Abstract
A CDCL SAT solver can backtrack a large distance when it learns a new clause, e.g, when the new learnt clause is a unit clause the solver has to backtrack to level zero. When the length of the backtrack is large, the solver can end up reproducing many of the same decisions and propagations when it redescends the search tree. Different techniques have been proposed to reduce this potential redundancy, e.g., partial/chronological backtracking and trail saving on restarts. In this paper we present a new trail saving technique that is not restricted to restarts, unlike prior trail saving methods. Our technique makes a copy of the part of the trail that is backtracked over. This saved copy can then be used to improve the efficiency of the solver’s subsequent redescent. Furthermore, the saved trail also provides the solver with the ability to look ahead along the previous trail which can be exploited to improve its efficiency. Our new trail saving technique offers different tradeoffs in comparison with chronological backtracking and often yields superior performance. We also show that our technique is able to improve the performance of stateoftheart solvers.
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1 Introduction
The vast majority of modern SAT solvers that are used to solve realworld problems are based on the conflictdriven clause learning (CDCL) algorithm. In a CDCL SAT solver, backtracking occurs after every conflict, where all literals from one or more decision levels become unassigned before the solver resumes making decisions and performing unit propagations. Traditionally, CDCL solvers would backtrack to the conflict level, which is the second highest decision level remaining in the conflict clause after conflict analysis has resolved away all but one literal from the current decision level [9]. Recently, however, it has been shown that partial backtracking [6] or chronological backtracking, Cbt, (i.e., backtracking only to the previous level after conflict analysis) [8, 11] can be effective on many instances. Partial backtracking has been used in the solvers that won the last two SAT competitions. Although chronological backtracking breaks some of the conventional invariants of CDCL solvers, it has been formalized and proven correct [8] (also see related formalizations [10, 12]).
The motivation for using Cbt is the observation that when a solver backtracks across many levels, many of the literals that are unassigned during the backtrack might be reassigned again in roughly the same order when the solver redescends. This observation was first made in the context of restarts by van der Tak et al. [14]. Their technique backtracks to the minimum change level, i.e., the first level at which the solver’s trail can change on redescent. However, their technique cannot be used when backtracking from a conflict: the solver’s trail is going to be changed at the backtrack level so the minimum change level is the same as the backtrack level.
Chronological backtracking or partial backtracking instead allows a reduction in the length of the backtrack by placing literals on the trail out of decision level order. By reducing the length of the backtrack the solver can keep more of its assignment trail intact. This can save it from the work involved in reconstructing a lot of its trail. Using Cbt is not a panacea however. Its application must be limited for peak effectiveness. This indicates that it is sometimes beneficial for the solver to backtrack fully and redo its trail, even if this takes more work. We will expand on why this might be the case below.
In this paper we present a new trail saving method whereby we save the backtracked part of the solver’s trail and attempt to use that information to make the solver’s redescent more efficient. Unlike Cbt, our trail saving method preserves the traditional invariants of the SAT solver and its basic version is very simple to implement. It allows the search to retain complete control over the order of decisions, but helps make propagation faster. We develop some enhancements to make the idea more effective, and demonstrate experimentally that it performs as well as and often better than chronological backtracking. We also show that with our enhancements we are able to improve the performance of stateoftheart solvers.
2 Background
SAT solvers determine the satisfiability of a propositional formula \(\mathcal {F}\) expressed in Conjunctive Normal Form (CNF). \(\mathcal {F}\) contains a set of variables V. A literal is a variable \(v\in V\) or its negation \(\lnot v\), and for a literal l we let \(\text {var}(l)\) denote its underlying variable. A CNF consists of a conjunction of clauses, each of which is a disjunction of literals. We often view a clause as being a set of literals and employ set notation, e.g., \(\ell \in C\) and \(C'\subset C\). We will assume that the reader is familiar with the basic operations of CDCL SAT solvers. A good source for this background is [13].
Trails. CDCL SAT solvers maintain a trail which is the sequence of literals that have currently been assigned true by the solver. During its operation a SAT solver will add newly assigned literals to the end of the trail, and on backtrack remove literals from the end of the trail. For convenience, we will regard literals as having been assigned true if and only if they are on the trail. So removing/adding a literal to the trail is equivalent to unassigning/assigning the literal \(\textsc {true} \).
A SAT solver’s trail satisfies a number of conditions. However, in this work we will need some additional flexibility in our definitions, as we will sometimes be working with trails that would never be constructed by a SAT solver. Hence, we define a trail to be a sequence of literals each of which is either a decision literal or an implied literal, and each of which has a reason. These two types of literals are distinguished by their reasons. Decision literals d have a null reason, \( reason (d) = \varnothing \). Implied literals l have as a reason a clause of the formula \(\mathcal {F}\), \( reason (l) = C\in \mathcal {F}\). (The clause \( reason (l)\) can be a learnt clause that has been added to \(\mathcal {F}\)).
If literal \(\ell \) is on the trail \(\mathcal {T}\) let \(\iota _{\mathcal {T}}(\ell )\) denote its index on the trail, i.e, \(\mathcal {T}[\iota _{\mathcal {T}}(\ell )] = \ell \). If x and y are both on the trail and \(\iota _{\mathcal {T}}(x) < \iota _{\mathcal {T}}(y)\) we say that x appears before y on the trail. For convenience, when the trail being discussed is clear from context we simply write \(\iota \) instead of \(\iota _{\mathcal {T}}\).
