Deductive Priority Ordering in StålmarckSAT

• Deductive Priority Ordering (DPO) is a heuristic that orders implication triplets in the Stålmarck Procedure by assigning each a deductive potential score based on variable frequency and rule-specific bonuses.
• The scoring mechanism combines a base frequency derived from variable occurrences with a bonus multiplier for structural rules, prioritizing triplets likely to trigger further deductions.
• Static pre-sorting using DPO improves solver efficiency, and when combined with Cardinality Driven Branching, it leads to faster solve times and higher instance success rates on benchmark tests.

Searching arXiv for the primary SAT-solving paper and for acronym-disambiguation papers on “DPO”. Deductive Priority Ordering (DPO) is a heuristic for ordering the application of simple rules in the Stålmarck Procedure, introduced in the Rust-based solver StalmarckSAT as one of two optimizations alongside Cardinality Driven Branching (CDB). In this setting, DPO does not alter the logical semantics of deduction; it assigns each implication triplet a deductive potential score, sorts triplets in descending order of that score, and applies simple rules to the highest-scoring triplets first. Its purpose is to prioritize rule applications most likely to trigger further deductions and thereby improve solve time in a deduction-based SAT-validity workflow (Leonov et al., 19 Sep 2025).

1. Position within the Stålmarck Procedure

The Stålmarck Procedure, as used here, is a SAT solver in the form of a validity checker for propositional formulas, relying on a normalized implication-based representation rather than direct CNF reasoning. A propositional formula is transformed in linear time using the rules

$\begin{enumerate} \item A \lor B \quad \Rightarrow \quad \neg A \rightarrow B \item A \land B \quad \Rightarrow \quad \neg (A \rightarrow \neg B) \item \neg \neg A \quad \Rightarrow \quad A \item \neg A \quad \Rightarrow \quad A \rightarrow \text{False} \end{enumerate}$

and then encoded as implication triplets of the form

$$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$

Complex formulas are decomposed using bridge variables; for example,

$(p \land \neg q) \lor r$

is encoded as

$\{(b_1, q, r), (b_2, p, b_1)\}.$

Deduction proceeds via seven simple rules over triplets $(x,y,z)$: $\begin{array}{rlrlrl} (1)\; & \cfrac{(0, y, z)}{\substack{y / 1 \quad z / 0}} & (2)\; & \cfrac{(x, y, 1)}{\substack{x / 1}} & (3)\; & \cfrac{(x, 0, z)}{\substack{x / 1}} \[1.2ex] (4)\; & \cfrac{(x, 1, z)}{\substack{x / z}} & (5)\; & \cfrac{(x, y, 0)}{\substack{x / \neg y}} & (6)\; & \cfrac{(x, x, z)}{\substack{x / 1 \quad z / 1}} \[1.2ex] (7)\; & \cfrac{(x, y, y)}{\substack{x / 1}} & & & & \end{array}$ These rules are sound local deductions derived from the implication truth table. However, they are incomplete; when no simple rule applies, the procedure uses the dilemma rule to branch on a literal, explore both truth assignments, and merge outcomes according to Stålmarck semantics. DPO operates strictly within the simple-rule phase: it changes the order in which candidate triplets are considered, not the rules themselves, not the branching semantics, and not the representation (Leonov et al., 19 Sep 2025).

This placement is central. DPO is neither a new inference rule nor a replacement for the dilemma rule. It is a propagation-order heuristic inside a proof system whose deduction phase already has a fixed rule set. A plausible implication is that DPO should be understood as an execution-policy refinement for a deduction engine, analogous in role to propagation or scheduling heuristics in other solver architectures, but specialized to implication triplets and the Stålmarck proof system.

2. Scoring triplets by deductive potential

DPO assigns a numeric score to each triplet before the main solving loop. The score combines two ingredients: a base frequency score derived from variable cardinalities and a deductive bonus determined by triplet structure. The paper’s core description is that DPO “intelligently selects the application order of simple rules” by prioritizing those “most likely to trigger further deductions” (Leonov et al., 19 Sep 2025).

