Forward & Backward Constraints Generation

• Forward and backward constraints generation is a dual approach that expands reachable states while restricting solutions to goal-relevant conditions.
• It spans multiple domains—from optimization and program analysis to text generation and CAD—by coupling forward approximations with backward resolvents or restrictions.
• The framework improves safety proofs, inverse constraint learning, and operator splitting by integrating precise forward propagation with controlled backward analysis.

Across several research literatures, forward and backward constraints generation denotes a family of methods in which constraints are produced, propagated, or enforced from initial conditions, from goals or bad states, or from both directions simultaneously. In optimization, the pattern appears as forward use of approximate operator values and backward use of resolvents or proximal maps; in program analysis, it appears as alternation between reachable-state approximations and goal-relevant predecessor approximations; in generative modeling, it appears as sentence or plan construction around a fixed anchor or goal-derived subgoal; and in control and safety, it appears as the distinction between forward failure semantics and backward reachable or reflected constraint sets. The literature therefore suggests an umbrella usage rather than a single canonical formalism (Bakhirkin et al., 2017, Mou et al., 2015, Garrett et al., 2016).

1. General structure and domain-specific meanings

A recurrent formal pattern is the decomposition of a problem into a forward component that expands feasible or reachable structure and a backward component that restricts attention to states, symbols, or actions relevant to a target condition. In some settings the two directions are alternated until a fixed point or refinement sequence is obtained; in others they are coupled around a pivot, such as a lexical constraint or a goal constraint. This suggests that the central issue is not directionality alone, but the generation of constraints that are informative under one-sided reasoning only in part.

Domain	Forward role	Backward role
Horn clauses	reachable atoms via post	goal-relevant predecessors via pre
Constrained text generation	generate suffix/right context	generate prefix/left context
Manipulation planning	executable forward search	goal-driven constraint identification
Parametric CAD	generate constraint sequence from geometry	not directly addressed
Safe learning/control	failure-set semantics	backward reachable or reflected unsafe set
Monotone operator splitting	forward approximate operator term	backward resolvent/projection step

The technical meaning of “constraint” varies accordingly. In constrained Horn clauses, constraints are predicates over atoms and transition relations; in lexical generation, they are required words or fragments; in CAD, they are geometric constraints and dimensions over primitives; in manipulation planning, they are condition constraints of actions and goal constraints over hybrid state variables; and in stochastic control, they may be terminal-state, initial-terminal, or law constraints (Bakhirkin et al., 2017, Liu et al., 2018, Casey et al., 17 Apr 2025, Garrett et al., 2016, Briand et al., 2019).

2. Abstract interpretation, reachability, and safety proofs

In Horn-clause analysis, forward and backward constraints generation is formalized through collecting semantics. The forward operator is

$\post{\TRel}{X} = \{ a' \mid \exists A \subseteq X.\,(A,a') \in \TRel \},$

and its least fixed point gives the forward collecting semantics. The backward side uses

$\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$

but the simple backward semantics is insufficient when merely intersected with forward reachability. The key refinement is the restricted pre-condition

${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$

which yields a combined forward-backward semantics in which backward propagation is explicitly restricted to the forward-reachable region $M$. The abstract iteration

$b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$

alternates forward approximations $d_i$ and backward approximations $b_i$, and each run refines the model of the original system. If some $d_k = \Dbot$, the goal is unreachable (Bakhirkin et al., 2017).

A related but distinct line is constraint-based reachability for iterative imperative programs. Instead of explicit forward or backward path exploration, the whole program is encoded as a constraint system whose solutions correspond to reachable behaviors. Exact filtering for constraints $c_1,\dots,c_m$ is modeled by operators $f_i(S)=\{s\mid s\in S\land c_i(s)\}$, and the loop operator $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$0 integrates forward clauses, backward clauses, and a join over suspended alternatives. The paper explicitly states that the first two guarded constraints implement forward analysis and the next two backward reasoning. Polyhedral abstraction is then used to summarize loop relations that are impractical for path-by-path unfolding, such as programs requiring hundreds of loop iterations before a target branch becomes reachable (Gotlieb et al., 2013).

In incremental safety proofs, forward and backward constraints are treated as proof obligations rather than only semantic operators. A safety problem is $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$1. The forward rules $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$2, $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$3, and $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$4 build an incremental proof from inductive invariants over reachable states. Backward reasoning is introduced by time reversal,

$\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$5

together with $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$6, $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$7, $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$8, and $\pre{\TRel}{A'} = \{ a \mid \exists A \subseteq A.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$9. In this framework, a backward-inductive invariant need not hold on all reachable states; it characterizes states on traces from bad states backwards. Prophecy extends the system further by adding witnesses for existentially quantified properties through a soundness safety problem and a witness-refined system. The paper proves that forward-backward steps can remove complex Boolean structure while prophecy eliminates quantifiers and quantifier alternations, and its case studies on Paxos, its variants, and Raft show that the intermediate formulas can be simpler than the final recovered safe inductive invariant (Frenkel et al., 16 Apr 2026).