Each literal \(\ell \in \mathcal {T}\) has a decision level \( decLvl (\ell )\) which is equal to the number of decision literals appearing on the trail up to and including \(\ell \) ; hence, \( decLvl (d)=1\) for the first decision literal \(d\in \mathcal {T}\). The set of literals on \(\mathcal {T}\) that have the same decision level forms a contiguous subsequence^{Footnote 1} that starts with a decision literal \(d_i\) and ends just before the next decision literal \(d_{i+1}\). We will often need to refer to different decision level subsequences of \(\mathcal {T}\). Hence, we let \(\mathcal {T}[[i]]\) denote the subsequence of literals at decision level i; and let \(\mathcal {T}[[i\ldots j]]\) denote the subsequence of literals at decision levels k for \(i\le k\le j\).
Definition 1
A clause C has been made unit by \(\mathcal {T}\) implying l when \(l\in C \wedge \bigl (\forall x\in C. x\ne l \rightarrow \lnot x \in \mathcal {T}\bigr )\). That is, all literals in C except l must have been falsified by \(\mathcal {T}\)
Now we define the following properties that a trail \(\mathcal {T}\) can have.

noncontradictory: A variable cannot appear in both polarities in the trail: \(l\in \mathcal {T}\rightarrow \lnot l \not \in \mathcal {T}\).

nonredundant: A literal can only appear once on \(\mathcal {T}\).

reasonsound: For each implied literal \(l\in \mathcal {T}\) we have that its reason clause \( reason (l)=C\) has been made unit by \(\mathcal {T}\) implying l, and for each \(x\in C\) with \(x\ne l\) we have that \(\lnot x\) appears before l on \(\mathcal {T}\): \(\forall l\in \mathcal {T}.\, reason (l)\ne \varnothing \rightarrow l\in reason (l) \wedge \bigl (\forall x\in reason (l).\, x\ne l \rightarrow \lnot x \in \mathcal {T}\wedge \iota (\lnot x) < \iota (l)\bigr )\).

propagationcomplete: Unit propagation has been run to completion at all decisions levels of \(\mathcal {T}\). This means that literals appear on \(\mathcal {T}\) at the first decision level they were unit implied. Formally, this can be captured by the condition: \(\forall i \in \{ decLvl (l)\,\,l\in \mathcal {T}\}. \bigl (\exists C\in \mathcal {F}. C \text{ is } \text{ made } \text{ unit } \text{ by } \mathcal {T}[[0\ldots i]]{ implying}l\bigr ) \rightarrow l\in \mathcal {T}[[0\ldots i]]\). Note that propagation completeness implies that \( reason (l)\ne \varnothing \) must contain at least one other literal \(y\ne l\) with \( decLvl (y) = decLvl (l)\).

conflictfree: No clause of F is falsified by \(\mathcal {T}\). Clauses \(C\in F\) falsified by \(\mathcal {T}\) are typically called conflicts.
In CDCL solvers using standard conflict directed backtracking all properties hold of the prefix of the solver’s trail consisting of all decisions levels but the deepest. The full trail might, however, contain a conflict at its deepest level so is not necessarily conflictfree. The full trail might also not be propagationcomplete, as unit propagation at the deepest level is typically terminated early if a conflict is found. It can further be noted that the first four properties imply that if a clause C is falsified at decision level k, then C must contain at least two literals at level k (otherwise C would have become unit at a prior level and then satisfied by making its last unfalsified literal true).
Standard Backtracking. In CDCL SAT solving the solver extends its trail by adding new decision literals followed by finding and adding all unit implied literals arising from that new decision. This continues until it reaches a decision level \(L_{\textit{deep}}\) where a conflict C is found.
In standard backtracking, the solver then constructs a new 1UIP clause by resolving away all but one literal at level \(L_{\textit{deep}}\) from the conflict C using the reason clauses of these literals. (As noted above C must contain at least two literals at level \(L_{\textit{deep}}\)). Hence, the new clause \(C_{\textit{1UIP}}\) will contain one literal \(\ell _{\textit{deep}}\) at level \(L_{\textit{deep}}\) and have all of its other literals a levels less than \(L_{\textit{deep}}\). The solver then backtracks to \(L_{\textit{back}}\) the second deepest level in \(C_{\textit{1UIP}}\). This involves changing \(\mathcal {T}\) to its prefix \(\mathcal {T}[[0\ldots L_{\textit{back}}]]\) (by our convention all literals removed from \(\mathcal {T}\) are now unassigned). The new clause \(C_{\textit{1UIP}}\) is made unit by \(\mathcal {T}[[0\ldots L_{\textit{back}}]]\) implying \(\ell _{\textit{deep}}\), so the solver then adds \(\ell _{\textit{deep}}\) to the trail and executes another round of unit propagation at level \(L_{\textit{back}}\), after which it continues by once again growing the trail with new decisions and unit implied literals until a new conflict or a satisfying assignment is found.