Let $freq(v)$ denote the frequency, or cardinality, of variable $v$ in the triplet set. For a triplet $(x,y,z)$, the base score is

$\text{base}(x,y,z) = freq(x) + freq(y) + freq(z).$

The deductive bonus multiplier is

$$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$0

and the DPO score is

$$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$1

The structural asymmetry in the multiplier is deliberate. Triplets of the form $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$2, corresponding to rule (6), deduce two assignments, $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$3 and $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$4, and therefore receive multiplier $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$5. Triplets of the form $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$6, corresponding to rule (7), deduce one assignment, $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$7, and receive multiplier $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$8. All other triplets receive no bonus beyond the base score. The heuristic therefore privileges two kinds of triplets simultaneously: those involving frequently occurring variables, and those whose associated simple rule has higher immediate deductive power (Leonov et al., 19 Sep 2025).

This scoring rule is intentionally coarse. It does not attempt to estimate future branching reduction, clause-learning analogues, or dynamic activation probability. Instead, it encodes a compact proxy for “deductive potential.” The underlying rationale is explicit: assignments on high-frequency variables should unlock more deductions across the triplet set, while triplets matching rule (6) or rule (7) should increase the number of known assignments quickly. This suggests a two-level heuristic bias: structural strength at the triplet level and connectivity strength at the variable level.

3. Static pre-sorting and solver integration

After scores are computed, the triplets are sorted in descending order and scanned in that fixed order during deduction. The mechanism is:

1. compute $$x \leftrightarrow y \rightarrow z \quad \text{represented as} \quad (x,y,z).$$9 for each triplet $(p \land \neg q) \lor r$0;
2. sort triplets in descending order of score;
3. during the solving loop, repeatedly scan the sorted list and apply any simple rule whose preconditions are satisfied.

In StalmarckSAT, DPO appears after normalization and after CDB preprocessing. The implementation pipeline is:

1. input CNF formula;
2. convert it to Stålmarck-style implications and encode as triplets with bridge variables;
3. perform CDB preprocessing to build a frequency map $(p \land \neg q) \lor r$1 and a descending-cardinality variable list;
4. compute DPO scores and sort triplets;
5. enter the main solving loop, applying simple rules in sorted triplet order until no rule applies, then using the dilemma rule if required (Leonov et al., 19 Sep 2025).

The paper emphasizes that DPO is static. Scores are computed once before solving begins and are not recomputed as assignments evolve. The same triplet order is reused recursively after dilemma branching. This is an important implementation property because it limits overhead to one frequency pass, one scoring pass, and one sort over the triplet list. The solver therefore pays the cost of prioritization once per instance rather than maintaining a dynamic priority queue.

The significance of this design is pragmatic rather than semantic. DPO preserves correctness because it changes only the order of rule consideration. It does not skip any rule, weaken any precondition, or introduce any new inference. A plausible implication is that DPO can be inserted into an existing Stålmarck-style implementation with relatively low engineering cost, provided the solver already materializes triplets and can compute variable frequencies.

4. Relationship to baseline rule application and to CDB

Without DPO, the baseline procedure repeatedly scans the triplet set and applies any simple rule whose preconditions match the current partial assignment. No special ordering of triplets is used beyond raw generation order or some fixed scheme. DPO replaces that neutral scan order with a score-driven priority order. Conceptually, the contrast is “apply simple rules as you encounter them” versus “apply the most promising simple rules first,” where “promising” means high variable frequency and strong deductive pattern (Leonov et al., 19 Sep 2025).

The second optimization in StalmarckSAT, Cardinality Driven Branching, affects a different control point. CDB is a branching heuristic for the dilemma rule: it prioritizes branching on variables appearing with greater frequency within the normalized triplet representation, “mirroring VSIDS heuristics in CDCL solvers.” Its preprocessing step computes the same $(p \land \neg q) \lor r$2 values later used by DPO. When branching is needed, CDB selects the unassigned original variable with highest cardinality, falling back to bridge variables only if all original variables are assigned (Leonov et al., 19 Sep 2025).

The division of labor is therefore precise. CDB affects which literal is branched on when simple deduction reaches a fixpoint. DPO affects which triplet is examined first during simple-rule propagation. They are logically independent but operationally synergistic because they share the same frequency map. One heuristic steers propagation toward high-impact triplets; the other steers search toward high-cardinality branching variables. The paper compares four configurations—baseline, DPO only, CDB only, and DPO + CDB—and reports that the combined system performs best (Leonov et al., 19 Sep 2025).

