Towards a Certifying Grounder

Tractability tests are often computed from input syntax: support-graph treewidth, local coefficient patterns, backdoor tests, or action-count bounds. Before such a test can be lower-bounded or made algorithmic, it must define a predicate on the exact-certification problem itself. Equivalent presentations must receive the same verdict. The semantic object is the correctness quotient, whose classes are states with the same correct outputs. Correctness-preserving presentation moves generate closure orbits. A target that changes inside one closure orbit has an orbit gap and fails descent. Exact closure-invariant classification is possible exactly when the positive and negative orbit hulls are disjoint; the positive hull is then the least exact classifier, and computable orbit representatives make the classifier algorithmic. The results separate three layers. The descent layer gives orbit-gap obstructions for raw local syntax, raw action and coordinate counts, and raw support-graph predicates. The post-descent complexity layer applies ordinary reductions to descended objects: graph-predicate lower bounds transfer through action-gap graph extraction, and Action-Gap-Treewidth is NP-complete when the width bound is part of the input. The certification layer asks whether a proxy descends: for split proxies $b\wedge\varphi(z)$, SAT reduces to non-descent and UNSAT reduces to descent. Positive regimes use quotient-preserving normalizations or catalogues before model checking; bounded quotient size, bounded full Gaifman treewidth of the constructed quotient, sparse unary-gap certificates, and strict-margin perturbation balls give explicit cost bounds after quotient construction.

GK is a query-directed first-order prover that extends ordinary resolution-based proof search with explicit positive and negative claims, numerical confidence values, and prioritized default rules with exceptions. It works directly with non-ground clauses, including equality and function terms. Candidate proofs are found by bounded first-order proof search; exception conditions of defaults are checked by further bounded searches, recursively when exceptions themselves depend on defaults. This avoids requiring a finite global grounding, while allowing incomplete searches to be reported as such. This paper adds structure-preserving quantitative reporting to that framework. Retained proof histories are used in two calculations. The first reconstructs the uncertain ground premises used by each proof and computes the probability that at least one retained proof is available, without counting shared premises independently. The second resolves positive and negative support at intermediate atoms before that support is propagated through later rules; the same calculation evaluates uncertain exception conditions for individual rule applications. Reports separate positive support, negative support, conflict, and ignorance and identify detected incomplete calculations or fallbacks. The implementation performs bounded reconstruction and dependency traversal after proof search and still requires no global grounding. Analytic examples and independent simulators reproduce the reference calculations on their stated fragments. Comparisons with probabilistic logic, probabilistic ASP, default logic, and goal-directed ASP identify cases of agreement, semantic difference, unsupported translation, and incomplete computation.

Constraint programming is a core technology for solving complex combinatorial problems in scheduling, planning, configuration, and verification. Trusting its results therefore demands guarantees at two levels: that reformulations applied beforehand are semantics-preserving, and that solvers produce correct answers. In this work, we introduce a framework that addresses both verification levels in the Lean theorem prover: it can be used to prove formulation-level properties, such as equivalence, equisatisfiability, and the correctness of symmetry-breaking constraints, parametrically for entire problem families; and to check solver-produced certificates for individual instances via translation backends to external formats such as MiniZinc, SMT-LIB, and OPB. Combining both levels yields an end-to-end workflow that establishes the satisfiability or unsatisfiability of a constraint problem without trusting the external solver. Experimental results show that our framework's verified symmetry breaking also pays off in practice: a single parametric proof per problem family, reused across all instance sizes, reduces solver search effort by a factor of up to 2x10^7, while the entire in-Lean certification stays affordable, taking at most a few minutes for our largest instances.