In standard backtracking, the difference between the backtrack level, \(L_{\textit{back}}\) and the current deepest level \(L_{\textit{deep}}\) can be very large. During its new descent from \(L_{\textit{back}}\) the solver can reproduce a large number of the same decisions and unit propagations, essentially wasting work. This potential inefficiency has been noted in prior work [6, 8, 11, 14].
In [14] a technique for reducing the length of the backtrack during restarts was presented. In restarts, the solver backtracks to level 0, and this technique involves computing a new deeper backtrack level \(M> 0\) for which it is known that on redescent the first \(M+1\) levels of the trail will be unchanged (except perhaps for the ordering of the literals). This technique removes the redundant work of reproducing the first M trail levels. When backtracking from a conflict, however, the trail will be changed at level \(L_{\textit{back}}\) (\(\ell _{\textit{deep}}\) will be newly inserted at this level). Hence this technique cannot reduce the length of the backtrack. In this paper we will show that although we have to backtrack to \(L_{\textit{back}}\) we can make the subsequent redescent much more efficient.
Chronological Backtracking. Chronological backtracking (Cbt) and partial backtracking in the context of clause learning solvers are alternatives to standard backtracking which allow the solver to execute a shorter backtrack. That is, with these techniques the solver can avoid having to go all the way back to the second deepest level in the learnt clause, as in standard backtracking.
Formalisms for partial backtracking in clause learning solvers have been presented in [10, 12]. In [6] practical issues of implementation were addressed, and experiments shown with a CDCL solver using partial backtracking. In [11] improved and more efficient implementation techniques were developed which allowed Cbt to make improvements to stateoftheart SAT solvers, and [8] presented additional implementation ideas and details along with correctness results for these methods.
The aim of partial backtracking is to reduce the redundant work that might be done by the SAT solver on its redescent from the backtrack level \(L_{\textit{back}}\). The technique allows the solver to backtrack to any level j in the range \(L_{\textit{back}}\le j \le L_{\textit{deep}}{}1\) (where \(L_{\textit{deep}}\) is the level the conflict was discovered). Nadel and Ryvchin [11] proposed to always backtrack chronologically to \(L_{\textit{deep}}{}1\) while Möhle and Biere [8] returned to the proposal of [6] of flexibly backtracking to any level in the allowed range. Note that the new learnt 1UIP clause \(C_{\textit{1UIP}}\) is made unit at every level in this range. So after backtracking to level j the newly implied literal \(\ell _{\textit{deep}}\) is added to the trail with \( reason (\ell _{\textit{deep}})=C_{\textit{1UIP}}\), and \( decLvl (\ell _{\textit{deep}})\) is set to \(L_{\textit{back}}\) (the second deepest level in \(C_{\textit{1UIP}}\)).
This means that the decision levels on the trail are no longer contiguous, as \(\ell _{\textit{deep}}\) has a different level than the other literals at level j (if \(j\ne L_{\textit{back}}\)). This change has a number of consequences for the SAT solver’s operation, all of which were described in [6]. Möhle and Biere [8] showed that despite these consequences partial backtracking can be made to preserve the soundness of a CDCL solver.
3 Chronological Backtracking Effects on Search
In this paper we present a new technique that allows the SAT solver to use standard backtracking, but also allows saving some redundant work on its redescent. Our method has more overhead than Cbt so the first question that must be addressed is why not just use chronological backtracking.
Although Cbt is able to avoid a lot of redundant work it also has other effects on the SAT solver search. These effects are sometimes detrimental to the solver’s performance and so it is not always beneficial to use Cbt. In fact, in both [11] and [8] it was found that fairly limited application of Cbt performed best. In [11] Cbt was applied only when the length of the standard backtrack, \(L_{\textit{back}}L_{\textit{deep}}\) was greater than a given threshold T. In their experiments they found that \(T=100\) was the best value, i.e., Cbt is done only on longer backtracks. In practice, this meant that Cbt was relatively infrequent; in our measurements with their solver only about 3% of the solver backtracks were Cbt backtracks. In [8] the value \(T=100\) was also applied. However, they introduced an additional technique to add some applications of Cbt when the length of the backtrack is less than T. This allowed [8] to utilize Cbt in about 15% of the backtracks.
Although it is difficult to know precisely why Cbt is not always beneficial, we can identify some different ways in which Cbt can affect the SAT solver’s search. With standard backtracking the literal \(\ell _{\textit{deep}}\) is placed on the trail at the end of \(L_{\textit{back}}\) and then unit propagated. This could impact the trail in at least the following ways. First, some literals might become unit at earlier levels. This could include decision literals becoming forced which might compress some decision levels together. Second, different decisions might be made due to changes in the variable scores arising from the newly learnt clause. And third, literals might be unit implied with different reasons. Cbt can change all of these things, each of which could have an impact on the future learnt clauses, and thus on the solver’s overall efficiency.
The second impact, changing variable scores, is partially addressed in [8] who utilize the ideas of [14] to backtrack to a level where the decisions would be unchanged. However, if the length of the backtrack is greater than 100 there could still be a divergence between the variable decisions generated in standard backtracking and Cbt. An argument is also given in [14] that the third impact, changing literal reasons, is not significant. However, the experiments in [14] were run before good notions of clause quality were known [1]. Our empirical results indicate that once clause quality is accounted for, changing the literal reasons can have a significant impact.
