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P RACTICAL A NALYSIS OF N ON - T ERMINATION IN L ARGE L OGIC P ROGRAMS Senlin Liang and Michael Kifer

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M OTIVATION High-level LP languages, e.g., SILK http://silk.semwebcentral.org http://silk.semwebcentral.org Flora-2 http://flora.sourceforge.net http://flora.sourceforge.net are designed to be suitable for knowledge engineers, who are not programmers KBs created by engineers typically are complex, large stress the capabilities of the underlying engine => Non-termination happens often – very hard to debug To address this, we developed Terminyzer – a non-Termination analyzer (this extends our previous work in PADL-13)

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O UTLINE Preliminaries: causes of non-termination, tabling, and forest logging Adding ids to rules Terminyzer: analyses causes, repairs problems Experiments Conclusion and future work

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C AUSES OF N ON -T ERMINATION Cause 1 : loops in SLD-resolution Example p(X) :- p(X). ?- p(a). Solution: tabling [SW12] Caches calls to subgoals, which cuts recursive loops If enough predicates are tabled then Each subgoal is tabled once Each answer is tabled once Evaluation terminates if there are finitely many subgoals and answers [SW12] Terry Swift and David S. Warren. XSB: Extending prolog with tabled logic programming. TPLP’12.

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C AUSES OF N ON -T ERMINATION ( CONT ’ D ) Cause 2: Engine supports tabling But the program generates infinitely many tabled subgoals Example p(X) :- p(f(X)). ?- p(a). Subgoals to be tabled: p(a), p(f(a)), p(f(f(a))),... Solution: subgoal abstraction [RS] Abstracts subgoals that are deeper than a threshold Assuming threshold = 2, p(f(f(f(a)))) would be abstracted to p(f(f(X))), X = f(a) Guarantees that there will be only a finite number of tabled subgoals [RS] F. Riguzzi and T. Swift. Terminating evaluation of logic programs with finite three-valued models. ACM on Computational Logic. To appear.

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C AUSES OF N ON -T ERMINATION ( CONT ’ D ) Cause 3: Engine supports both tabling and subgoal abstraction But infinitely many answers Example p(a). p(f(X)) :- p(X). ?- p(X). Answers to be derived: p(a), p(f(a)), p(f(f(a))), … Solution: does not exist Halting problem is undecidable Whether a program has a finite number of answers is undecidable We can only try to help the user to deal with the issue If user really intended the program to be the way it is We need a way to limit output. E.g., bounded rationality [BS13] Our focus : Unexpected non-termination (ie, when it’s a bug) [BS13] B. Grosof and T. Swift. Radial restraint: A semantically clean approach to bounded rationality for logic programs. AAAI’13

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T ABLING AND F OREST L OGGING Tabling needs no introduction Forest logging is a new tracing facility in XSB EventsLogs Calls to tabled subgoals E.g. parent calls child tabled_call(child,parent,status,timestamp) neg_tabled_call(child,parent,status,timestamp) Answer derivations E.g. ansr is derived for sub new_answer(ansr,sub,timestamp) new_delayed_answer(ansr,sub,delayed_lits,timestamp) Return answers to consumers E.g. ansr for child is retuned to parent answer_return(ansr,child,parent,timestamp) delayed_answer_return(ansr,child,parent,timestamp) Subgoal completions E.g. sub is completed completed(sub,scc_num,timestamp) completed(sub,early_completed,timestamp) Other eventsIrrelevant to our discussion where Timestamp preserves the order of events status = new, complete, incomplete

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A DDING ID S TO R ULES – K EY I NSIGHT Add unique ids to rules s.t. tabled subgoals remember their host rules: Each tabling declaration :- table p/n is changed to :- table p/(n+1) For a query ?- p(x 1, …, x n ) If p/n is tabled, then Change it to ?- p(x 1, …, x n, Newvar) Chop off the last arguments of returned answers Otherwise, the query stays the same

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T ERMINYZER O VERVIEW Two versions Version 1 (most precise) requires: Tabling Forest logging Subgoal abstraction Version 2 requires only Tabling Forest logging Currently only XSB has all three features, but: Several systems have tabling Forest logging info exists internally in all of them – just needs to be exposed to the user => at least Version 2 is easily portable

