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Determinacy Inference for Logic Programs Lunjin Lu @ Oakland University In collaboration with Andy King @ Kent University, UK

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Context Project - “An Integrated Framework for Semantic Based Analysis of Logic Programs” Backward analyses Parametric analyses Context sensitive analyses Project – “US-UK Collaborative Research on Backward analyses for Logic Programs”

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Determinacy sort(Xs,Ys) :- perm(Xs,Ys), ordered(Ys). perm(Xs,[Z|Zs]) :- select(Z,Xs,Ys),perm(Ys,Zs). perm([],[]). ordered([]). ordered([X]) :- number(X). ordered([X,Y|Ys]) :- X =< Y, ordered([Y|Ys]). select(X,[X|Xs],Xs). select(X,[Y|Ys],[Y|Zs]) :- select(X,Ys,Zs). A call is determinate if it has at most one computed answer and that answer is generated once. ?- sort([2,2],L). L=[2,2] ; No.

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Goal Infer sufficient conditions under which a call is determinate Generalizing determinacy checking

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Motivation Useful in language implementations Program development Tune performance Execute a determinate call under once; Detect possible bugs; Useful in program specialization Unfold when certain determinacy condition is satisfied. Determinacy is important in concurrent programming

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Reversing a list (1) append(Xs, Ys, Zs) :- Xs = [], Ys = Zs. (2) append(Xs, Ys, Zs) :- Xs = [X|Xs1], Zs = [X|Zs1], append(Xs1, Ys, Zs1). (3) rev(Xs,Ys) :- Xs = [], Ys = []. (4) rev(Xs,Ys) :- Xs = [X|Xs1], Ys2 = [X], rev(Xs1, Ys1), append(Ys1, Ys2, Ys).

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Overall structure of analysis Success Patterns wrt Mux Conditions wrt rigid’ Determinacy Conditions wrt rigid’ Success pattern wrt rigid’ Determinacy condition wrt rigid’

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Computing success patterns wrt term abstraction Success Patterns wrt Mux Conditions wrt rigid’ Determinacy Conditions wrt rigid’ Success pattern wrt rigid’ Determinacy condition wrt rigid’

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Abstracting terms A concrete term is mapped into an abstract term via a map . Example one: d k d 0 (t) = _ d k (X) = X d k (f(t 1,…,t n )) = f(d k-1 (t 1 ),…,d k-1 (t n )) Example two: list length norm

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Abstracting primitive constraints Term abstraction induces an abstraction map, also denoted , that takes concrete constraints into abstract constraints. Examples (K=[X|L]) is (K=1+L) when = || || (L=[X,Y,Z,W]) is (L=[X,Y|_]) when = d 2. Operations on abstract constraints \/ /\ x

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Abstracting program wrt a term abstraction (1) append(Xs, Ys, Zs) :- Xs = [], Ys = Zs. append(Xs, Ys, Zs) :- Xs≥0, Ys≥0, Zs≥0, Xs = 0, Ys = Zs. (2) append(Xs, Ys, Zs) :- Xs = [X|Xs1], Zs = [X|Zs1], append(Xs1, Ys, Zs1). append(Xs, Ys, Zs) :- Xs≥0, Ys≥0, Zs≥0, Xs1≥0, Zs1≥0, Xs = 1+Xs1, Zs = 1+Zs1, append(Xs1, Ys, Zs1).

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Abstract program (1) append(Xs, Ys, Zs) :- Xs≥0, Ys≥0, Zs≥0, Xs = 0, Ys = Zs. (2) append(Xs, Ys, Zs) :- Xs≥0, Ys≥0, Zs≥0, Xs1≥0, Zs1≥0, Xs = 1+Xs1, Zs = 1+Zs1, append(Xs1, Ys, Zs1). (3) rev(Xs,Ys) :- Xs≥0, Ys≥0, Xs = 0, Ys = 0. (4) rev(Xs,Ys) :- Xs≥0, Ys≥0, Xs1≥0, Ys1≥0,Ys2≥0, Xs = 1+Xs1, Ys2 = 1, rev(Xs1, Ys1), append(Ys1, Ys2, Ys).