The first impact is worth discussing since it was mentioned in [6] but not in the subsequent works. This is the issue of changing the decision levels of literals on the trail. Cbt computes the decision level of each implied literal based on the decision levels of the literals in its reason, but it does not go backwards to change the decisions levels of literals earlier on the trail.
Example 1
For example, suppose that \((x, \lnot y)\in \mathcal {F}\), the literal x is a decision literal on the trail with \( decLvl (x)=2\), and that the solver is currently at level 150 where it encounters a conflict. If this conflict yields the unit clause (y), standard backtracking would backtrack to level 0, where x would be implied. On redescent, x would no longer form a new decision level and it would not appear in any new clauses (as it is entailed by \(\mathcal {F}\)). Cbt, on the other hand, would backtrack to level 149. On its trail x would still be at level 2. Until a backtrack past level 2 occurs, learnt clauses might contain \(\lnot x\), and thus have level 2 added to their set of levels (potentially changing their LBD score). Only when backtrack past level 2 occurs would x be restored to its correct level 0, and it would require inprocessing simplifications to remove x from the learnt clauses.
In sum, although these impacts of Cbt on the SAT solver’s search might or might not be harmful to the SAT solver, they do exist. In fact, there are two pieces of evidence that these impacts can sometimes be harmful. First, as mentioned above, previous work found that it is best to only apply Cbt on large backtracks where it has the potential to save the most work. If there were no harmful effects it would always be effective to apply Cbt. And second, in our empirical results below we show that our new trail saving technique, which always uses standard backtracking, can often outperform Cbt. Although our technique reduces the solver’s work on redescent it does not completely eliminate it like Cbt does. Hence its superior performance can only occur if Cbt is sometimes harmful.
It is possible to combine Cbt with our trail saving technique to reduce the amount of work required whenever the solver performs nonchronological backtracking. However, Cbt greatly reduces the potential savings that could be achieved by our method since most of its nonchronological backtracks are relatively short (less than threshold T levels). In our preliminary experiments this combination did not seem promising.
Nevertheless, there is good evidence that Cbt can improve SAT solver performance.^{Footnote 2} Hence, it should be that it is better to perform Cbt in some branches. Hence, an interesting direction for future work would be to develop better heuristics about when to use Cbt in a branch and when to use standard backtracking augmented by our trail saving method.
4 Trail Saving
Our approach is to save the trail \(\mathcal {T}\) on backtrack, and to use the saved trail \(\mathcal {T}_{ save }\) when the solver redescends to improve the efficiency of propagations without affecting the decisions the solver wants to make. The saved trail \(\mathcal {T}_{ save }\) also provides a secondary “lookahead mechanism” that the SAT solver can exploit as it redescends.
Suppose that the solver is at \(L_{\textit{deep}}\) where it has encountered a conflict. From the 1UIP clause it learns, \(C_{\textit{1UIP}}\), it now has to backtrack to \(L_{\textit{back}}\). This is accomplished by calling backtrack(\(L_{\textit{back}}\)), shown in Fig. 1, which saves the backtracked portion of the trail.
Note that backtrack does not save the deepest level of \(\mathcal {T}\). The full \(\mathcal {T}\) contains a conflict (at its deepest level). Hence the solver will never reproduce all the same levels, and it would be useless to save all of them. Note also that in addition to saving the literals in \(\mathcal {T}_{ save }\) we also save the clause reason of the unit implied literals in a separate \( reason_{save} \) vector. Finally, we see that after backtrack the first literal on \(\mathcal {T}_{ save }\) is a decision literal: it is the first literal of \(\mathcal {T}\) at decision level \(L_{\textit{back}}+1\). Literals will be removed from \(\mathcal {T}_{ save }\) during its use, but always in units of complete decision levels. So \(\mathcal {T}_{ save }[0]\) will always be a (previous) decision literal.
After backtrack the solver will add \(\ell _{\textit{deep}}\) to the end of the updated \(\mathcal {T}\) with \( reason (\ell _{\textit{deep}})\) \(=C_{\textit{1UIP}}\) and then invoke unit propagation. \(\mathcal {T}_{ save }\) is exploited during propagation by the version of propagate shown in Fig. 1, which will initially be invoked with the argument \(\iota (\ell _{\textit{deep}})\) (i.e., the trail index of the newly added implicant). The saved trail will be continually consulted during the solver’s descent whenever unit propagation is performed. When backtrack occurs \(\mathcal {T}_{ save }\) will be overwritten to store the new backtracked portion of \(\mathcal {T}\).
\(\mathcal {T}_{ save }\) is consulted in the procedure \(\textsc {useSavedTrail} \) (Fig. 1). This procedure tries to add saved implied literals and their reasons to the solver’s trail, when these implications are valid. We will show below that those implications that are added are in fact valid. We do not interfere with the solver’s variable decisions. Instead we opportunistically test to see if literals implied on \(\mathcal {T}_{ save }\) are valid implications for the solver given the solver’s current decisions.