A common misconception is that backward reasoning in verification is merely inverse execution. The Horn-clause and incremental-proof literatures instead treat it as a controlled approximation of goal-relevant or error-trace-relevant states, often with explicit restrictions to forward-reachable regions or witness-augmented traces (Bakhirkin et al., 2017, Frenkel et al., 16 Apr 2026).

3. Bidirectional generation in language and manipulation planning

In constrained sentence generation, the central problem is to generate a sentence containing a required lexical item at an arbitrary position. A standard left-to-right LLM

${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$0

does not guarantee the appearance of a desired word. The backward-and-forward language-modeling approach instead chooses a split word ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$1 and generates a backward sequence ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$2 and a forward sequence ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$3. The paper proposes synchronous and asynchronous variants; the asynchronous model first generates the left half and then the right half conditioned on the generated prefix. Reported perplexities show that asyn-B/F performs best among the backward/forward variants, and the model guarantees inclusion of the required word because generation starts from that word rather than hoping to insert it later (Mou et al., 2015).

BFGAN generalizes the same structural idea to lexically constrained sentence generation with adversarial training. A backward generator ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$4 produces the left half in reverse order,

${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$5

while a forward generator ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$6 generates the right half conditioned on the generated prefix. The framework adds a CNN-based discriminator ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$7 that provides the reward ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$8, with policy-gradient training, curriculum learning over constraint difficulty, mixed cross-entropy and policy-gradient updates, teacher forcing after each policy update, and beam-search-based reward estimation rather than Monte Carlo rollouts. On the CSMC human evaluation, BFGAN beat BF-MLE in 76% of comparisons and grid beam search in 82%; against human-written samples, 56% were ties and 6% of BFGAN outputs were judged better than human (Liu et al., 2018).

In manipulation planning, the hybrid backward-forward (HBF) algorithm uses backward identification of constraints to guide a forward search over an infinite action space in a hybrid configuration space. The backward component builds a reachability graph from goal constraints ${}^{R}\TRel A' = \{ a \mid \exists A \subseteq R.\,\exists a' \in A'.\ (A, a') \in \TRel \land a \in A \},$9, repeatedly asking which actions could produce a needed constraint and which condition constraints those actions require. Procedures such as s-intersection, s-transition, and s-roadmap generate witness states and action edges, and every newly discovered action condition is pushed back onto the agenda. The forward side is a persistent enforced hill-climbing search that samples only those actions currently supported by the reachability graph. The paper states that the planner is probabilistically complete if the samplers are probabilistically complete and shows that it can construct long manipulation plans requiring both prehensile and nonprehensile actions in cluttered environments (Garrett et al., 2016).

These language and planning methods share a non-classical use of backward information. In neither case is the backward phase a simple reversal of a forward decoder or executor. Instead, it generates anchor-conditioned context or goal-supporting subgoals that make the forward phase computationally and combinatorially tractable (Mou et al., 2015, Garrett et al., 2016).

4. Constraint generation and solver alignment in parametric CAD

In parametric CAD, the problem is to generate a sequence of sketch constraints and dimensions that fully constrains a given geometry while preserving design intent. The constraint vocabulary includes Coincident, Horizontal, Vertical, Parallel, Perpendicular, Tangent, Midpoint, Equal, Offset, Concentric, and dimensions such as Diameter, Radius, Distance, Angle, and Length. The model predicts a sequence $M$0 autoregressively from a sketch geometry query $M$1, and a CAD constraint solver then classifies the result as fully constrained, under-constrained, over-constrained, not solvable, and stable. “Design alignment” is defined as the application of alignment methods from generative modeling to ensure that outputs maintain design intent, with the solver providing the reward or feedback signal rather than human preference (Casey et al., 17 Apr 2025).

The paper studies supervised fine-tuning, preference-based optimization, and reinforcement learning. Expert Iteration samples $M$2 candidate sequences at temperature $M$3, rejects sequences that are under-constrained, over-constrained, or unsolvable, and distills the remaining samples with cross-entropy. Direct Preference Optimization ranks pairs by the percentage of fully-constrained curves. Reinforcement-learning variants include ReMax, RLOO, and GRPO, all initialized from the SFT model. Reward components include $M$4, $M$5, penalties for instability, not solvable, over-constrained, and other failures, together with token-level $M$6 penalties for constraints that the solver identifies as problematic when applied one by one (Casey et al., 17 Apr 2025).

Method	Fully constrained rate
Vitruvion base	8.87%
SFT	34.24%
Iterative DPO	64.91%
Expert Iteration	71.70%
ReMax	79.84%
RLOO	93.05%
GRPO	91.59%

The same study reports Pass@1 and Pass@8 for the success condition “fully constrained, not over-constrained, solvable, and stable at 4 bins,” with RLOO achieving 83.57 and 84.96 and GRPO 81.49 and 83.42. At the same time, it explicitly notes that the method is primarily about forward constraint generation, not backward inference: it does not introduce a dedicated reverse-mapping module, does not reconstruct a symbolic constraint set from arbitrary CAD geometry in a backward setting, and only uses solver interactions for evaluation, preprocessing, and token-level blame assignment. This is an important boundary on the scope of the term in CAD (Casey et al., 17 Apr 2025).

