All-Termination(SCP)
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1 All-Teriatio(SCP) Aaro Turo Northeaster Uiversity (joit work with Pete Maolios) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 1
2 Why, soeties I've believed as ay as six ipossible thigs before breakfast. -- Quee of Hearts, Alice i Woderlad Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 2
3 Why, soeties I've believed as ay as six ipossible thigs before breakfast. -- Quee of Hearts, Alice i Woderlad Sloga 0 All-Teriatio: how to solve six ipossible probles before luch Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 3
4 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 4
5 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 5
6 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? 1 (i, x, xs) = i Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 6
7 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? 1 (i, x, xs) = i 2 (i, x, xs) = legth xs Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 7
8 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? 1 (i, x, xs) = i 2 (i, x, xs) = legth xs 3 (i, x, xs) = i + legth xs Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 8
9 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? 1 (i, x, xs) = i 2 (i, x, xs) = legth xs 3 (i, x, xs) = i + legth xs Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 9
10 (defie (isert i x xs) (cod ((<= i 0) (cos x xs)) ((epty? xs) (list x)) (else (cos (car xs) (isert (- i 1) x (cdr xs)))) How do we prove that isert teriates? 1 (i, x, xs) = {i} Measured sets 2 (i, x, xs) = {xs} for isert 3 (i, x, xs) = {i, xs} Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 10
11 Measured sets iductio schees [Boyer&Moore, 1979] To prove i,x,xs :: φ(i,x,xs), show: Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 11
12 Measured sets iductio schees [Boyer&Moore, 1979] To prove i,x,xs :: φ(i,x,xs), show: Measured set {i} φ(0,x,xs) [ y,ys :: φ(i,y,ys)] φ(i+1,x,xs) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 12
13 Measured sets iductio schees [Boyer&Moore, 1979] To prove i,x,xs :: φ(i,x,xs), show: Measured set {i} φ(0,x,xs) [ y,ys :: φ(i,y,ys)] φ(i+1,x,xs) Measured set {xs} φ(i,x,il) [ j,y :: φ(j,y,xs)] φ(i,x,cos(z,xs)) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 13
14 Measured sets iductio schees [Boyer&Moore, 1979] To prove i,x,xs :: φ(i,x,xs), show: Measured set {i} φ(0,x,xs) [ y,ys :: φ(i,y,ys)] φ(i+1,x,xs) Measured set {xs} φ(i,x,il) [ j,y :: φ(j,y,xs)] φ(i,x,cos(z,xs)) Measured set {i,xs} φ(0,x,il) [ y :: φ(i,y,xs)] φ(i+1,x,cos(z,xs)) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 14
15 Life with a theore prover: Defie fuctios Kow fuctios Rewritig Make cojectures Iductio Provig egie Kow facts User Theore prover Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 15
16 Life with a theore prover: Defie fuctios Prove teriatio Make cojectures Kow fuctios Rewritig Iductio Provig egie Kow facts User Theore prover Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 16
17 Life with ACL2: Defie fuctios Prove teriatio Make cojectures Kow fuctios Rewritig Iductio Provig egie Kow facts User Theore prover Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 17
18 Life with ACL2 Seda: Defie fuctios Teriatio aalysis Kow fuctios Make cojectures Rewritig Iductio Provig egie User Kow facts Theore prover Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 18
19 A better life ACL2 Seda: Defie fuctios All-Teriatio aalysis Kow fuctios Make cojectures Rewritig Iductio Provig egie User Kow facts Theore prover Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 19
20 Sloga 1 Teriatio is ot a yes/o questio it's ultiple choice Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 20
21 The rest of the talk: All-Teriatio(T ) defiitio research progra Poly-tie size-chage teriatio (SCP) All-Teriatio(SCP) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 21
22 Teriatio aalysis Teriatio udecidable Soud, icoplete aalyses: T : Progras Bool predicate such that if T(P) the P teriates o all iputs Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 22