\(\mathcal {T}_{ save }[0]\) is always a (previous) decision literal d with \( reason_{ save }(d) =\varnothing \). Note that, since new literals (e.g., \(\ell _{\textit{deep}}\)) have been added to \(\mathcal {T}\), d might now be an implied literal on \(\mathcal {T}\) (i.e., \( reason (d)\ne \varnothing \)) even though before the backtrack it was previously a decision (i.e., \( reason_{ save }(d) =\varnothing \)). If d has not been assigned \(\textsc {true} \) by the solver (i.e., \(\lnot d \in \mathcal {T}\)), we cannot add any implied literals below it on \(\mathcal {T}_{ save }\) to \(\mathcal {T}\) as these implied literals depend on d being assigned \(\textsc {true} \). In this case we stop looking for more literals to add to \(\mathcal {T}\) (line 18).
On the other hand if d has been made true by the solver we can continue to add all of the implied literals below it (up to but not including the next decision literal on \(\mathcal {T}_{ save }\)) to \(\mathcal {T}\) (line 24), reusing their saved reasons. Any literals that have already been made true by the solver can be skipped (line 20). Finally, if we encounter a literal that has already been falsified by the solver, then its saved reason clause must be falsified by the solver and we can return it as a conflict (line 21). If a conflict is encountered we leave \(\mathcal {T}_{ save }\) unchanged by resetting idx to zero. Otherwise, idx will be the number of literals at the front of \(\mathcal {T}_{ save }\) that have been moved to \(\mathcal {T}\) (or skipped over since they are already on \(\mathcal {T}\)). We then remove the first idx literals from \(\mathcal {T}_{ save }\) (line 27), and return the conflict (equal to \(\varnothing \) if no conflict was found).
Example 2
Figure 2 provides an example of how \(\mathcal {T}_{ save }\) is used. Initially the literals \(l_1\) to \(l_{14}\) are on the solver’s \(\mathcal {T}\), and \(\mathcal {T}_{ save }\) is empty. This is shown in the first two lines of the figure. In the figure the superscript on the literals indicates their decision level, and a superscripted \(*\) indicates that the literal is a decision. Hence \(l_1^{1*}\) indicates that \( decLvl (l_1)=1\) and that \(l_1\) is a decision.
Then a conflict is found at level 6 and the 1UIP clause \((\lnot l_1, \lnot l_3, \lnot l_{12})\) is learnt. Thus the solver will backtrack to level 2, where it will add \(\lnot l_{12}\) as a unit implicant. The next two lines show \(\mathcal {T}\) and \(\mathcal {T}_{ save }\) right after the backtrack to level 2: the backtracked levels have been copied into \(\mathcal {T}_{ save }\) omitting the conflict level 6.
The new unit \(\lnot l_{12}\) is now added to \(\mathcal {T}\) and unit propagation performed adding \(l_7\) and \(l_9\) to level 2. Since the first literal on \(\mathcal {T}_{ save }\), \(l_5\), has \( reason_{ save }(l_5) =\varnothing \) (\(l_5\) was a decision on \(\mathcal {T}\) at the time backtrack occurred) and is not yet \(\textsc {true} \), \(\mathcal {T}_{ save }\) is not helpful at this stage. The status of \(\mathcal {T}\) and \(\mathcal {T}_{ save }\) at this point is shown in the figure.
After unit propagation is finished the solver makes a new decision, which happens to be (but is not forced to be) \(l_5\). Now \(\mathcal {T}_{ save }\) can be used: \(l_5\) is true so it is removed, \(l_6\) is unassigned so it is added to \(\mathcal {T}\), \(l_7\) is true and so removed, \(l_8\) is unassigned so it is added to \(\mathcal {T}\), \(l_9\) is true and removed, and finally \(l_{10}\) and \(l_{11}\) are unassigned and so are added to \(\mathcal {T}\). In this example, \(\mathcal {T}_{ save }\) is emptied, and cannot contribute more to \(\mathcal {T}\).
All of these units are added to \(\mathcal {T}\) before the solver starts to unit propagate \(l_5\). Since, new literals have been added to \(\mathcal {T}\) before \(l_5\) the solver must propagate \(l_5\) and all of the literals that follow it before making its next decision.
As noted in the previous example unit propagation has to be rerun on all saved literals added to \(\mathcal {T}\) from \(\mathcal {T}_{ save }\). Thus our technique, unlike Cbt, does not completely remove the overhead of reproducing the trail on the solver’s redescent. Nevertheless, trail saving improves the efficiency of this redescent in three different ways. First, by adding more forced literals to the trail before continuing propagating the next literal, propagation can potentially gain a quadratic speedup [2, 5]. Second, propagation does not need to examine the reason clause of the added literals. If these literals were not added by useSavedTrail, propagation would have to traverse each of these reason clauses to determine that they have in fact become unit. Third, when a conflict is returned by useSavedTrail all further propagations can be halted. The added literals and their reasons will be sufficient to perform clause learning from the conflict returned by useSavedTrail. Since trail saving can sometimes save hundreds or thousands of literals at a time these savings can in sum be significant.
