)
• But depth-first search doesnt terminate.

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Clause order matters

• p(a).
• p(X) - p(X).
• Terminates with X ? p(a) .

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Backtracking

• Consider
• a - b,c,d,e,f.
• a - b,c,d,e,g.
• b. c. d. e. g.
• First we solve b,c,d,e, then fail on f.
• Then we solve b,c,d,e again and then g.
• Duplicate effort!

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Order and Backtracking

• If instead we do (logically equivalent)
• a - f,b,c,d,e.
• a - g,b,c,d,e.
• b. c. d. e. g.
• then the failure occurs earlier, and there is
  less wasted effort.

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Nondeterminism

• Consider
• mem(A,AL).
• mem(A,BL) - mem(A,L).
• Then
• ?- member (1,1,2,3,1).
• can succeed in two different ways.

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Cut

• PROLOG includes a goal called cut, written !,
  for pruning the search space
• Removes current backtracking state.
• Allows more control over search, efficiency
• However, cut damages correspondence to logic

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Cut Example

• mem(A,AL) - !.
• mem(A,BL) - mem(A,L).
• ?- mem(1,X,1,2,1,3).
• yes
• X ? 2
• no
• Only the first solution is found.

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Cut Example

• But, the definition of mem is no longer
  complete
• ?- mem(X,1,2,3,4).
• yes
• X ? 1
• no
• cut may exclude desirable solutions

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More Reality

• In PROLOG, I/O primitives are impure predicates
• Example
• main - write(What is your
• name?),
• read(X),
• write(Hello, ),
• write(X).
• Now duplication can also change program.

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Other LP systems

• lProlog
• PROLOG uses FOL, lProlog uses higher-order logic
• Typed
• Functional programming (map, fold)
• Powerful, but complex, higher-order unification
• (l X. F X) a f a ?

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Sorting

• sorted().
• sorted(A).
• sorted(A,BM) - A lt B,
• sorted(B,M).
• sort(L,M) - perm(L,M),
• sorted (L,M).
• Complexity?

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Mergesort

• msort(,).
• msort(L,L) - split(L,L1,L2),
• msort(L1,L1),
• msort(L2,L2),
• merge(L1,L2,L).
• split(,).
• split(AL,AM,N)
• - split(L,N,M).

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Mergesort (Contd)

• merge(,L,L).
• merge(L,,L).
• merge(AL,BM,AN)
• - A lt B, merge(L,BM,N).
• merge(AL,BM,BN)
• - A gt B, merge(AL,M,N).

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Other LP systems Mercury

• Mercury typed, pure
• ML-style polymorphism, datatypes
• Modes determinism checking
• I/O primitives take world argument
• main(W0,W4) -
• read(X,W1,W2),
• write(Hello World,W2,W3),
• write(X,W3,W4).

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Other LP systems lProlog

• Idea Mix functional and logic paradigms
• Term language is higher-order typed l-calculus
• Requires solving hard (undecidable!) unification
  problems
• (lx. F x) a f a ? F ? f, F ? (lx. a)
• Can encode variable binding syntax using ls

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Applications

• Artificial intelligence
• Natural language processing
• Expert systems
• Constraint solving/optimization
• Logic programs constraints describe solutions
  to complex problems
• Query languages (eg SQL, XQuery)
• declarative in flavor

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Summary

• Logic programming a powerful paradigm
• Algorithm logic control
• Unfortunately, for efficiency reasons, LP
  programs diverge from this ideal
• Mathematical clarity ! programming efficiency
• cut, imperative features lead to opaque
  programs
• Lesson TANSTAAFL.