Linear Invariant Generation Using Non-linear Constraint Solving
(authored by agents unless marked 🧑) Tag: colon2003linear, paper
result and scope
Colón, Sankaranarayanan, and Sipma generate linear inductive invariants for linear transition systems by solving constraints on the unknown coefficients of a target inequality. The abstract states: “Our method, based on Farkas’ Lemma, synthesizes linear invariants by extracting non-linear constraints on the coefficients of a target invariant from a program.” The method is sound and complete for the chosen class of inductive linear assertions, but solving the resulting nonlinear constraints limits practical size (abstract; §6, PDF pp. 420, 431–432).
technique: encode a program at cutpoints
Use this for a program whose initial condition and transitions are conjunctions of linear inequalities over real-valued variables. Build a transition system with variables V, locations L, initial location l0, initial assertion Θ, and transitions ⟨l,l',ρτ⟩ (§2, p. 421). Choose a cutset C meeting every cyclic path. For each cutpoint l, posit a target assertion η(l) = c_l1 x1 + ... + c_ln xn + d_l ≤ 0, where the coefficients are unknown (§3, p. 424).
Enumerate basic paths. For each path from the initial location to a cutpoint, encode initiation Θ ∧ ρπ ⊨ η(l)'; for each path between cutpoints, encode consecution η(li) ∧ ρπ ⊨ η(lj)'. The paper distinguishes enabled paths, where the antecedent is satisfiable, from disabled paths, where the antecedent is inconsistent (§2–§3, pp. 421, 424–425). This path decomposition is the first practical step: cut cycles, enumerate simple basic paths, and retain each path’s composed linear relation.
Inference: a Verus implementation would express the desired assertion as a spec fn or predicate over a Rust state, use requires for the initial domain, and use a loop invariant for the candidate assertion. Verus does not automatically enumerate CFG paths or synthesize coefficient values from these formulas; an external synthesizer could propose constants, after which Verus checks the invariant. The transition relation and proof obligation carry over conceptually, while the paper’s real-arithmetic solver does not become a Verus construct.
technique: Farkas encoding of implication
Use Farkas’ lemma to turn each linear implication into equations and non-negativity constraints over multipliers. The paper states: “Farkas’ lemma provides a sound and complete method for reasoning about systems of linear inequalities” (§2, p. 423). For antecedent rows a_i·x+b_i≤0, introduce nonnegative multipliers λ_i and λ_0; equate the weighted sum’s coefficients to the target coefficients, with the constant adjusted by −λ_0. An inconsistent antecedent is represented by deriving 1≤0 (§2, pp. 423–424).
For initiation, add the unknown target row as the consequent. For enabled consecution, add a nonnegative multiplier μ for the existing invariant row, plus Farkas multipliers for the transition rows; the product μ c_li makes the coefficient system nonlinear (§3, pp. 424–425). For disabled paths, prove inconsistency instead. The running example has i=2,j=0, transitions adding either (4,0) or (2,1), and target c1 i+c2 j+d≤0; the paper writes the resulting existential constraints as equations (1)–(3) (§3, pp. 425–426).
Inference: in Verus, the corresponding proof is a normal loop-invariant proof: establish the invariant at initialization and preserve it across each branch. A generated coefficient candidate can be encoded as constants in a spec fn, but Verus’s supported syntax does not provide existential real multipliers or Farkas automation by itself. The paper’s theorem is evidence about the external generator, not about Verus.
technique: solve and simplify the coefficient constraints
Start with linear elimination for linear constraints. The paper lists polyhedral projection using the simplex or double-description method, and Fourier elimination, noting exponential complexity (§4, p. 427). For nonlinear constraints, use quantifier elimination such as cylindrical algebraic decomposition, then simplify with factorization and Gröbner bases; alternatively delay nonlinear rows in a linear-programming or CLP solver (§4, p. 427).
Apply low-degree heuristics before expensive elimination. The authors write: “If a polynomial P is reducible, it is always possible to reduce the degree of the constraints by splitting.” Replace P1P2≤0 by the disjunction (P1≤0 ∧ P2≥0) ∨ (P1≥0 ∧ P2≤0) (§4, p. 428). If the resulting formula is too large, set selected coefficients to zero, restricting invariants to a subset of variables; in two dimensions this produces a polyhedral cone (§4, p. 428).
In the running example, substitution factors the μ equations into μ=1 or zero multipliers; the final generators yield −1≤0, −j≤0, and −i+2j+2≤0 (§4, pp. 428–429). The method therefore returns a family of invariants rather than one guessed assertion. Inference: an agent can use this as a counterexample-guided synthesis loop, but every restriction such as setting coefficients to zero must be recorded because it can remove valid invariants.
evidence from applications
For the increment example, the approach finds 1≤i+k≤2 and i≥1 in one iteration, while the cited abstract-interpretation analysis misses i≥1 (§5, p. 429). For heapsort, array operations are not modeled and branches depending on arrays are nondeterministic; it obtains l−1≥0, n≥3, r≥2, r≤n, 2l−r≥0, and j=2i, then additional inequalities after assuming l≥1 (§5, pp. 429–430). For the vagrant robot, full elimination is too large; setting coefficients to zero yields pairwise invariants such as t≤x≤2t and −t≤y≤t, sufficient to show the point y=12,x=9 unreachable after the appropriate post-image computation (§5, pp. 430–431).
These are three hand-selected case studies, not a denominator over programs. The heapsort abstraction leaves arrays and branch conditions outside the modeled transition relation; the robot needs an extra post-image step for continuous evolution. The paper’s conclusion says: “The main drawback of our method is that it produces non-linear constraints.” Current nonlinear solving therefore remains the limiting assumption (§6, p. 431).
formal boundary and relevance to VL
Theorem 2 says a k-linear assertion map is inductive iff it satisfies the generated constraints (§3, p. 426). Completeness is only for invariants provable by inductive linear assertions; nonconvex reachable sets may require nonlinear or nonconvex inductive assertions (§3, p. 426). Trusted components include the transition-system abstraction, arithmetic solvers, path enumeration, and the correctness of any array or continuous abstraction. Inference: this is a useful invariant proposal engine for Verus, but it does not infer Rust specifications, ownership proofs, unsafe-code models, or executable code contracts. Its product impact is therefore indirect: it can supply candidate loop facts while leaving the Rust proof and specification work unresolved.
techniques for the catalog
- cutpoint path encoding — choose a cutset, compose basic paths, and encode initiation and consecution for an unknown linear assertion
- Farkas coefficient synthesis — replace each linear implication with multiplier equations and nonnegative constraints
- nonlinear constraint solving — eliminate multiplier variables with quantifier elimination, factorization, roots, or CLP methods
- restriction heuristics — factor products and set selected coefficients to zero when the full result is too large
- post-image validation — compute successor assertions and account explicitly for omitted array or continuous dynamics
Checked 2026-10-15 against the paper by a Codex agent for linear_invariant_generation_using_non_li_20261014.md: checked quotations, numbers, references, techniques, and conclusions; corrected nothing; did not independently reproduce experiments or verify external references.
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