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T ERMINYZER O VERVIEW Suppose a query does not terminate, Terminyzer then Analyzes an execution forest log Determines the causes of non-termination The exact sequence of unfinished tabled subgoals, and the host rule id of each subgoal Subgoals forming recursive cycles Rectifies some causes of misbehavior (heuristically)

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C ALL S EQUENCE A NALYSIS Identifies the exact sequence of unfinished calls and their host rule ids that lead to a non-termination Find unfinished subgoals – whose answers have not been completely derived – by unfinished(Child,Parent,Timestamp) :- tabled_call(Child,Parent,new,Timestamp), not_exists(completed(Child,SCCNum,Timestamp1)). unfinished(child,parent,timestamp) says that Subgoal parent calls subgoal child, and it happened at timestamp Neither child nor parent have been completely evaluated Sort unfinished calls by their timestamps Host rule ids are kept in the last arguments of child- subgoals not_exists is the XSB well-founded negation operator; existentially quantifies SCCNum and Timestamp1.

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C ALL S EQUENCE A NALYSIS ( CONT ’ D ) Example 1 @!r1 p(a).@!r5 r(X) :- r(X). @!r2 p(f(X)) :- q(X).@!r6 r(X) :- p(X), s(X). @!r3 q(b).@!r7 s(f(b)). @!r4 q(g(X)) :- p(X).?- r(X). where @!ruleid is the syntax to assign rule ids Its unfinished calls – the red ones form a non- terminating loop unfinished(r(_h9900,_h9908), root, 0) – root is for intial query unfinished(r(_h9870,r5), r(_h9870,_h9889), 8) unfinished(r(_h9840,r5), r(_h9840,r5), 11) unfinished(p(_h9810,r6), r(_h9810,r5), 12) unfinished(q(_h9780,r2), p(_h9780,r6), 16) unfinished(p(_h9750,r4), q(_h9750,r2), 20) unfinished(q(_h9720,r2), p(_h9720,r4), 24) Next, we will find recursive cycles

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C ALL S EQUENCE A NALYSIS ( CONT ’ D ) Unfinished calls can be represented using an u nfinished- call graph UCG = (N,E) N : the set of unfinished subgoals E : { (parent,child) |unfinished(child,parent,ts) is true} For the above example unfinished(r(_h9900,_h9908), root, 0) unfinished(r(_h9870,r5), r(_h9870,_h9889), 8) unfinished(r(_h9840,r5), r(_h9840,r5), 11) unfinished(p(_h9810,r6), r(_h9810,r5), 12) unfinished(q(_h9780,r2), p(_h9780,r6), 16) unfinished(p(_h9750,r4), q(_h9750,r2), 20) unfinished(q(_h9720,r2), p(_h9720,r4), 24) where: Each node is represented by the timestamp when it is first called -1 represents root Edges are labeled with timestamps of calls Loops in UCG represent recursive cycles However, not all cycles are causing non-termination E.g. [8, 8] Can be represented as thousands of subgoals

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C ALL S EQUENCE A NALYSIS ( CONT ’ D ) Assume all predicates are tabled + subgoal abstraction Theorem (Soundness of the call sequence analysis) If there are unfinished calls in a query’s complete trace, then Call sequence analysis finds the exact sequence of unfinished calls that caused non-termination, and The ids of the rules that issued these calls Theorem (Completeness of the call sequence analysis) If the evaluation of a query does not terminate, then There is at least one loop in the UCG for its complete trace, and the loop’s subgoals are responsible for generating infinite number of answers, and The last argument of each of these subgoals specifies the rule ids from whose bodies these subgoals were called. T he complete trace is infinite due to non-termination so, practically speaking We work with only a prefix of the trace by limiting term depth/size or execution time/space It may produce false negatives, but they are also useful for identifying computational bottlenecks.

