ttt一种非CDCL框架SAT求解器.基本概念——Ternary Tree Solver (tts-4-0)

Ternary Tree Solver (tts-4-0)

Ivor Spence School of Electronics, Electrical Engineering and Computer Science Queen’s University Belfast

i.spence@qub.ac.uk

31st May 2007

 

无法与cdcl和sls现代求解器相比，但在手工编写的基准测试最差性能(如hgen8, holen, xor-chain等)求解中表现良好。

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1 Introduction

The Ternary Tree Solver (tts) algorithm is a complete, deterministic solver for CNF satisfiability.

This note describes the operation of version 4.x. Version 4.0 was entered into the SAT 2007 competition.

The solver is very loosely based on the well-known Davis-Putnam model。

The solver has five phases, namely: Minimization; Variable ordering; Tree building; Tree walking, Rebuilding.

• •  Minimization;
  • Variable ordering;
  • Tree building;
  • Tree walking,
  • Rebuilding.

2.Minimization 在BCP中包含化简

Here, if possible, the problem is first partitioned into disjoint sub-problems in which any variable can be reached from any other. The rest of the algorithm processes each sub-problem separately. 

译文：在这里，如果可能的话，首先将问题划分为不相交的子问题，其中任何变量都可以从任何其他变量得到。剩下的算法分别处理每个子问题

Any clauses that are tautologies, i.e. contain both v and v‘, are removed.重言式去除。

If all occurrences of a particular variable are of the same sign it is safe to assign the corresponding value to the variable and hence remove any clauses containing it. 纯文字规则去除。

This is combined with unit clause propagation。

3 Variable ordering

Unlike most Davis-Putnam solvers the variables are processed according to a static ordering.

The overall performance depends critically on this ordering, in which variables that occur in the same clause should be processed near to each other.

译文：总体性能主要依赖于这种顺序，在这种顺序中，发生在同一子句中的变量应该被相邻地处理。

 

If the variables are regarded as nodes and the clauses as hyperedges, this corresponds to the minimum linear arrangement problem for hypergraphs.

译文：如果将变量视为节点，子句视为超边，则对应于超图的最小线性排列问题。

 

A perfect solution to this problem is known to be NP-hard, and so an approximation algorithm is used. Note that this approximation affects the overall performance, but not the correctness of the solver.

译文：注意，这种近似会影响总体性能，但不会影响求解器的正确性.

The approximation algorithm used for small inputs (of the order of fewer than 1000 literals) combines:

• • 1. Simulated Annealing - this is generally regarded as providing the best approximations for MLA, but the execution time is significant;
  • 2. A local search to see whether the simulated annealing result can be improved.

译文：局部搜索，看看模拟退火结果是否可以改进。

For larger inputs this algorithm is too slow and a more direct algorithm is used in which variables are chosen in turn according to weights which are derived from the number of clauses in common with variables already chosen. This is much faster so at least the solver has a chance of processing larger, easier inputs but does not give such a good ordering.

译文：对于较大的输入，这种算法太慢，因此使用了一种更直接的算法，在这种算法中，变量是根据权重依次选择的，权重是由与已经选择的变量相同的子句的数量导出的。

It should be noted that this phase of the overall solver is the most recently developed and is where most of the current research effort is devoted. In particular, although reducing the linear arrangement certainly improves the overall performance it is not known whether this is exactly the correct metric.

译文：特别是，尽管减少线性排列肯定会提高整体性能，但不知道这是否是正确的度量标准。

4 Tree building

At the heart of the algorithm is a 3-tree which represents the proposition to be solved. Each node of this tree corresponds to a proposition and each level corresponds to a variable according to the variable ordering which was determined in the previous phase. The tree is constructed as follows:

译文：该算法的核心是一个表示待解命题的3-树。该树的每个节点对应一个命题，每层对应一个变量，按照前一阶段确定的变量排序。树的构造如下:

• The root of the tree corresponds to the proposition to be solved.树的根对应于待解决的命题。

 

• Each node of the tree has three children, left, middle and right. The left child consists of those clauses from the current proposition that contain the literal v except that the v is removed (where v is the current level). The right child consists of clauses that contain v‘ except that the v’ is removed. The middle child consists of those clauses that don’t contain v or v’. 译文：树的每个节点都有三个子节点，左、中、右。

 

• • When the removal of a v or v‘ leaves a clause empty this generates the proposition false. When there are no clauses to be included in a child this generates the proposition true.

