When Goals Collide: Synthesizing the Best Possible Outcome

Opening — Why this matters now

Most AI control systems are still designed around a brittle assumption: either the agent satisfies everything, or the problem is declared unsolvable. That logic collapses quickly in the real world. Robots run out of battery. Services compete for shared resources. Environments act adversarially, not politely. In practice, goals collide.

The paper Multi-Property Synthesis tackles this gap head-on. Instead of asking whether all goals can be achieved, it asks a more operational question: which combinations of goals can be guaranteed, and which of those combinations are maximal? That shift sounds modest. It isn’t. It fundamentally changes how finite-horizon autonomous systems reason about trade-offs.

Background — Context and prior art

Finite-trace temporal logic (LTLf) has become a standard tool for planning and controller synthesis when tasks are meant to finish. Traditional LTLf synthesis reduces a specification to a reachability game: reach an accepting state, then stop. Clean. Elegant. And unforgiving.

The problem is over-subscription. Many realistic specifications decompose naturally into multiple goals—reach region A, collect resource B, maintain safety constraint C—but the conjunction of all of them may be unrealizable. Classical solutions fall back to enumeration: try every subset of goals, synthesize a controller, repeat. This is conceptually simple and computationally brutal.

Prior work explored partial satisfaction, preferences, or quantitative relaxations. Most of it either assumes non-adversarial environments or optimizes a scalar score. What has been missing is a qualitative, adversarial, finite-horizon method that computes all achievable trade-offs at once.

Analysis — What the paper actually does

The core move is to lift LTLf synthesis from single goals to sets of goals without exploding the computation.

Instead of solving one synthesis game per subset of goals, the authors build a single product game that tracks progress toward all properties simultaneously. A state in this game encodes where the agent stands with respect to each individual goal automaton.

They then define a fixed-point computation over pairs:

• a product-game state, and
• a set of goals claimed to be achievable from that state.

The base case is obvious: if a state already satisfies a set of goals, that set is achievable. The induction step mirrors classic controllable-predecessor reasoning: the agent must choose an action such that, regardless of the environment’s response, it moves to a state where the same goal set remains achievable.

Crucially, the construction respects monotonicity: if a set of goals is achievable, all of its subsets are too. This allows the algorithm to focus on maximal realizable sets, pruning dominated goal combinations on the fly.

The result is not just a decision procedure. It synthesizes concrete finite-state strategies that guarantee the chosen maximal set of goals and terminate once achieved.

Symbolic synthesis — Where the real leverage comes from

The explicit construction is already conceptually satisfying. The symbolic version is where things become sharp.

Each goal is assigned a Boolean variable. A valuation over these variables represents a goal set. Instead of enumerating subsets, the algorithm encodes realizability as a Boolean formula with implications of the form:

if goal i is claimed, then its automaton must be in an accepting state.

This single trick captures exponentially many goal combinations compactly. The same symbolic fixed-point computation used for single-goal LTLf synthesis now operates over states and goal variables together. No new operators. No bespoke machinery.

At the end, the system produces a Boolean formula characterizing exactly which goal sets are achievable from the initial state. Extracting an optimal outcome becomes a matter of selecting a satisfying assignment with maximal cardinality.

Findings — What the experiments show

Across a wide range of benchmarks—counters, robot navigation, deep temporal patterns—the symbolic multi-property approach consistently outperforms enumeration-based baselines.

The headline result is not asymptotic complexity—both approaches remain 2EXPTIME-complete—but shared structure. One fixed point replaces thousands or millions of redundant synthesis calls.

Implications — Why this matters beyond theory

This work quietly reframes synthesis as a design-space exploration tool rather than a yes/no oracle.

For practitioners, it enables:

• principled trade-offs in over-subscribed planning,
• controllers that degrade gracefully instead of failing outright,
• precomputed adaptability: if the environment turns out to be less adversarial than assumed, the agent can safely pursue larger goal sets.

For governance and assurance, it offers something rarer: transparent limits. You don’t just know what the system can do—you know what it cannot guarantee, and why.

Conclusion — A better question than “is it realizable?”

Multi-property synthesis replaces a brittle binary with a structured frontier of possibilities. The technical contribution is a fixed-point computation. The conceptual contribution is more subtle: it treats failure not as an endpoint, but as a space to reason within.

That is a healthier model for real AI systems—finite, constrained, and forced to choose.

Cognaptus: Automate the Present, Incubate the Future.