23 Teriatio aalysis Teriatio udecidable Soud, icoplete aalyses: T : Progras Bool predicate such that if T(P) the P teriates o all iputs Restricted teriatio aalysis such that T : Progras 2 Variables Bool if T(P,V) the V is a easured set for P Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 23
24 Measured sets are upward-closed: if U V ad U is a easured set for P the so is V Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 24
25 Measured sets are upward-closed: if U V ad U is a easured set for P the so is V All-Teriatio(T) aalysis All-Teriatio(T)(P) iial{v T(P,V)} where T is a restricted teriatio aalysis. Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 25
26 Measured sets are upward-closed: if U V ad U is a easured set for P the so is V All-Teriatio(T) aalysis All-Teriatio(T)(P) iial{v T(P,V)} where T is a restricted teriatio aalysis. Warig All-Teriatio(T)(P) ca be expoetial i P. Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 26
27 Theore: if T is i PSPACE the AllTeriatio(T) is i PSPACE. Proof: All-Teriatio(T)(P): for each V vars(p) if T(P,V) the iial := true for each U V if T(P,U) the iial := false if iial the output(v) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 27
28 Research progra Begi with stadard teriatio aalysis, A Defie restricted versio, T, so that [ V :: T(P,V)] A(P) Istruet A to produce a certificate Ipleet All-Teriatio(T)(P) by ruig A o P to produce certificate extractig easured sets fro certificate Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 28
29 Sloga 2 All-Teriatio does ot icrease power it eriches results Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 29
30 The rest of the talk: All-Teriatio(T ) Poly-tie size-chage teriatio (SCP) All-Teriatio(SCP) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 30
31 Size-chage teriatio [Lee et al, POPL01] works by aalyzig a safe abstractio of the progra. ack(0,) = +1 ack(,0) = 1 ack(-1, 1) ack(,) = 2 ack(-1, 3 ack(, -1)) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 31
32 Size-chage teriatio [Lee et al, POPL01] works by aalyzig a safe abstractio of the progra. 1 ack 2 3 ack(0,) = +1 ack(,0) = 1 ack(-1, 1) ack(,) = 2 ack(-1, 3 ack(, -1)) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 32
33 Size-chage teriatio [Lee et al, POPL01] works by aalyzig a safe abstractio of the progra. ack(0,) = +1 ack(,0) = 1 ack(-1, 1) ack(,) = 2 ack(-1, 3 ack(, -1)) 1 ack Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 33
34 Size-chage teriatio [Lee et al, POPL01] works by aalyzig a safe abstractio of the progra. ack(0,) = +1 ack(,0) = 1 ack(-1, 1) ack(,) = 2 ack(-1, 3 ack(, -1)) 1 ack Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 34
35 Size-chage teriatio [Lee et al, POPL01] works by aalyzig a safe abstractio of the progra. ack(0,) = +1 ack(,0) = 1 ack(-1, 1) ack(,) = 2 ack(-1, 3 ack(, -1)) 1 ack Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 35
36 1 ack Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 36
37 1 ack Call sequece Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 37
38 1 ack Call sequece Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 38
39 Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 39 ack Call sequece
40 Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 40 ack Call sequece
41 Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 41 ack Call sequece Thread
42 Progra P is size-chage teriatig for graph G if: each ifiite path i G has a thread w/ ifiite descet Theore [Lee et al, POPL01] Decidig size-chage teriatio is PSPACE-coplete Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 42
43 Poly-tie size-chage aalysis Call site C is a achor iff: Passig through C ifiitely ofte etails ifiite descet Algorith [Be-Ara, Lee 2007]: SCP(G): for each H i SCC(G) A := FidAchors(H) if epty(a) or SCP(H-A) = false the retur false retur true Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 43
44 The rest of the talk: All-Teriatio(T ) Poly-tie size-chage teriatio (SCP) All-Teriatio(SCP) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 44
