Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning

Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is proved that DLL proof search algorithms that use clause learning based on unit propagation can be polynomially simulated by regular WRTI. More generally, non-greedy DLL algorithms with learning by unit propagation are equivalent to regular WRTI. A general form of clause learning, called DLL-Learn, is defined that is equivalent to regular WRTL. A variable extension method is used to give simulations of resolution by regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and non-greedy DLL algorithms with learning by unit propagation can use variable extensions to simulate general resolution without doing restarts. Finally, an exponential lower bound for WRTL where the lemmas are restricted to short clauses is shown.

💡 Research Summary

The paper introduces two refined resolution proof systems that incorporate lemmas and a weakened resolution rule, called w‑resolution. The systems are named WR‑TL (resolution trees with lemmas) and WR‑TI (resolution trees with input lemmas). In WR‑TL each derived clause may be reused as a leaf (a lemma) anywhere in the tree, while WR‑TI restricts lemmas to clauses that appear as the root of an input sub‑proof, mirroring the effect of unit‑propagation (also called input resolution).

The authors first show that, when no regularity condition is imposed, both WR‑TL and WR‑TI are polynomially equivalent to ordinary dag‑resolution. This establishes that the lemma mechanism can replace the sharing of dag‑resolution without a blow‑up in size.

When the regularity condition (no variable is resolved more than once on any path) is added, a striking separation appears: regular dag‑resolution cannot polynomially simulate regular WR‑TL or regular WR‑TI. The paper constructs families of formulas for which any regular dag‑resolution refutation must be exponentially larger than the corresponding regular WR‑TL/WR‑TI refutation. The separation stems from the fact that lemmas allow a clause derived once to be reused on many branches, something regular dag‑resolution cannot do because of the variable‑reuse restriction.

Next, the paper connects these proof systems to modern SAT solvers based on the DLL (Davis‑Logemann‑Loveland) backtracking algorithm with clause learning. A general framework called DLL‑L‑UP is defined, capturing all learning strategies that rely on unit propagation (including first‑UIP, RelSat, and cut‑based clauses). The authors prove that for any such learning strategy a DLL search tree can be transformed into a regular WR‑TI proof with only a polynomial increase in size. Conversely, any regular WR‑TI proof can be simulated by a “non‑greedy” DLL algorithm that continues decision branching even after unit propagation has already produced a contradiction. This establishes an exact correspondence between the space of regular WR‑TI proofs and the behavior of clause‑learning DLL solvers.

A more general learning model, DLL‑Learn, is then introduced. DLL‑Learn is shown to be polynomially equivalent to regular WR‑TL. The key difference from WR‑TL is the use of w‑resolution instead of the ordinary resolution rule; w‑resolution incorporates a restricted weakening step, allowing the inference of a clause even when the resolved variable does not appear in both premises. This mirrors the “pool resolution” system of Van Gelder, but replaces the degenerate inference with w‑resolution. The equivalence implies that any proof generated by DLL‑Learn can be captured by a regular WR‑TL proof and vice‑versa.

The paper also addresses the simulation of full (non‑regular) resolution without the need for restarts. By employing a variable‑extension technique—adding fresh variables together with simple clauses that do not affect satisfiability—the authors show that regular WR‑TI can p‑simulate arbitrary resolution proofs. Unlike earlier proof‑trace extensions, the number of new variables depends only on the original variable count, not on the unknown size of the resolution refutation, making the construction more practical.

Finally, the authors investigate the impact of limiting the size of lemmas. They prove an exponential lower bound for WR‑TL when lemmas are restricted to clauses shorter than n/2 on the pigeon‑hole principle formulas PHPₙ. The bound matches the known lower bound for ordinary tree‑resolution, showing that short lemmas do not help. In contrast, allowing longer lemmas yields regular resolution refutations of size 2^{Ω(n)} for PHPₙ, which can be simulated by DLL‑Learn. This demonstrates that the ability to learn large clauses can give clause‑learning DLL algorithms a super‑polynomial speed‑up over algorithms restricted to short clauses.

Overall, the paper provides a comprehensive theoretical bridge between clause‑learning SAT solvers and refined resolution proof systems. By introducing w‑resolution and lemma‑based trees, it clarifies which proof families are accessible to DLL algorithms, quantifies the power gained by learning larger clauses, and offers new simulation techniques (variable extensions) that avoid restarts. These results deepen our understanding of SAT solver complexity and suggest directions for designing more powerful learning strategies.