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A NSWER F LOW A NALYSIS Recall that not all cycles in UCG are causing non- termination, so we need to refine call sequence analysis Answer flow analysis does precisely that: it identifies the cycles that actually cause non- termination Non-termination happens if and only if a subset of subgoals keeps: Receiving answers from producers, Deriving new answers, and Returning answers to callers Answer flow analysis looks for repeated patterns of answer returns

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A NSWER F LOW A NALYSIS ( CONT ’ D ) Compute answer-flow patterns (AFP) Answer-return sequence (ARS): the sequence of (child,parent) pairs where child returns answers to parent Candidate AFP : a sequence cafp s.t. cafp 2+ is a suffix of ARS AFP: the shortest candidate AFP cafp s.t. its repetition forms the maximal suffix of ARS among all candidate AFP’s In previous Example 1 ARS = [ (p(_h599,r4),q(_h599,r2)), (q(_h619,r2),p(_h619,r4)), (p(_h639,r4),q(_h639,r2)), (q(_h659,r2),p(_h659,r4)), (p(_h679,r4),q(_h679,r2)), (q(_h699,r2),p(_h699,r4)), (p(_h719,r4),q(_h719,r2)), (q(_h739,r2),p(_h739,r4)), (p(_h759,r4),q(_h759,r2)), (q(_h779,r2),p(_h779,r4))]. where (child,parent) indicates child returns answers to parent Candidate AFPs are: cafp1 = [(p(_h759,r4),q(_h759,r2)), (q(_h779,r2),p(_h779,r4))] cafp2 = cafp1 cafp1 AFP is cafp1 AFP captures information flow pattern without redundancy @!r2 p(f(X)) :- q(X). @!r4 q(g(X)) :- p(X).

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A NSWER F LOW A NALYSIS ( CONT ’ D ) An AFP can be represented as an answer-flow graph AFG = (N,E), where N: the set of subgoals in afp E: {(child,parent) | (child,parent) ∈ afp} Loops in AFG represent cycles that cause non- termination

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A NSWER F LOW A NALYSIS ( CONT ’ D ) As before, we assume all predicates are tabled + subgoal abstraction Theorem (Soundness of the answer flow analysis) If the complete trace of a query has an AFP then the query does not terminate. Theorem (Completeness of the answer flow analysis) If the query evaluation does not terminate, then: There is an AFP in its complete trace, AFG = (N, E) contains at least one loop, Every sub ∈ N appears in at least one loop, and Each edge (sub1,sub2) ∈ E, where sub1 = pred(..., ruleid), tells us that sub2 calls sub1 from the body of a rule whose id is ruleid.

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M ORE ON UCG AND AFG Theorem (Relationship between UCG and AFG) Consider the UCG and AFG for a non- terminating forest log, we have: nodes(AFG) ⊂ nodes(UCG) edges(AFG) ⊂ reverse-edges(UCG) loops(AFG) ⊆ loops(UCG) Theorem (No false results for finite traces) If the evaluation of a query, Q, terminates, then both the UCG and the AFG for Q’ s trace are empty.

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A UTO -R EPAIR OF R ULES A query does not terminate if It has infinitely many answers, or It has a finite number of answers, but one of its subqueries has an infinite number of them In this case: a different evaluation order may terminate the query This case is targeted by our auto-repair heuristic For each unfinished(child,parent,timestamp) We know the host rule for this call, and the common set of the unbound arguments of parent and child – the arguments whose bindings are to be derived Thus, to reduce the possibility that parent receives infinite number of bindings from child, one can delay issuing a child -call from its host rule until these arguments are bound

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A UTO -R EPAIR OF R ULES ( CONT ’ D ) Example @!r1 p(a).@!r5 r(X) :- r(X). @!r2 p(f(X)) :- q(X).@!r6 r(X) :- p(X), s(X). @!r3 q(b).@!r7 s(f(b)). @!r4 q(g(X)) :- p(X).?- r(X). Its unfinished calls are: unfinished(r(_h9900,_h9908), root, 0) unfinished(r(_h9870,r5), r(_h9870,_h9889), 8) unfinished(r(_h9840,r5), r(_h9840,r5), 11) unfinished(p(_h9810,r6), r(_h9810,r5), 12) unfinished(q(_h9780,r2), p(_h9780,r6), 16) unfinished(p(_h9750,r4), q(_h9750,r2), 20) unfinished(q(_h9720,r2), p(_h9720,r4), 24) Applying auto-repair @!r1 p(a).@!r5 r(X) :- wish(ground(X))^r(X). @!r2 p(f(X)) :- wish(ground(X))^q(X).@!r6 r(X) :- wish(ground(X))^p(X), s(X). @!r3 q(b).@!r7 s(f(b)). @!r4 q(g(X)) :- wish(ground(X))^p(X).?- wish(ground(X))^r(X). Then the query will terminate with X = f(b)