This interaction matters because it situates DPO within a broader solver-control architecture rather than as an isolated propagation tweak. A plausible implication is that DPO benefits most when the solver’s branching behavior is also informed by the same triplet-derived structural information.

5. Empirical performance

The evaluation uses 1000 KSAT benchmarks, each with 50 variables, 218 clauses, and clause length 3. The environment is a server with 2.6 GHz AMD CPUs, 4 CPU cores allocated per benchmark, a 30-minute time limit per benchmark, and a 32 GB memory limit. StalmarckSAT is tested in four modes: baseline, DPO only, CDB only, and DPO + CDB. The reported metrics are the number of instances solved within the time limit and the average solve time on solved instances (Leonov et al., 19 Sep 2025).

Configuration	Solved Instances	Average Solve Time (seconds)
Baseline	503	650.37
DPO only	612	557.63
CDB only	797	376.68
DPO + CDB	864	290.56

These results establish three points. First, DPO alone improves both solved-instance count and average solve time relative to the baseline: from 503 to 612 solved instances and from 650.37 to 557.63 seconds. Second, CDB has a larger standalone effect than DPO in this benchmark suite. Third, DPO remains additive in the presence of CDB: combining both raises solved instances from 797 to 864 and lowers average solve time from 376.68 to 290.56 (Leonov et al., 19 Sep 2025).

The paper also notes cactus plots showing that both strategies “lead to an increase in solved instances in less time, with combining the two strategies yielding the best results.” No statistical significance tests or variance estimates are reported, so the evidence is comparative rather than inferential. Even so, within the reported setup, DPO is empirically useful both as an independent heuristic and as part of a combined propagation-plus-branching strategy.

6. Limitations, theoretical status, and terminological ambiguity

The principal limitation identified for DPO is its static nature. Deductive potential is evaluated once before solving, and scores are never updated as assignments change or as triplets become more or less relevant. The authors explicitly state: “we plan to explore making DPO dynamic by continually re-evaluating deductive potential instead of using static pre-scoring” (Leonov et al., 19 Sep 2025). This suggests an unresolved trade-off between responsiveness and overhead. Static scoring is lightweight, but it may become suboptimal in later solving stages when the active deduction frontier has shifted.

The paper does not provide formal complexity bounds or proofs that DPO reduces the number of deduction steps. Nor does it present explicit edge-case analyses showing when DPO may hurt performance. Its theoretical status is therefore that of a semantics-preserving ordering heuristic with empirical support rather than a proved improvement. Because DPO changes only ordering, not inference content, correctness and completeness of the underlying Stålmarck Procedure are preserved as long as the base procedure is sound and complete. What remains heuristic is only the operational claim that one fixed ordering is better than another.

A separate source of confusion is terminological. The acronym “DPO” is already heavily overloaded in several research areas. In the RLHF and LLM-alignment literature, “DPO” ordinarily denotes Direct Preference Optimization rather than Deductive Priority Ordering; one paper states that “in the RLHF/LLM context, DPO uniformly means Direct Preference Optimization” (Su et al., 5 Feb 2025). Related work includes GraphDPO, a graph-structured generalization of pairwise Direct Preference Optimization (Liu et al., 8 May 2026). Outside SAT solving, priority orderings also appear in default logic as strict orders over default rules, including Baader and Hollunder, Brewka, and lexicographic prioritized default logics (Rintanen, 2011). None of these uses should be conflated with the StålmarckSAT heuristic.

In the specific sense established by StalmarckSAT, Deductive Priority Ordering names a triplet-ordering strategy for simple-rule propagation in implication-triplet proof search. Its importance lies in being concrete, easily implementable, and empirically beneficial without modifying the proof system’s logic. More broadly, it represents an attempt to modernize the Stålmarck Procedure by importing heuristic control ideas—frequency-sensitive prioritization and structurally informed scheduling—into a solver architecture centered on implication triplets rather than clauses (Leonov et al., 19 Sep 2025).