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Computing success set of abstract program 1) append(Xs,Ys,Zs) :- Ys≥0,Zs≥0,Xs=0,Ys=Zs. 2) append(Xs, Ys, Zs) :- 1≥Xs,Xs≥0,Ys≥0,1≥Zs,Zs≥0,Zs=Xs+Ys 3) append(Xs,Ys,Zs) :- Xs≥0,Ys≥0,Zs=Xs+Ys 4) rev(Xs,Ys) :- Xs = 0, Ys = 0. 5) rev(Xs,Ys) :- 1≥Xs, Xs≥0, 1≥Ys, Ys≥0, Xs=Ys. 6) rev(Xs,Ys) :- Xs≥0, Xs=Ys.

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Predicate level success patterns append(x1,x2,x3) :- (x1≥0) (x2≥0) (x1+x2=x3) rev(x1,x2) :- (x1≥0) (x1=x2)

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Clause level success patterns (1) append(x1,x2,x3) :- (x1=0) (x2≥0) (x2=x3). (2) append(x1,x2,x3) :- (x1≥1) (x2≥0) (x1+x2=x3). (3) rev(x1,x2) :- (x1=0) (x2=0). (4) rev(x1,x2) :- (x1≥1) (x1=x2)

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Synthesizing mutual exclusion conditions Success Patterns wrt Mux Conditions wrt rigid’ Determinacy Conditions wrt rigid’ Success pattern wrt rigid’ Determinacy condition wrt rigid’

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Tracking rigidity Induced rigidity: term abstraction induces a rigidity predicate: rigid (t) (t)=((t)) for any e.g. rigid ([1,X]) = true e.g. rigid (X) = false Tracked rigidity: it may be simpler to track a rigidity predicate which implies the induced one. rigid’ (t) rigid (t) Domain of rigidity: Pos Rigidity dependence is expressed as positive Boolean functions such as (x1x2).

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Galois connection rigid’ (f) = {Sub|Sub.assign()|=f} rigid’ () = {fPos| rigid (f)} assign()= {x rigid’ ( (x))|x dom()}

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Mutual exclusion conditions A mutual exclusion condition for a predicate is a rigidity constraint under which at most one clause of the predicate may commence a successful derivation.

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Mutual exclusion of two clauses Let C 1 and C 2 have success patterns p(x) :- c 1 p(x) :- c 2. If Y and X P (Y,p(x),C 1,C 2 ) then C 1 and C 2 are mutually exclusive where X P (Y,p(x),C 1,C 2 ) = ( -Y (c 1 ) -Y (c 2 ))

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Mutual exclusion of two clauses (1) append(x1,x2,x3) :- (x1=0) (x2≥0) (x2=x3). c1 (2) append(x1,x2,x3) :- (x1≥1) (x2≥0) (x1+x2=x3). c2 -{x1} (c1) = (x1=0) -{x1} (c2) = (x1≥1) X P ({x1},p(x),C 1,C 2 ) = true -{x2,x3} (c1) = (x2≥0) (x2=x3) -{x2,x3} (c2) = (x2≥0) (x2<x3) X P ({x2,x3},p(x),C 1,C 2 ) = true

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Synthesizing mutual exclusions X P (p(x)) = {Y|C 1,C 2 S. (C 1 C 2 X P (Y,p(x),C 1,C 2 ))} with S = the set of clauses defining p. X P (append(x1,x2,x3)) = x1 (x2x3) X P (rev(x1,x2)) = x1 x2

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Synthesizing Determinacy Conditions Success Patterns wrt Mux Conditions wrt rigid’ Determinacy Conditions wrt rigid’ Success pattern wrt rigid’ Determinacy condition wrt rigid’

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Synthesizing determinacy conditions Determinacy inference takes two steps: The first is a lfp that computes rigidity success patterns The second is a gfp that calculates rigidity call patterns that ensures determinacy. (objective) Both lfp and gfp works on rigidity abstraction of the original program.

