A Goal-Independent Suspension Analysis for Logic Programs with Dynamic Scheduling

1
A Goal-Independent Suspension Analysisfor Logic
Programs with Dynamic Scheduling

• Andy King
• Computing Laboratory
• University of Kent at Canterbury
• Samir Genaim
• Universita degli Studi di Verona
• Italy

Abstract. The talk will describe a
goal-independent analysis that infers a class of
goals which ensure that a ccp program, or a logic
program with delay, can be executed without
suspension. The crucial point is that the
analysis does not verify that an (abstract) goal
does not lead to non-suspension but infers
(abstract) goals that do not lead to
non-suspension. Put simply, the analysis is to
suspension analysis what termination inference
Mesnard and Neumerkel01 is to termination
analysis. The analysis has applications in
debugging and program optimisation.
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Background

• Since the early days of abstract interpretation,
  the connection between backward reasoning and
  debugging has been known Cousot82, Bourdoncle93
• Forward reasoning is usually applied in ccp and
  clp to verify that certain properties hold
  Ciao-Prolog
• Forward analysis checks that a certain property
  holds for a given (abstract) query
• In the context of clp, backward analysis infers
  properties that the query must satisfy for the
  program to satisfy certain requirements

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Forward suspension analysis

• Suspension analysis Codognet90, Codish94 checks
  that a class of goals will not lead to suspension
• These analyses use local fixpoints Codognet90
  or star-abstractions to avoid the state-space
  explosion that arises from goal interleaving
• In the context of the ccp language Janus, Debray
  Debray96 checks that a program does not suspend
  by attempting to schedule atoms left-to-right

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Backward suspension analysis

• The advantage of backward suspension analysis is
  that the programmer need not rerun the analysis
  for different (abstract) queries
• The programmer need not even be aware of how to
  invoke the program, thus zero effort to apply.
• Analysis infers where depth-first scheduling can
  be (locally) applied without compromising
  suspension
• Conditions sufficient for depth-first scheduling
  are reported to the programmer these conditions
  guarantee non-suspension and hint at where
  efficiency can be improved Parlog

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Example analysis for inorder (Prolog with delay
style)

• inorder(nil,).
• inorder(tree(L,V,R),I) -
• app(LI,VRI,I),
• inorder(L,LI),
• inorder(R,RI).
• - block app(-, ?, -).
• app(, X, X).
• app(XXs, Ys, XZs) -
• app(Xs,Ys,Zs).

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Example analysis for inorder (ccp style)

• inorder(T, I) -
• true
• T nil,
• I true.
• inorder(T, I) -
• true
• T tree(L,V,R),
• A VRI
• app(LI,A,I),
• inorder(L,LI),
• inorder(R,RI).

app(L, Ys, A) - nonvar(L) V nonvar(A) L
, A Ys true. app(L, Ys, A) -
nonvar(L) V nonvar(A) L XXs, A
XZs app(Xs,Ys,Zs).
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Pos abstraction for inorder

• inorder(T, I) -
• true
• T /\ I.
• inorder(T, I) -
• true
• T ? (L /\ V /\ R),
• A ? (V /\ RI)
• app(LI,A,I),
• inorder(L,LI),
• inorder(R,RI).

app(L, Ys, A) - L \/ A L /\ (A ? Ys)
true. app(L, Ys, A) - L \/ A L ? (X /\
Xs), A ? (X /\ Zs) app(Xs,Ys,Zs).
Note that asks are abstracted from below whereas
tells are abstracted from above
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lfp calculation

• The success patterns of the ccp program (and thus
  the Prolog with delay program) are described by
  the lfp of the abstract Pos program
• A success pattern is an atom with distinct
  variables for arguments paired with a Pos formula
  over those variables
• The lfp of the Pos program can be computed in
• TP-style to give
• F inorder(x1, x2)- x1 ? x2,
• app(x1, x2, x3) - (x1 /\ x2) ?
  x3
• Observe that F faithfully describes the grounding
  behaviour of inorder and app

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gfp calculation

• A gfp is computed to characterise the safe call
  patterns of the program
• A call pattern has the same syntactic form as a
  success pattern
• Iteration commences with D0 top and
  incrementally strengthens the call pattern
  formulae until they are safe, that is, they
  describe queries that do not violate the ask
  constraints

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gfp calculation (under the microscope, part I)

• Di1 is computed from Di by considering each
• clause p(x) - d f p1(x1), pn(xn) in the
  abstract program and calculating a formula that
  (possibly) strengthens the call pattern
• Specifically, let fi denote the success pattern
  formula for pi(xi) in F and let di denote the
  call pattern formula for pi(xi) in Di
• Compute e /\i 1n (di gt fi) which describes
  the grounding behaviour of the compound goal
  p1(x1), pn(xn),
• Calculate e' /\i 1n di which describes a
  groundness property sufficient for scheduling the
  compound goal depth-first without suspension.

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gfp calculation (under the microscope, part II)

• The formula e gt e' then describes a grounding
  property which, if satisfied when the compound
  goal is called, ensures the goal can be scheduled
  depth-first without suspension
• Similarly, g d /\ (f gt (e gt e')) describes a
  grounding property which ensures that the ask is
  satisfied and the body atoms can be scheduled
  depth-first without suspension
• Variables not present in p(x), Y y1,..yn say,
  are then eliminated by g' forallY1(... forallYn
  (g))
• A safe calling mode for this particular clause is
  then given by g'

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gfp calculation (under the microscope, part III)

• This procedure generates the following Di
  sequence that converges from above
• D0 top
• D1 inorder(x1, x2)- true,
• app(x1, x2, x3) - x1 \/ x3
• D2 inorder(x1, x2)- x1 \/ x2 ,
• app(x1, x2, x3) - x1 \/ x3
• D3 D2 so that the gfp is reached and checked
  in 2 iterations

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Pre-requisites for backward suspension analysis

• Logical implication lt coincides with the
  intuitionistic implication for Pos
• Any (abstract) domain equipped with
  intuitionistic implication is condensing (and
  vice versa) Giacobazzi98
• Backwards suspension analysis can be applied with
  any condensing abstract domain

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Future work

• Weaken requirements for ccp
• Relate to Hoarau and Mesnard99 after TPLP paper
  is accepted
• Run larger programs through http//www.cs.bgu.ac.i
  l/cgi-bin/genaim/susweb.cgi