4.1 Correctness
Now we will prove that our use of \(\mathcal {T}_{ save }\) preserves the SAT solver’s soundness. In particular, \(\mathcal {T}_{ save }\) is only used in the procedure useSavedTrail, in which it either adds new literals to the solver’s trail, or returns conflict clauses to the solver. Hence, we only need to show that these new literals are in fact unit implied and the conflicts are in fact falsified by the solver’s trail. Since both \(\mathcal {T}\) and \(\mathcal {T}_{ save }\) are sequences of literals (with associated reasons) we can consider their concatenation denoted as \(\mathcal {T}+ \mathcal {T}_{ save }\).
Theorem 1
If \(\mathcal {T}+ \mathcal {T}_{ save }\) is reason sound (Sect. 2) then the following holds. If the first i literals on \(\mathcal {T}_{ save }\) are all in \(\mathcal {T}\) (\(\forall j. 0\le j< i. \mathcal {T}_{ save }[j]\in \mathcal {T}\)) and \(\mathcal {T}_{ save }[i]=l\) is an implied literal with \( reason_{ save }(l) = C\), then C has been made unit by \(\mathcal {T}\) implying l.
Proof:
Since \(\mathcal {T}+ \mathcal {T}_{ save }\) is reason sound, every literal in C other than l appears negated before l in the sequence \(\mathcal {T}+ \mathcal {T}_{ save }\). Thus for \(x\in C\) we have \(\lnot x \in \mathcal {T}\) or \(\lnot x \in \mathcal {T}_{ save }[0]\ldots \mathcal {T}_{ save }[i1]\). But in the later case we also have \(\lnot x \in \mathcal {T}\). \(\square \)
This theorem shows that useSavedTrail ’s processing is sound. In this procedure, an implied literal from \(\mathcal {T}_{ save }\) is added to \(\mathcal {T}\) (line 24) only when all prior literals on \(\mathcal {T}_{ save }\) are already on \(\mathcal {T}\) (i.e., previously on \(\mathcal {T}\) or already added to \(\mathcal {T}\)). Thus each new addition is sound given the inductive soundness of the previous additions, with the base case covered by Theorem 1. If l is to be added, the theorem shows that every other literal in \( reason_{ save }(l) \) has been falsified by \(\mathcal {T}\). Hence if l is also falsified by \(\mathcal {T}\) then \( reason_{ save }(l) \) is a clause that is falsified by \(\mathcal {T}\), thus it is a sound conflict for the solver.
Now we only have to show that \(\mathcal {T}+\mathcal {T}_{ save }\) is always reason sound during the operation of the solver.
Proposition 1
If \(\mathcal {T}+\mathcal {T}_{ save }\) is reason sound then \(\mathcal {T}'+\mathcal {T}_{ save }'\) is reason sound in all of the following cases.

1.
\(\mathcal {T}_{ save }[0]\in \mathcal {T}\), \(\mathcal {T}'=\mathcal {T}\), and \(\mathcal {T}_{ save }' = \mathcal {T}_{ save }.\mathrm {removeFront()}\).

2.
\(\mathcal {T}' = \mathcal {T}+\mathcal {T}_{ save }[0]\) and \(\mathcal {T}_{ save }' = \mathcal {T}_{ save }.\mathrm {removeFront()}\).

3.
\(\mathcal {T}' = \mathcal {T}+ \mathcal {T}_{ new }\) and \(\mathcal {T}_{ save }' = \mathcal {T}_{ save }'\) and \(\mathcal {T}'\) is reason sound.

4.
We also have that \(\mathcal {T}\) is reason sound if \(\mathcal {T}\) was generated by the solver.
Proof:
(1) \(\mathcal {T}_{ save }[0]\) already appears earlier in the \(\mathcal {T}\) so it can be removed without affecting the soundness of any reason following it. (2) is obvious as the sequence is unchanged. (3) the reasons in \(\mathcal {T}+\mathcal {T}_{ new }\) are sound by assumption. Those in \(\mathcal {T}_{ save }\) remain sound as they depend only on the literals in \(\mathcal {T}\) and prior literals on \(\mathcal {T}_{ save }\), both of which are unchanged. (4) is obvious from the operation of unit propagation in the solver. \(\square \)
Theorem 2
\(\mathcal {T}+\mathcal {T}_{ save }\) is always reason sound during the operation of the solver.
Proof:
\(\mathcal {T}_{ save }\) starts off being empty, so \(\mathcal {T}+\mathcal {T}_{ save }= \mathcal {T}\) is reason sound as it was generated by the solver (4). In procedure backtrack \(\mathcal {T}+\mathcal {T}_{ save }\) is set to a trail that was previously generated by the solver (4). The solver can add to \(\mathcal {T}\) by decisions and propagations without using \(\mathcal {T}_{ save }\). In this case \(\mathcal {T}' = \mathcal {T}+\mathcal {T}_{ new }\), and \(\mathcal {T}'\) is reason sound by (4), thus the new \(\mathcal {T}'+\mathcal {T}_{ save }\) is reason sound by (3). Finally, in procedure useSavedTrail either (a) literals at the front of \(\mathcal {T}_{ save }\) are discarded since they already appear on \(\mathcal {T}\), or (b) literals are moved from \(\mathcal {T}_{ save }\) to \(\mathcal {T}\). Under both of these changes \(\mathcal {T}+\mathcal {T}_{ save }\) remains reason sound by (1) and (2). \(\square \)
4.2 Enhancements
We developed three enhancements of the base trail saving method described above. In this section we present these enhancements.
