Expressiveness of Communication in Answer Set Programming

Answer set programming (ASP) is a form of declarative programming that allows to succinctly formulate and efficiently solve complex problems. An intuitive extension of this formalism is communicating ASP, in which multiple ASP programs collaborate to solve the problem at hand. However, the expressiveness of communicating ASP has not been thoroughly studied. In this paper, we present a systematic study of the additional expressiveness offered by allowing ASP programs to communicate. First, we consider a simple form of communication where programs are only allowed to ask questions to each other. For the most part, we deliberately only consider simple programs, i.e. programs for which computing the answer sets is in P. We find that the problem of deciding whether a literal is in some answer set of a communicating ASP program using simple communication is NP-hard. In other words: we move up a step in the polynomial hierarchy due to the ability of these simple ASP programs to communicate and collaborate. Second, we modify the communication mechanism to also allow us to focus on a sequence of communicating programs, where each program in the sequence may successively remove some of the remaining models. This mimics a network of leaders, where the first leader has the first say and may remove models that he or she finds unsatisfactory. Using this particular communication mechanism allows us to capture the entire polynomial hierarchy. This means, in particular, that communicating ASP could be used to solve problems that are above the second level of the polynomial hierarchy, such as some forms of abductive reasoning as well as PSPACE-complete problems such as STRIPS planning.

💡 Research Summary

This paper conducts a systematic investigation of how communication among Answer Set Programming (ASP) components influences the expressive power and computational complexity of the overall system. The authors focus on two distinct communication mechanisms.

The first mechanism, termed simple communication, restricts each component to a simple ASP program—i.e., a program whose answer sets can be computed in polynomial time. Components may ask questions of the form Q:l, where Q denotes another component and l is a literal. The truth of Q:l depends on whether l belongs to some answer set of component Q. By encoding a Boolean formula into a network of such simple programs—variables, clauses, and a coordinating component that asks the appropriate questions—the authors show that deciding whether a particular literal appears in any answer set of the communicating system is NP‑hard. Consequently, the decision problems “does an answer set exist?”, “is a literal true in some answer set (brave reasoning)?” and “is a literal true in all answer sets (cautious reasoning)” become NP‑complete, even though each individual program lies in P. This demonstrates that the mere ability to exchange queries lifts the overall complexity one level up the polynomial hierarchy.

The second, more powerful mechanism is multi‑focused communication. Here the components are ordered, and each component in the sequence may focus on the remaining models by discarding those that it finds unsatisfactory before the next component is considered. This leader‑follower scheme corresponds to a successive minimisation over the answer sets of the components. The authors prove that, by arranging a suitable number of focus stages, any problem in the entire polynomial hierarchy (PH) can be simulated. Specifically, a Σ_k^P problem (alternating ∃/∀ quantifiers of depth k) can be represented by a chain of k components where the odd‑indexed components enforce existential choices and the even‑indexed components enforce universal choices. Dually, Π_k^P problems are captured by reversing the order of existential/universal stages. Hence, the multi‑focused framework captures PH completely.

Moreover, the authors extend the framework to allow disjunctive component programs and show that, when combined with multi‑focused communication, the system can express PSPACE‑complete problems. They illustrate this by encoding STRIPS planning, a classic PSPACE‑complete task, as a multi‑focused communicating ASP system: the first leader eliminates plans that violate the goal, subsequent leaders prune infeasible intermediate states, and the final leader verifies the remaining plan. The paper also presents a construction that simulates negation‑as‑failure using only simple programs and question‑answer interactions, thereby isolating the source of the complexity increase to the communication layer rather than to the internal expressiveness of the components.

A significant conceptual contribution is the clarification of global vs. local minimality. Traditional ASP semantics rely on a global minimal model; in a multi‑agent setting this is ambiguous. The multi‑focused approach resolves the ambiguity by applying local minimality at each focus stage, which collectively yields a globally minimal solution. The authors formalize this via a novel notion of P‑situated literals and extended situated literals, define a reduct operation that simultaneously removes negation‑as‑failure and non‑local situated literals, and prove that an interpretation is an answer set iff it is a minimal model of each component’s reduct.

The paper also provides comprehensive complexity tables for various combinations of component types (simple, normal, disjunctive) and reasoning modes (existence, brave, cautious, multi‑focused existence). All proofs are relegated to an appendix, ensuring reproducibility.

In summary, the work demonstrates that communication alone can dramatically increase the expressive power of ASP: simple query‑based interaction raises the complexity to NP, while ordered multi‑focused interaction reaches the full polynomial hierarchy and even PSPACE. These results open new avenues for designing multi‑agent declarative systems where the communication protocol itself serves as a powerful computational resource.