 

In summary, the middle child consists of those clauses not containing the current variable, the left child is what remains after setting v to false and the right child is what remains after setting v to true. If the current variable is not contained anywhere in the proposition then no children are created at this stage - children are only assigned when the current variable is relevant. The construction of this tree can be carried out quickly.

A hash table of derived propositions is maintained during the tree building process. This is ensures that if different partial assignments lead to the same proposition only one corresponding node is created. The data structure thus contains cycles and is no longer a tree, but can be interpreted as a tree for the purposes of the next stage.

译文：在树的构建过程中，维护导出命题的哈希表。这确保了如果不同的部分分配导致相同的命题，那么只创建一个相应的节点。因此，数据结构包含循环，不再是树，但可以被解释为树，以供下一阶段使用。

5 Tree walking

This phase is where the bulk of the computation occurs. Starting from the root, the walk has in principle a false/true choice to make at each level, representing an assignment to the corresponding variable, which would by itself lead to 2ⁿ routes (where n is the number of variables).

Each node of the tree represents only a portion of the proposition, and so a set of nodes is maintained to record progress. For the false branch from a particular set of nodes, the new set of nodes consists of the union of all left and middle children of the current set. For the true branch, the new set consists of all right and middle children.

There are three outcomes which can result in a path being pruned, i.e. abandoned before the full depth of variables has been explored:

• • When a set of nodes contains false, no further computation is performed on that branch because there can be no satisfying assignment built from the choices up to this point;

 

• • If a set contains all true nodes a satisfying assignment has been found, regardless of the choice of values for subsequent variables; 

 

• • When a set of nodes has been found to correspond to an unsatisfiable proposition a record of this is made. If a subsequent request is made for the same set (or indeed a superset) it is known immediately that this is unsatisfiable without having to repeat the previous analysis. This corresponds to clause memoization and in this form is perhaps the main contribution of this solver.

 

 

6 Rebuilding

If any of the sub-problems is found to be unsatisfiable then the overall problem is unsatisfiable. Otherwise, if all the sub-problems have been found to be satisfiable, some of the actions of the initial minimization have to be undone to construct the overall model. Variables removed because they only occur with one sign are inserted with the appropriate value and the models for each of the sub-problems are renumbered as required.

 

7 Conclusions

An upper-bound for the complexity of this algorithm appears to be the sum of the number of sets of nodes at each level, which is exponential in the number of nodes at that level. This number of nodes is at most the corresponding cut of the hypergraph and can be less than that. Improving the results from the variable ordering phase is expected to be the best way to improve the overall algorithm.

 

8 Change log

8.1 4.1 → 4.2

The generation of certificates of unsatisfiability, which had been present in an earlier version, was re-introduced. There is a version of the solver (ttsp) which can generate Allen van Gelder’s RPT proofs or tts-specific PRT proofs (for which a checker, checkprt, is also included)

8.2 4.0 → 4.1

Some minor changes to the c code to permit compilation in 64-bit mode. This mode means that pointers consume more memory and it is an open question whether the solver performs better in 64 or 32 bit mode.

8.3 3.0 → 4.0

In version 3.0 the tree was built with a new node for child and there was a separate phase during which nodes which represented the same proposition were merged. In version 4.0 a hash table of propositions is used so that only one node is ever created for a given proposition even if it can be derived in different ways by partial assignment of different clauses.

In version 3.0 if a proposition did not contain a given variable the corresponding tree node had ”true” for it’s ”left” and ”right” children and a node representing the same proposition again for it’s ”both” child. In version 4.0 the number of nodes is significantly reduced by recording the first variable which is relevant for each node and only creating child nodes when the level in the tree corresponds to this variable.

In version 3.0 the static variable ordering was determined using simulating annealing to find a minimum linear arrangement. This produced good results but the time taken increased rapidly with the number of variables. In version 4.0, if the size of the input exceeds a given threshold a more direct algorithm is used which does not produce such a good ordering for the tree walk but is much faster.