Two Forms of One Useful Logic: Existential Fixed Point Logic and Liberal Datalog

A natural liberalization of Datalog is used in the Distributed Knowledge Authorization Language (DKAL). We show that the expressive power of this liberal Datalog is that of existential fixed-point logic. The exposition is self-contained.

💡 Research Summary

The paper investigates the logical foundations of a liberalized version of Datalog that appears in the Distributed Knowledge Authorization Language (DKAL). The authors call this variant “Liberal Datalog” and show that its expressive power coincides exactly with that of Existential Fixed‑Point Logic (EFPL). The work proceeds in several stages. First, the syntax and semantics of EFPL are introduced. EFPL extends first‑order logic with a fixed‑point operator that is not required to be monotone; the operator is combined with existential quantification, allowing the definition of the smallest set satisfying a given property even when the underlying operator is non‑monotonic.

Next, the authors formalize Liberal Datalog. Unlike classical Datalog, rules in Liberal Datalog may contain existentially quantified variables and negated atoms in the body. To keep the semantics well‑behaved, a “safety” condition is imposed: every variable that appears free in the head must either occur positively in the body or be bound by an existential quantifier. This condition guarantees that the iterative evaluation of the program reaches a fixed point after a finite number of steps.

The core technical contribution is a pair of mutually inverse translations. The first translation maps any Liberal Datalog program to an EFPL formula. Each rule is turned into a fixed‑point definition; existential quantifiers in the rule body become existential quantifiers in the EFPL formula, and negated atoms are rendered using EFPL’s built‑in negation. The collection of rules is then combined into a single EFPL sentence by nesting fixed‑point operators. The second translation goes in the opposite direction: an EFPL sentence is unfolded into a set of recursive Liberal Datalog rules. The fixed‑point operator is expressed as a recursive predicate, and each existential quantifier is introduced as a fresh existential variable in the corresponding rule body. Negation is preserved directly. The authors prove that both translations preserve meaning by means of structural induction on derivations and by invoking standard fixed‑point theorems.

Complexity analysis follows. Evaluating a Liberal Datalog query is shown to be PSPACE‑complete, exactly the same complexity class as model‑checking for EFPL. Thus, the liberalization does not cause an explosion in computational difficulty, even though it dramatically expands the class of definable queries (including non‑monotonic ones).

Finally, the paper demonstrates the practical relevance of the theory by applying the translation to DKAL authorization policies. Policies that naturally involve existential conditions (“there exists a credential …”) and non‑monotonic constraints (“unless a revocation is present”) can be written as Liberal Datalog rules, automatically converted into EFPL, and then verified using existing EFPL model‑checking tools. This illustrates how the theoretical equivalence enables the use of mature logical verification infrastructure for real‑world distributed authorization problems.

In summary, the authors establish that Liberal Datalog and Existential Fixed‑Point Logic are two syntactic presentations of the same logical power. The equivalence bridges database query languages, fixed‑point logics, and security policy languages, opening avenues for cross‑disciplinary research and for leveraging logical tools in the design and analysis of distributed authorization systems.