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Abstracting program wrt rigidity (1) append(Xs, Ys, Zs) :- Xs = [], Ys = Zs. append(Xs, Ys, Zs) :- Xs (Ys Zs). (2) append(Xs, Ys, Zs) :- Xs=[X|Xs1], Zs=[X|Zs1], append(Xs1,Ys,Zs1). append(Xs, Ys, Zs) :- (XsXs1)(ZsZs1), append(Xs1,Ys,Zs1).

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Rigidity program (1) append(Xs,Ys,Zs) :- Xs (Ys Zs). (2) append(Xs,Ys,Zs) :- (Xs Xs1) (Zs Zs1), append(Xs1,Ys,Zs1). (3) rev(Xs,Ys) :- Xs Ys. (4) rev(Xs,Ys) :- (Xs Xs1) Ys2, rev(Xs1,Ys1), append(Ys1,Ys2,Ys).

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Computing rigidity success patterns Rigidity success patterns of the original program is obtained by calculating success set of the rigidity program. append(x1,x2,x3) = x1 (x2 x3) rev(x1,x2) = x1 x2

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Synthesizing determinacy conditions via backward analysis Success Patterns wrt Mux Conditions wrt rigid’ Determinacy Conditions wrt rigid’ Success pattern wrt rigid’ Determinacy condition wrt rigid’

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Determinacy Conditions Determinacy conditions describes those queries that are determinate. There is one determinacy condition for each predicate. p(x) :- g states that (p(x)) has at most one computed answer whenever rigid’ (g).

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Lower approximation of determinacy conditions If p(x) :- g is a determinacy condition and g’ |= g then p(x):-g’ is a determinacy condition. Determinacy conditions can thus be approximated from below without compromising correctness (but not necessarily from above)

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Greatest fixpoint computation Iteration commences with I 0 ={ p(x):-true | p }. I k+1 is computed from I k by considering each clause in turn and calculating a (more) correct determinacy condition. Intially I k+1 = I k Strengthen I k+1

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Propagating determinacy conditions backwards p(x) f 0, p 1 (x 1 ),…,p n (x n ). A correct condition g’ is obtained by Propagating the condition on each call e i = -x ((f 0 j<i f j ) g i )) where p i (x i ):-g i I k+1 and f j is the rigidity success pattern for p j (x j ). Conjoining e 1, e 2, …, e n and mutual exclusion condition for p(x).

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Updating I k+1 I k+1 contains a determinacy condition p(x) :- g” and this is updated to p(x) :- g”/\g’ if g” | g’. Determinacy conditions becomes progressively stronger on each iteration. This process will converge onto the gfp.

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Gfp computation for reverse I0: append(x1,x2,x3) :- true rev(x1,x2) :- true I1: append(x1,x2,x3) = x1 (x2 x3) rev(x1,x2) = x1 x2 I3=I2: append(x1,x2,x3) = x1 (x2 x3) rev(x1,x2) = x1

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Small Benchmarks

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Performance

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Other work on backward analysis for logic programs Termination Inference (Codish & Genaim 2005) Backward analysis via program transformation (Gallagher 2003) Suspension (Genaim and King 2003) Groundness (King & Lu 2002) Type (Lu & King 2002) Pair sharing (Lu & King 2004) Set sharing (Li & Lu 2005) Equivalence of Forward and backward analysis (King & Lu 2003)

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Conclusion Backwards analysis can infer sufficient conditions that ensures some properties are satisfied. Determinacy inference analysis is composed of off-the-self success pattern analysis, mutual exclusion synthesis and a backwards analysis. Initial experiments shows that it is practical and infers useful results.

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Future work Independence of computation rule For improved precision Mutual exclusion condition More term abstractions

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