Saving the Trail over Multiple Backtracks. It can often be the case that when the solver finds a conflict and backtracks to \(L_{\textit{back}}\) it might immediately find a another conflict at \(L_{\textit{back}}\) causing a further backtrack. In the procedure backtrack every backtrack causes \(\mathcal {T}_{ save }\) to be overwritten. Hence, in these cases most of the trail will not be saved—only the portion from the last backtrack. Our first extension addresses this potential issue and also provides more general trail saving in other contexts as well.
This extension is simply to add the latest backtrack to the front of \(\mathcal {T}_{ save }\) leaving all of the previous contents of \(\mathcal {T}_{ save }\) intact. Specifically, we replace line 3 of backtrack by the new line:
3. \(\mathcal {T}_{ save }= \mathcal {T}[[L_{\textit{back}}{+}1\ldots L_{\textit{deep}}{}1]] + \mathcal {T}_{ save }\)
It is not difficult to show that this change preserves soundness. Only Theorem 2 is potentially affected. However, we know that \(\mathcal {T}_{ save }\) is unchanged at the level at which a conflict occurs: either the conflict is detected without consulting \(\mathcal {T}_{ save }\) or if the conflict comes from \(\mathcal {T}_{ save }\) then useSavedTrail leaves \(\mathcal {T}_{ save }\) unchanged (line 21). Hence, at the level before the conflict occurred we have inductively that \(\mathcal {T}[[0\ldots L_{\textit{deep}}{}1]]+\mathcal {T}_{ save }\) was reason sound, and hence so is \(\mathcal {T}'+\mathcal {T}_{ save }'\) with \(\mathcal {T}' = \mathcal {T}[[0\ldots L_{\textit{back}}]]\) and \(\mathcal {T}_{ save }' = \mathcal {T}[[L_{\textit{back}}{+}1\ldots L_{\textit{deep}}{}1]] + \mathcal {T}_{ save }\).
When adding to the front of \(\mathcal {T}_{ save }\) in this manner \(\mathcal {T}_{ save }\) can grow indefinitely. So we prune \(\mathcal {T}_{ save }\) when it gets too large by (a) removing \(\mathcal {T}_{ save }[i]\) if \(\mathcal {T}_{ save }[i] = \mathcal {T}_{ save }[j]\) for some \(j<i\) (\(\mathcal {T}_{ save }[i]\) is redundant), and (b) removing the suffix of \(\mathcal {T}_{ save }\) starting at \(\mathcal {T}_{ save }[i]\) when \(\mathcal {T}_{ save }[j] = \lnot \mathcal {T}_{ save }[i]\) for some \(j<i\) (\(\mathcal {T}_{ save }[i]\) will never be useful as its negation, \(\mathcal {T}_{ save }[j]\), would have to be added to \(\mathcal {T}\) first). In this way \(\mathcal {T}_{ save }\) need never become larger than the number of variables in \(\mathcal {F}\).
Lookahead for Conflicts. In useSavedTrail we stop adding literals from \(\mathcal {T}_{ save }\) to \(\mathcal {T}\) once we reach a decision literal d on \(\mathcal {T}_{ save }\) that is not yet on \(\mathcal {T}\) (line 18 of useSavedTrail). This is done so that the solver has full control over variable decisions without interference from the trail saving mechanism (unlike the case with Cbt). However, another option would be to force the solver to use d as its next decision literal, which would then allow us to further add all of d’s implied literals on \(\mathcal {T}_{ save }\) onto \(\mathcal {T}\). This can be done for the first k decisions on \(\mathcal {T}_{ save }\) for any k. But in general, we do not want to remove the solver’s autonomy by forcing it to make potentially different decisions than it might have wanted to.
However, if there is a literal \(l\in \mathcal {T}_{ save }\) for which \(\lnot l\in \mathcal {T}\), we can observe that forcing the solver to make all of the decisions of \(\mathcal {T}_{ save }\) that lie above l will immediately generate a conflict in the solver: \( reason_{ save }(l) \) will be falsified. In fact, in this situation we would not even need to perform unit propagation over the literals added from \(\mathcal {T}_{ save }\); the literals and their reasons obtained from \(\mathcal {T}_{ save }\) would be sufficient to perform 1UIP learning from \( reason_{ save }(l) \).
We experimented with this “lookahead for conflicts” idea using various values of k. We found that \(k=2\), i.e., forcing up to two decisions from \(\mathcal {T}_{ save }\) to be made by the solver if this yields a conflict, often enhanced the solver’s performance. Limiting the lookahead to only one decision level of \(\mathcal {T}_{ save }\) was not as good, and looking ahead more than 2 decisions of \(\mathcal {T}_{ save }\) also degraded performance. This provides some evidence that taking too much control away from the solver and forcing it to make too many decisions from \(\mathcal {T}_{ save }\) can lead to conflicts that are not as useful to the solver.
