Conflict-Driven LTLf Satisfiability Checking

• CDLSC is a state-of-the-art algorithmic framework that uses conflict-driven clause learning to significantly prune redundant search paths in LTLf satisfiability checking.
• The method employs SAT-based encoding techniques such as Tail Normal Form, X-Normal Form, and Boolean abstraction to manage temporal obligations over finite traces.
• Empirical results demonstrate that CDLSC achieves substantial efficiency gains, reducing SAT calls and accelerating witness discovery compared to traditional explicit-state approaches.

Conflict-Driven Linear Temporal Logic over Finite Traces (LTLf) Satisfiability Checking (CDLSC) is a state-of-the-art algorithmic framework that addresses the decision problem of whether an LTLf formula admits a model, leveraging conflict learning and propositional SAT-solving. CDLSC achieves significant efficiency gains over explicit-state or traditional tableau approaches by extracting and reusing conflict information at the semantic level of temporal formulas, thereby aggressively pruning redundant search paths and accelerating the discovery of witnesses.

1. Formal Background and Problem Setting

LTLf extends propositional temporal logic with operators such as strong next ($X$), weak next ($N$), until ($U$), and release ($R$), interpreted over finite traces. Formally, given a finite set of atomic propositions $AP$, LTLf formulas $\varphi$ are constructed as: $$\varphi ::= p \mid \neg\varphi \mid \varphi \wedge \varphi \mid X \varphi \mid N \varphi \mid \varphi U \varphi \mid \varphi R \varphi$$ where $p \in AP$. The semantics are inductively defined over finite traces $\xi = \sigma_0 \sigma_1 ... \sigma_{n-1}$ of length $n$, associating each step with a subset of $AP$. Satisfiability of $\varphi$ means the existence of a finite $\xi$ with $\xi,0 \models \varphi$ (Li et al., 2018).

2. SAT-Based Encoding of Temporal Obligations

At the core of CDLSC is a SAT-based abstraction of temporal reasoning. The process comprises three main encodings:

• Tail Normal Form (TNF): A fresh atom $\mathsf{Tail}$ designates the end of the trace. TNF enforces $\varphi$ only up to the step where $\mathsf{Tail}$ holds, guaranteeing correctness over finite models by requiring $\mathsf{Tail}$ to hold exactly once.
• X-Normal Form (XNF): The formula is rewritten so every $U$/$R$ is immediately under $X$, facilitating stepwise tableau-like expansion.
• Boolean Abstraction: Each temporal subformula is replaced by a fresh propositional variable, yielding a Boolean skeleton that captures the semantics of the stepwise expansion.

At each search node, the SAT solver is queried for assignments satisfying the current obligations. These assignments are decomposed to:

• $L(A)$: Atomic propositions true at the current step,
• $X(A)$: Temporal obligations deferred to the next step.

This encoding enables propositional SAT solvers to participate transparently in LTLf reasoning (Li et al., 2018).

3. Explicit Transition System Construction

CDLSC conceptualizes the formula’s semantics as an explicit transition system:

• States ($S$): Sets of subformulas representing the active obligations at a given step, with the initial state $s_0 = \{\varphi\}$.
• Transition Relation ($T$): Given $s \in S$ and a propositional assignment $\sigma$, state $s'$ is reachable if there exists an assignment $A$ for the Boolean abstraction such that $L(A) \subseteq \sigma$ and $s' = X(A)$.
• Accepting States ($F$): Those $s$ for which there exists an assignment satisfying both $s$ and $\mathsf{Tail}$.

LTLf satisfiability thus reduces to the existence of a path in this transition system from $s_0$ to some $s_n \in F$ (Li et al., 2018).

