[Paper] Multi-Property Synthesis

Source: arXiv - 2601.10651v1

Overview

The paper tackles LTLf (Linear Temporal Logic over finite traces) synthesis when a system must satisfy multiple specifications that may conflict. Instead of trying every possible subset of requirements, the authors devise a single fixed‑point computation that directly discovers the largest set of goals that can be realized together and automatically generates a strategy for it. The result is a symbolic algorithm that can handle exponentially many goal combinations with dramatically lower runtime.

Key Contributions

• Unified Fixed‑Point Computation: Computes, in one pass, the mapping from game states to maximally realizable goal sets, avoiding costly enumeration of subsets.
• Boolean Goal Variables: Introduces symbolic Boolean variables for each property, enabling compact representation of all goal combinations.
• Monotonicity Exploitation: Leverages the monotone nature of realizability (adding goals never makes a realizable set unrealizable) to prune the search space dramatically.
• Fully Symbolic Algorithm: Implements the approach using BDD/SMT‑style symbolic data structures, making it scalable to large state spaces.
• Empirical Speedup: Shows up to 100× faster performance compared with naïve enumeration baselines on benchmark suites from AI planning and reactive synthesis.

Methodology

1. Problem Formalization – The synthesis task is modeled as a product game between the system and its environment, where each vertex carries a set of Boolean goal variables (one per LTLf property).
2. Goal‑Set Lattice – All possible subsets of goals form a lattice ordered by inclusion. Realizability is monotone on this lattice: if a set G is realizable, any subset of G is also realizable.
3. Symbolic Fixed‑Point Engine – The algorithm iteratively computes a greatest fixed point over the product game using symbolic transition relations. At each iteration it updates a goal‑realizability relation R(s, g) that says “from state s we can guarantee the goals described by Boolean vector g”.
4. Monotone Propagation – By propagating only maximal goal vectors upward in the lattice, the engine avoids storing every combination explicitly; instead, it stores a compact set of maximal vectors per state.
5. Strategy Extraction – Once R stabilizes, a concrete strategy is extracted by picking, for each reachable state, an action that leads to a successor with a superset of the current goal vector.

The whole pipeline is implemented with standard symbolic back‑ends (BDD libraries, SAT/SMT solvers), so developers can plug it into existing verification toolchains.

Results & Findings

• Scalability: The symbolic method scales linearly with the number of states and only logarithmically with the number of properties, thanks to the compact maximal‑goal representation.
• Correctness: For every benchmark, the synthesized strategy achieves the largest realizable subset of goals, matching the optimal result obtained by exhaustive enumeration.
• Memory Footprint: Symbolic representation reduces memory usage by up to 90 % compared with storing all subsets explicitly.

Practical Implications

• Robust Reactive Controllers: Engineers building autonomous agents (robots, drones, UI bots) can now automatically generate controllers that gracefully degrade when not all safety or performance requirements can be met simultaneously.
• Feature‑Toggle Synthesis: In complex software product lines, the technique can synthesize runtime policies that enable the maximal set of optional features without violating core invariants.
• Rapid Prototyping: Because the algorithm runs in seconds on realistic models, developers can iterate on specifications and instantly see which combinations are feasible, fostering a more interactive design workflow.
• Integration Path: The approach fits into existing model‑checking pipelines (e.g., Spot, NuSMV) and can be exposed as a library function synthesize_maximal(LTLfSpecs, SystemModel) → Strategy.

Limitations & Future Work

• Finite‑Trace Assumption: The method is tailored to LTL over finite traces; extending to infinite‑trace LTL (ω‑regular) would require additional handling of fairness and liveness.
• Symbolic Backend Dependency: Performance hinges on the efficiency of the underlying BDD/SMT engine; pathological cases with highly non‑monotone transition relations may still cause blow‑up.
• Goal Interaction Modeling: The current monotonicity assumption treats goals as independent; richer interactions (e.g., mutually exclusive goals) could be captured with more sophisticated lattice structures.

Future research directions include adapting the fixed‑point framework to infinite‑trace synthesis, exploring hybrid symbolic‑explicit approaches for highly dynamic environments, and building higher‑level DSLs that let developers express multi‑property requirements without delving into LTL syntax.

Authors

• Christoph Weinhuber
• Yannik Schnitzer
• Alessandro Abate
• David Parker
• Giuseppe De Giacomo
• Moshe Y. Vardi

Paper Information

• arXiv ID: 2601.10651v1
• Categories: cs.AI, cs.LO
• Published: January 15, 2026
• PDF: Download PDF