Recently, the iterative approach named linear tabling has received considerable attention because of its simplicity, ease of implementation, and good space efficiency. Linear tabling is a framework from which different methods can be derived based on the strategies used in handling looping subgoals. One decision concerns when answers are consumed and returned. This paper describes two strategies, namely, lazy and eager strategies, and compares them both qualitatively and quantitatively. The results indicate that, while the lazy strategy has good locality and is well suited for finding all solutions, the eager strategy is comparable in speed with the lazy strategy and is well suited for programs with cuts. Linear tabling relies on depth-first iterative deepening rather than suspension to compute fixpoints. Each cluster of inter-dependent subgoals as represented by a top-most looping subgoal is iteratively evaluated until no subgoal in it can produce any new answers. Naive re-evaluation of all looping subgoals, albeit simple, may be computationally unacceptable. In this paper, we also introduce semi-naive optimization, an effective technique employed in bottom-up evaluation of logic programs to avoid redundant joins of answers, into linear tabling. We give the conditions for the technique to be safe (i.e. sound and complete) and propose an optimization technique called early answer promotion to enhance its effectiveness. Benchmarking in B-Prolog demonstrates that with this optimization linear tabling compares favorably well in speed with the state-of-the-art implementation of SLG.
The DRA method @cite_15 is also iteration based, but it identifies looping clauses dynamically and iterates the execution of looping clauses to compute fixpoints. While in linear tabling iteration is performed on only top-most looping subgoals, in DRA iteration is performed on every looping subgoal. In ET* @cite_2 , every tabled subgoal is iterated even if it does not occur in a loop. Besides the difference in answer consumption strategies and optimizations, the linear tabling scheme described in this paper differs from the original version @cite_33 @cite_18 in that followers fail after they exhaust their answers rather than steal their pioneers' choice points. This strategy is originally adopted in the DRA method.
@cite_15 @cite_18 @cite_33 @cite_2
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Early resolution mechanisms proposed for tabling such as OLDT rely on suspension and resumption of subgoals to compute fixpoints. Recently, a new resolution framework called linear tabling has emerged as an alternative tabling method. The idea of linear tabling is to use iterative computation rather than suspension to compute fixpoints. Although linear tabling is simple, easy to implement, and superior in space efficiency, the current implementations are several times slower than XSB, the state-of-the-art implementation of OLDT, due to re-evaluation of looping subgoals. In this paper, we present a new linear tabling method and propose several optimization techniques for fast computation of fixpoints. The optimization techniques significantly improve the performance by avoiding redundant evaluation of subgoals, re-application of clauses, and reproduction of answers in iterative computation. Our implementation of the method in B-Prolog not only consumes an order of magnitude less stack space than XSB for some programs but also compares favorably well with XSB in speed. Tabled evaluations ensure termination of logic programs with finite models by keeping track of which subgoals have been called. Given several variant subgoals in an evaluation, only the first one encountered will use program clause resolution; the rest uses answer resolution. This use of answer resolution prevents infinite looping which happens in SLD. Given the asynchronicity of answer generation and answer return, tabling systems face an important scheduling choice not present in traditional top-down evaluation: How does the order of returning answers to consuming subgoals affect program efficiency. Delaying-based tabling mechanisms, such as the one adopted in XSB, are non-linear in the sense that the computation state of delayed calls has to be preserved. In this paper, we present the implementation of a linear tabling mechanism. The key idea is to let a call execute from the backtracking point of a former variant call if such a call exists. The linear tabling mechanism has the following advantages over non-linear ones: (1) it is relatively easy to implement; (2) it imposes no overhead on standard Prolog programs; and (3) the cut operator works as for standard Prolog programs and thus it is possible to use the cut operator to express negation-as-failure and conditionals in tabled programs. The weakness of the linear mechanism is the necessity of re-computation for computing fix-points. However, we have found that re-computation can be avoided for a large portion of calls of directly-recursive tabled predicates. We have implemented the linear tabling mechanism in B-Prolog. Experimental comparison shows that B-Prolog is close in speed to XSB and outperforms XSB when re-computation can be avoided. Concerning space efficiency, B-Prolog is an order of magnitude better than XSB for some programs. This paper examines techniques for finding falsifying trajectories of hybrid systems using an approach that we call trajectory splicing. Many formal verification techniques for hybrid systems, including flowpipe construction, can identify plausible abstract counterexamples for property violations. However, there is often a gap between the reported abstract counterexamples and the concrete system trajectories. Our approach starts with a candidate sequence of disconnected trajectory segments, each segment lying inside a discrete mode. However, such disconnected segments do not form concrete violations due to the gaps that exist between the ending state of one segment and the starting state of the subsequent segment. Therefore, trajectory splicing uses local optimization to minimize the gap between these segments, effectively splicing them together to form a concrete trajectory. We demonstrate the use of our approach for falsifying safety properties of hybrid systems using standard optimization techniques. As such, our approach is not restricted to linear systems. We compare our approach with other falsification approaches including uniform random sampling and a robustness guided falsification approach used in the tool S-Taliro. Our preliminary evaluation clearly shows the potential of our approach to search for candidate trajectory segments and use them to find concrete property violations.