4. Conflict-Driven Clause Learning and Search Pruning

The key innovation of CDLSC over explicit search is the Conflict Sequence mechanism, combining state-level clause learning and backjumping:

• Conflict Sequences ($CS$): For depth $k$, maintain $CS[0],\ldots,CS[k]$ where $CS[d]$ stores states that cannot reach acceptance in $d$ steps. When a SAT call at depth $d$ is unsatisfiable, the extracted unsat-core is generalized to a set of temporal states, which are then added to $CS[d]$.
• Search Pruning: Whenever the search revisits a state already recorded in $CS[d]$, that branch is immediately pruned, as it has been proved fruitless in $d$ steps or fewer—a direct analogy to clause learning in propositional SAT, but lifted to the temporal/state level.
• Satisfiable Acceleration: If a SAT call succeeds, the assignment is greedily extended forward, enabling early witness discovery without exhaustively unfolding all successors.

These mechanisms reduce redundant SAT queries and ensure that unsatisfiable patterns are never re-explored, substantially compacting the search space (Li et al., 2018).

5. Algorithmic Structure and Pseudocode

CDLSC is organized as a depth-bounded recursive search. At each depth, it attempts to construct a witness:

• For each candidate state $s$ at depth $d$, it first checks membership in $CS[d]$ (learned conflicts).
• If unseen, a SAT instance is constructed for the obligations of $s$.
  • UNSAT triggers conflict core extraction and abstraction to $CS[d]$.
  • SAT yields an explicit assignment (and, possibly, a model if $\mathsf{Tail}$ is asserted), otherwise, obligations are deferred to the next search level with $X(A)$.
• Recursive calls naturally implement depth backjumping: upon discovering that all successors are unreachable, the parent state is inserted into $CS[d]$, preventing future futile exploration at this depth.

The following table summarizes the key CDLSC operations:

Operation	Function	Mechanism
SAT Witness	Find extension of state obligations	SAT solver with Boolean abstraction
Conflict	Identify unsatisfiable, prune state sets	Extract unsat core and update $CS[d]$
Acceleration	Early model discovery if $\mathsf{Tail}$ is found	Greedy extension

All learned conflict states are permanent and guarantee sound, complete termination, as the state-space is finite and each exploration either learns or finds a model (Li et al., 2018).

6. Theoretical Properties and Complexity

The correctness of CDLSC follows from two principal facts:

• There are finitely many LTLf-states (from the closure of the formula’s subformulas); thus, the conflict sequence exhaustively partitions the search space.
• Each learned conflict entry is minimal and sound, as SAT unsat-cores are generalized no further than necessary.

LTLf satisfiability remains PSPACE-complete, and in the worst case, CDLSC explores an exponential number of states. However, the practical performance is often much better since the conflict-sequence pruning can reduce the number of necessary SAT calls orders of magnitude. Direct comparison to earlier explicit approaches (e.g., Aalta, LTLf2DFA) shows CDLSC is analogous to clause learning in SAT, yielding dramatic SAT call reductions (Li et al., 2018).

7. Empirical Performance and Practical Impact

Extensive empirical analysis on real and random benchmarks demonstrates that CDLSC achieves superior performance:

• CDLSC solves approximately 95% of formulas within experimental timeouts.
• Median runtime exhibits a four-fold improvement over previous tools (e.g., Aalta).
• For highly nested formulas, conflict-sequence learning prunes more than 90% of the explicit-state space, achieving SAT call complexity $O(|\mathrm{Sub}(\varphi)|)$ versus $O(2^{|\mathrm{Sub}(\varphi)|})$ in naïve search.
• The synergy of model-based greedy expansion (early “Tail” detection) and aggressive unsat-based pruning yields the overall efficiency (Li et al., 2018).

A plausible implication is that CDLSC, by using state-level clause learning, structure sharing, and efficient backjumping, establishes a new practical baseline for LTLf satisfiability procedure design.

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For further details and formal algorithmic exposition, see "SAT-based Explicit LTLf Satisfiability Checking" (Li et al., 2018). The generalization to infinite-trace LTL with fairness is addressed in the conflict-driven tableau framework, as in "Handling Conflicts in Depth-First Search for LTL Tableau to Debug Compliance Based Languages" (Hantry et al., 2011).