A Fragment of Dependence Logic Capturing Polynomial Time
In this paper we study the expressive power of Horn-formulae in dependence logic and show that they can express NP-complete problems. Therefore we define an even smaller fragment D-Horn* and show that over finite successor structures it captures the complexity class P of all sets decidable in polynomial time. Furthermore we study the question which of our results can ge generalized to the case of open formulae of D-Horn* and so-called downwards monotone polynomial time properties of teams.
💡 Research Summary
The paper investigates the expressive power of Horn‑style formulas within dependence logic (DL) and establishes a precise connection between logical fragments and classical complexity classes. It begins by recalling that full DL, interpreted via team semantics, is known to capture NP: any NP‑complete problem can be expressed as a DL sentence. Building on this foundation, the authors introduce Horn‑formulas into DL, showing that the combination of dependence atoms and Horn clauses suffices to encode canonical NP‑complete problems such as 3‑SAT, graph coloring, and Hamiltonian cycle. The encoding works by constructing a team whose rows correspond to variable assignments and whose columns encode the constraints; each Horn clause enforces a local consistency condition, while dependence atoms preserve the required functional relationships between variables.
Having demonstrated that DL‑Horn is already as expressive as NP, the authors then define a much tighter fragment, denoted D‑Horn*. D‑Horn* imposes two syntactic restrictions: (i) every antecedent contains at most one quantified variable together with dependence atoms, and (ii) the consequent is a single positive literal without any negation. This yields a highly disciplined form of Horn clause that dramatically limits the combinatorial explosion typical of unrestricted DL.
The core technical contribution is the proof that over finite successor structures—structures consisting of a finite domain equipped with a successor function S(x)=x+1—D‑Horn* exactly captures the class P. The proof proceeds in two directions. First, the authors show that any D‑Horn* sentence can be evaluated in polynomial time on such structures. The argument exploits the linear order induced by the successor relation: the team can be viewed as a sequence of configurations of a deterministic Turing machine, and each Horn clause corresponds to a local transition rule that can be checked in constant time per configuration. Because the number of configurations is polynomial in the size of the input, the overall evaluation stays within P.
Second, they prove the converse: every language decidable in polynomial time on a finite successor structure can be translated into an equivalent D‑Horn* sentence. The translation uses a log‑space encoding of the Turing machine’s tape and state, then expresses the machine’s step‑wise evolution as a collection of Horn clauses that relate the current configuration (captured by dependence atoms) to the next one (captured by the consequent). The successor function provides the necessary “next‑cell” reference, allowing the clauses to simulate the transition function of the machine faithfully. The construction yields a D‑Horn* formula whose size is polynomial in the original machine’s description, establishing that D‑Horn* is at least as expressive as P.
Beyond closed sentences, the paper explores two extensions. First, it considers open formulas of D‑Horn*, which allow free variables and thus teams that represent partial assignments or database relations. The authors discuss whether the polynomial‑time capture persists in this more general setting. Preliminary observations suggest that certain open‑formula fragments remain within P, while others may jump to NP‑hardness, depending on how the free variables interact with dependence atoms. Second, the authors study “downward monotone polynomial‑time properties” of teams: properties that are preserved under taking sub‑teams and that can be decided in polynomial time. They show that many natural monotone queries (e.g., inclusion, functional dependency checks) fit into the D‑Horn* framework, but a full characterization of the monotone‑P class remains open.
The paper concludes by highlighting the significance of pinpointing a logical fragment that exactly matches P on a natural class of finite structures. This result bridges descriptive complexity and dependence logic, demonstrating that syntactic restrictions (Horn‑style, limited quantification) can align logical expressiveness with computational tractability. Future work is outlined: extending the analysis to other ordered structures (trees, graphs), refining the classification of open‑formula fragments, and investigating practical applications such as database query optimization where dependence atoms naturally model functional dependencies.