One useful logic that defines its own truth

Existential fixed point logic (EFPL) is a natural fit for some applications, and the purpose of this talk is to attract attention to EFPL. The logic is also interesting in its own right as it has attractive properties. One of those properties is rather unusual: truth of formulas can be defined (given appropriate syntactic apparatus) in the logic. We mentioned that property elsewhere, and we use this opportunity to provide the proof.

💡 Research Summary

The paper “One Useful Logic That Defines Its Own Truth” by Andreas Blass and Yuri Gurevich investigates Existential Fixed‑Point Logic (EFPL), a variant of first‑order logic that omits universal quantification and adds a least‑fixed‑point operator applied only to positively occurring relation symbols. The authors first recall the definition of EFPL: formulas are built from atomic predicates using conjunction, disjunction, negation (restricted to atomic formulas with “negatable” relation symbols), and the existential quantifier; the new constructor “LET Π THEN ϕ” introduces a finite set Π of logic rules of the form P(⃗x) ← δ, where P is a positive relation symbol and δ is any EFPL formula possibly containing P. Each rule defines a monotone operator Γ on the space of interpretations of the positive symbols; by the Knaster‑Tarski theorem Γ has a least fixed point, which is used to evaluate the body ϕ of the induction assertion.

The authors then summarize known properties of EFPL. Over structures equipped with a successor relation (or equivalently a linear order), EFPL captures polynomial‑time computability: every property decidable in polynomial time can be expressed by an EFPL formula, and conversely every EFPL formula can be evaluated in polynomial time on finite structures. The set of logically valid EFPL formulas is recursively enumerable and complete for r.e.; the set of satisfiable formulas is also r.e.-complete, while the set of formulas true in all finite structures is co‑r.e.-complete. EFPL enjoys a finite‑model property: if a formula holds in a (possibly infinite) structure, it already holds in some finite substructure. Moreover, any monotone induction defined by an EFPL program stabilizes after at most ω steps; thus no transfinite induction is needed. Truth of EFPL formulas is preserved under homomorphisms, and any EFPL formula that is equivalent to a first‑order formula must already be equivalent to an existential first‑order formula.

The central contribution of the paper is to show that EFPL can formalize its own truth definition. To achieve this, the authors introduce a ternary predicate Sat(ϕ, Π, s), where:

• ϕ is an EFPL formula (possibly containing extra relation symbols not in the ambient vocabulary);
• Π is a finite set of logic rules whose head symbols exactly cover all extra relation symbols appearing in ϕ or in Π itself;
• s is an assignment of elements of the underlying structure X to all free individual variables occurring in ϕ or Π.

The intended meaning of Sat(ϕ, Π, s) in a structure X is: compute the least fixed point of the monotone operator induced by Π (using the current interpretation of the extra symbols), then evaluate ϕ under the assignment s, interpreting the extra symbols by that fixed point. The authors stress that no “clever tricks” are used; Sat is a straightforward formalization of the informal semantics described in earlier work on EFPL.

To make this definition possible inside EFPL, the authors assume a finite auxiliary vocabulary Υ that contains:

1. A copy N of the natural numbers, with constant 0 and unary successor S, allowing the encoding of lists, tuples, and the indexing of arguments of function and relation symbols.
2. A negatable equality predicate.
3. Sufficient symbols to encode syntactic objects (formulas, rules, assignments) as EFPL terms (for example, using Gödel‑number‑like encodings or explicit list constructors).

With these resources, the authors show how to represent atomic formulas, conjunctions, disjunctions, negations, and existential quantifications as EFPL terms, how to encode a rule as a pair (head, body), and how to encode a program Π as a finite list of such pairs. Assignments are encoded as finite maps from variable names to natural numbers.

The definition of Sat proceeds by structural recursion on the encoded formula ϕ:

• Atomic case: If the atom is of a positive relation symbol P, Sat checks whether the tuple of arguments (decoded from the term) belongs to the least‑fixed‑point interpretation of P obtained from Π. If the atom uses a negatable relation symbol, Sat checks the opposite condition.
• Negation: Allowed only on atomic formulas; Sat simply flips the result of the atomic case.
• Conjunction / Disjunction: Sat evaluates the two sub‑formulas recursively and combines the results with ∧ or ∨, respectively.
• Existential quantifier: Sat iterates over all natural numbers n (using the built‑in successor and zero) and checks Sat on the sub‑formula with the quantified variable bound to n; the existential is true if any iteration succeeds. This iteration is expressed inside EFPL by a small auxiliary program that searches up to a given bound, exploiting the lemma that EFPL can simulate bounded universal quantification.

Because the least‑fixed‑point of Π stabilizes after at most ω steps, the recursive evaluation of Sat never requires transfinite induction; the whole construction stays within the expressive power of EFPL. The authors provide an inductive proof that, for any structure X, any EFPL formula ϕ, any program Π, and any assignment s, the truth value of Sat(ϕ, Π, s) in X coincides with the intended semantic truth of ϕ under the least‑fixed‑point interpretation of the extra symbols.

The paper concludes by discussing the significance of this self‑referential capability. It shows that EFPL is not only a useful query language (equivalent to Datalog on relational vocabularies) and a natural capture of polynomial‑time computation, but also a logic that can internalize its own meta‑theory. This opens the door to formal verification of EFPL programs within EFPL itself, to building interpreters, proof systems, or even higher‑order reasoning frameworks without leaving the logic. The result thus strengthens the theoretical standing of EFPL and suggests new avenues for applications in program verification, database theory, and authorization languages.