45 A (aïve!) restricted versio of SCP Let restrict(g, V) be G, but with oly size-chage edges relatig variables i V. Theore: if 1. G is a valid aotated call graph for P 2. SCP(restrict(G,V)) the V is a easured subset for P. Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 45
46 Good, but... Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 46
47 Good, but... Theore: Decidig V :: SCP(restrict(G,V)) is NP-hard. Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 47
48 Good, but... Theore: Decidig V :: SCP(restrict(G,V)) is NP-hard. Corollary: It is ot true that SCP(G) iff V :: SCP(restrict(G,V)). What's goig o? Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 48
49 Good, but... Theore: Decidig V :: SCP(restrict(G,V)) is NP-hard. Corollary: It is ot true that SCP(G) iff V :: SCP(restrict(G,V)). What's goig o? No-ootoicity: V W ad SCP(restrict(G,V)) does ot iply SCP(restrict(G,W)) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 49
50 Sloga 3 Noootoicity eas trouble Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 50
51 What we do Istruet SCP to produce a achor tree Achor tree is a certificate of teriatio Trasfor achor tree to boolea costrait syste φ φ captures which variables are required for the teriatio proof sall thread preservers are allowed! φ = O( G ) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 51
52 Fidig the iial solutios Costraits φ are dual-hor: ca fid ψ that is equisatisfiable to φ cojuctio of clauses, each clause a disjuctio of literals at ost oe egative literal per clause i solutios to φ ca be foud fro ψ efficietly Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 52
53 Theore: After coputig φ, we ca fid k eleets of All-Teriatio(SCP)(P) i tie O( G k) Pay-as-you-go algorith Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 53
54 Sloga 4 To wi, istruet ad extract Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 54
55 Preliiary experietal results ACL2 has a large regressio suite: 100MB 11,000 fuctio defiitios (each of which ust be proved teriatig) Code ragig fro bit-vector libraries to odel checkers Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 55
56 Preliiary experietal results We have ipleeted, for ACL2, Poly-tie size-chage (SCP) Exp-tie size-chage (SCT) All-Teriatio(SCT) We have ot yet ipleeted All-Teriatio(SCP) Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 56
57 Preliiary experietal results The setup: we ra CCG + All-Teriatio(SCT) o the etire regressio suite. Nuber of fuctios: 11,000 Proved teriatig: 98% (ote: sae as CCG+SCT) Multiarguet fuctios: Proved teriatig 1728 With otrivial cores 90% With ultiple cores 7% Maxiu core cout 3 Ruig tie (ot icludig CCG): 30 secods Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 57
58 Preliiary experietal results The setup: we ra CCG + All-Teriatio(SCT) o the etire regressio suite. Nuber of fuctios: 11,000 Proved teriatig: 98% (ote: sae as CCG+SCT) Multiarguet fuctios: Proved teriatig 1728 With otrivial cores 90% With ultiple cores 7% Maxiu core cout 3 Ruig tie (ot icludig CCG): 30 secods Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 58
59 Preliiary experietal results The setup: we ra CCG + All-Teriatio(SCT) o the etire regressio suite. Nuber of fuctios: 11,000 Proved teriatig: 98% (ote: sae as CCG+SCT) Multiarguet fuctios: Proved teriatig 1728 With otrivial cores 90% With ultiple cores 7% Maxiu core cout 3 (the k paraeter) Ruig tie (ot icludig CCG): 30 secods Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 59
60 Future work Ipleet All-Teriatio(SCP) Exted our prototype to the ACL2 Seda Will help our fresha users at Northeaster Study All-Teriatio(T) for additioal T e.g. depedecy-pair teriatio aalysis Explore ew applicatios of easured subsets We've got a few i id, but wat to hear yours Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 60
61 Cotributio recap Proposed the All-Teriatio(T) proble Studied All-Teriatio(SCP) Sloga recap Teriatio is ot a yes/o questio All-teriatio icreases richess, ot power Noootoicity eas trouble To wi, istruet ad extract Mitchfest 2009 Aug 24 Aaro Turo All Teriatio(SCP) 61
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