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T ABLED E NGINES WITHOUT S UBGOAL A BSTRACTION Additional cause of non-termination: infinite number of subgoals Steps Compute the sequence of unfinished subgoals Compute simplified subgoal sequence (SSS) out of unfinished subgoal sequence Each unfinished subgoal, predicate(…, ruleid), is simplified to predicate(ruleid) Find the SSS pattern, as in the case of answer flow pattern SSS pattern contains the predicates and their rule ids that recursively call one another to form increasingly deep subgoals

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T ABLED E NGINES WITHOUT S UBGOAL A BSTRACTION ( CONT ’ D ) Example @!r1 p(a).@!r4 r(X) :- r(X). @!r2 p(X) :- q(f1(X)).@!r5 r(X) :- p(X), s(X). @!r3 q(X) :- p(f2(X)).@!r6 s(a). ?- r(a). Its unfinished calls are: unfinished(r(a,_h46), root, 0). unfinished(r(a,r4), r(a,_h27), 8). unfinished(r(a,r4), r(a,r4), 11). unfinished(p(a,r5), r(a,r4), 12). unfinished(q(f1(a),r2), p(a,r5), 16). unfinished(p(f2(f1(a)),r3), q(f1(a),r2), 19). unfinished(q(f1(f2(f1(a))),r2), p(f2(f1(a)),r3), 22). unfinished(p(f2(f1(f2(f1(a)))),r3), q(f1(f2(f1(a))),r2), 25). unfinished(q(f1(f2(f1(f2(f1(a))))),r2), p(f2(f1(f2(f1(a)))),r3), 28). unfinished(p(f2(f1(f2(f1(f2(f1(a)))))),r3), q(f1(f2(f1(f2(f1(a))))),r2), 31). unfinished(q(f1(f2(f1(f2(f1(f2(f1(a)))))),r2), p(f2(f1(f2(f1(f2(f1(a)))))),r3), 34). …… SSS = [root, r(_), r(r4), r(r4), p(r5), q(r2), p(r3), q(r2), p(r3), q(r2), p(r3), q(r2)] SSS pattern = [p(r3), q(r2)] – says that it is predicate p of r3 and predicate q of r2 that recursively call each other, thus forming increasingly deep nested subgoals

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S TATUS Unfinished-call/answer flow implemented in SILK and Flora-2 SILK has a GUI, Flora-2’s underway Rule Ids are crucial for practicality Auto-repair: not implemented yet

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E XPERIMENTS System Dual core 2.4GHz Lenovo X200 with 3GB RAM Ubuntu 11.04 with Linux kernel 2.6.38 Small programs They took a tiny fraction of a second to analyze Correctness of analyses is manually verified Large programs: one biology ontology from SILK KB size: Flora-2 program with 4,774 rules and 919 facts Compiled into XSB’s 5,500+ rules and 1,000+ facts Logs produced until evaluation consumed all memory Size: ~2GB Number of records: ~14M Took 170 seconds

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C ONCLUSIONS Terminyzer – a tool for analyzing non-termination Future work: Implement auto-repair better auto-repair algorithms Comparison with others: All other work deals with underpowered logic engines that are so last Century (Prolog) Or with trying to find sufficient conditions for termination (different focus)

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Thank you!

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F OREST L OGGING – E XAMPLE :- table path/2. edge(1,2). edge(1,3). edge(2,1). path(X,Y) :- edge(X,Y). path(X,Y) :- edge(X,Z), path(Z,Y). ?- path(1,Y).

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