Reason Quality. The saved trail can be thought of as remembering the solver’s recent trajectory. Sometimes we want to follow the past trajectory, but perhaps sometimes we do not. In particular, when adding literals from \(\mathcal {T}_{ save }{}\) to \(\mathcal {T}\) we can examine the quality of the saved reasons to see if they are worth using. Once we encounter a literal with a low quality saved reason we stop adding literals from \(\mathcal {T}_{ save }\) to the solver’s trail. In particular, we can change lines 24–25 of useSavedTrail to the following:
Note that the solver will still set the unadded literals as they are unit implied by \(\mathcal {T}\), but it might be able to find better reasons for these implicants. There is of course no guarantee that better reasons will be found, but our empirical results show that sometimes this does happen. We experimented with two quality metrics, clause size and clause LBD, obtaining positive results with both. These results also provides evidence against the argument given in [14] that changing literal reasons is not impactful. With an appropriate clause quality metric the changing of literal reasons can have an impact.
5 Experiments and Results
We implemented our techniques in two different SAT solvers, MapleSAT and Cadical,^{Footnote 3} both of which have finished at or near the top of SAT competitions for the past several years [3, 4]. We then ran each solver on the 800 total benchmark instances used in the main tracks of the 2018 SAT Competition and 2019 SAT Race. The experiments were executed on a cluster of 2.7 GHz Intel cores with 5000 s CPU time and 7 GB memory limits for each instance. We chose not to output or verify the proofs generated by any of the solvers. The Par2 scores obtained and total instances solved by each solver are reported in Figs. 3, 5, and 6. We also show the cactus plot of the new version of cadical in Fig. 4.
In Fig. 3 we used the newest version of cadical (downloaded as of January 1, 2020) as the baseline solver, in Fig. 5 we used the version of cadical published in [8] as the baseline, and in Fig. 6 we used MapleLCMDist [7, 15] as the baseline. Each of the baselines were run with standard nonchronological backtracking. We then refer to versions of each solver with additional features implemented on top by adding suffixes. “chrono” refers to the solver with Cbt enabled (using the solver’s default settings), “trail” refers to the baseline with plain trail saving added (as described in Fig. 1), “trailmultipleBT” refers to the baseline with trail saving plus the first enhancement of saving over multiple backtracks, “trailmultipleBTlookahead” also adds the enhancement of lookahead for conflicts by 2 decision levels, and “trailmultipleBTlookaheadreason” also adds the final enhancement to cease trail saving once a reason of “low quality” is reached. For more details on the enhancements, please see Sect. 4.2.
Interestingly, Cbt made the newest version of cadical perform worse than the baseline (Fig. 3). This demonstrates that Cbt is not always beneficial. Trail saving alone did not impact the performance of this solver significantly, but adding all of the enhancements on top of trail saving resulted in solving six more instances and yielding a better Par2 score than the baseline. The key enhancement for this solver seemed to be the last one where we stop using the saved trail once we detect a reason of “low quality”. We tried both clause size and lbd as the clause quality metric, and both yielded a positive gain, with clause size being slightly more effective.
The version of cadical used in Fig. 5 did show benefits from Cbt in agreement with previously published results [8]. Trail saving alone did not significantly impact this solver, but adding all of the enhancements on top of trail saving resulted in solving the same number of instances as the solver with Cbt did, albeit with a slight increase in the Par2 score.
MapleLCMDist (in Fig. 6) is another solver that benefited from Cbt. In this solver trail saving alone solved two more instances than the solver with Cbt did. Adding the first two enhancements on top of trail saving resulted in solving only one more instance but yielded a better Par2 score than the solver with Cbt. Adding the last enhancement of ceasing trail saving on a “low quality” reason made the performance worse, whether clause size or lbd was used as the clause quality metric. This suggests that the enhancements to trail saving have different impacts on different solvers.
6 Conclusion
We have shown that our trail saving technique can speed up two stateoftheart SAT solvers, cadical and MapleSAT, as or more effectively than chronological backtracking can. We also introduced three enhancements one can implement when using a saved trail and demonstrated experimentally that these enhancements can sometimes improve a solver’s performance by a significant amount. We have shown that trail saving and all enhancements we proposed are sound.
There are many avenues that can be pursued in future work, such as using the saved trail to help make inprocessing techniques faster or using the saved trail to learn multiple clauses from a single conflict. It is also possible to combine trail saving with chronological backtracking, but it would require further work to determine whether or not this would be useful and how to best approach it.
Notes
 1.
 2.
Cbt can also be extremely useful in contexts where each descent can be very expensive, e.g., when doing theory propagation in SMT solving, or component analysis in #SAT solving. In these cases, Cbt, by avoiding backtracking and subsequent redescent, has considerable potential for improving solver performance.
 3.
Our implementation is available at https://github.com/rgh000/cadicaltrail.
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Hickey, R., Bacchus, F. (2020). Trail Saving on Backtrack. In: Pulina, L., Seidl, M. (eds) Theory and Applications of Satisfiability Testing – SAT 2020. SAT 2020. Lecture Notes in Computer Science(), vol 12178. Springer, Cham. https://doi.org/10.1007/9783030518257_4
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