5. Operator splitting and stochastic systems

In monotone operator splitting, forward and backward constraints generation appears as a decomposition of the target operator into an extended sum

$M$7

rather than a classical sum $M$8. The extended forward-backward algorithm addresses

$M$9

with iteration

$b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$0

The forward term is therefore an approximate selection from the $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$1-enlargement of $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$2, while the backward term is the resolvent of $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$3. Under

$b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$4

together with hypotheses $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$5 and $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$6, the weighted Cesàro averages $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$7 converge weakly to a zero of $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$8. In the constrained minimization case $b_0,\, d_1,\, b_1,\, d_2,\, b_2,\ldots$9, the method becomes

$d_i$0

which the paper describes as a projected approximate subgradient scheme. Its significance lies in solving constrained convex minimization without qualification condition, since $d_i$1 even when the classical sum rule fails (Lassonde et al., 2012).

In controlled time-symmetric forward-backward doubly stochastic differential equations, forward and backward constraints are built into the state dynamics and admissible set. The state $d_i$2 evolves under a time-symmetric FBDSDE with controls $d_i$3, $d_i$4, and $d_i$5, cost

$d_i$6

and expectation constraints

$d_i$7

Using the terminal perturbation method and Ekeland’s variational principle, the paper derives a stochastic maximum principle with adjoint equations, a Hamiltonian

$d_i$8

and the pointwise optimality condition

$d_i$9

Here, forward and backward constraint propagation is literal: perturbing one boundary control affects all four coupled state processes (Ji et al., 2010).

A related stochastic maximum-principle result addresses a forward-backward stochastic control system in which the forward state is constrained at terminal time by $b_i$0 almost surely and the backward state is governed by a quadratic-growth BSDE. A key step is the backward reformulation that treats the terminal state as the control variable $b_i$1, transforming the original constrained forward problem into a backward-form control system over an admissible set $b_i$2 with $b_i$3 almost surely. BMO martingale estimates, terminal perturbation, and Ekeland’s variational principle then yield the maximum principle

$b_i$4

The paper interprets this as a sample-wise convex constraint condition and applies it to recursive utility maximization with bankruptcy prohibition (Ji et al., 2020).

Forward and backward law constraints in stochastic analysis provide a further variant. In the forward case, the law $b_i$5 or conditional law $b_i$6 is constrained to remain in

$b_i$7

with reflection along the normal vector $b_i$8. In the backward case, the law of $b_i$9 is constrained analogously. The paper proves existence and uniqueness via penalization, studies interacting particle approximations, and connects the forward and backward formulations to PDEs on Wasserstein space: a Neumann problem in the forward case and an obstacle problem in the backward one (Briand et al., 2019).

6. Backward reachability, learned constraints, and conceptual limits

In inverse constraint learning, the forward/backward distinction becomes a substantive claim about what is actually learned from safe demonstrations. The paper argues that ICL does not recover the failure set

$d_k = \Dbot$0

itself, but the backward reachable tube

$d_k = \Dbot$1

Its central theorem shows that exact entropy-regularized ICL recovers

$d_k = \Dbot$2

because safe expert demonstrations do not distinguish between states where failure has already occurred and states from which failure is inevitable under the data-collection dynamics. This yields a forward/semantic interpretation—“this state is the obstacle”—and a backward/dynamic interpretation—“if you are here, your future is doomed.” The learned constraint is therefore dynamics-conditioned rather than dynamics-independent (Qadri et al., 26 Jan 2025).

The practical consequences are two-sided. Positively, ICL can function as a data-driven estimator of a BRT and can improve sample efficiency in downstream safe policy search. Negatively, constraints learned under one dynamics model do not transfer naively to another. The paper’s agile and non-agile Dubins-car-style systems illustrate the point: a constraint learned from the agile system underestimates unsafe states for the non-agile system, whereas a constraint learned from the non-agile system is overly conservative for the agile one (Qadri et al., 26 Jan 2025).

Taken together, these literatures indicate that forward and backward constraints generation is best understood as a cross-domain methodological principle. Forward constraints approximate reachable, feasible, or constructible structure; backward constraints approximate goal-supporting, error-trace, predecessor, or inevitable-failure structure. In some domains the two are alternated until a refined fixed point is obtained, in others they are coupled around an anchor, and in still others only the forward side is genuinely generative while the backward side appears only as semantic interpretation or solver feedback. This suggests that the unifying theme is not symmetry for its own sake, but the use of bidirectional constraint information when purely forward or purely backward reasoning is too weak, too expensive, or too indirect.