LPC(ID): A Sequent Calculus Proof System for Propositional Logic Extended with Inductive Definitions
The logic FO(ID) uses ideas from the field of logic programming to extend first order logic with non-monotone inductive definitions. Such logic formally extends logic programming, abductive logic programming and datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. The goal of this paper is to study a deductive inference method for PC(ID), which is the propositional fragment of FO(ID). We introduce a formal proof system based on the sequent calculus (Gentzen-style deductive system) for this logic. As PC(ID) is an integration of classical propositional logic and propositional inductive definitions, our sequent calculus proof system integrates inference rules for propositional calculus and definitions. We present the soundness and completeness of this proof system with respect to a slightly restricted fragment of PC(ID). We also provide some complexity results for PC(ID). By developing the proof system for PC(ID), it helps us to enhance the understanding of proof-theoretic foundations of FO(ID), and therefore to investigate useful proof systems for FO(ID).
💡 Research Summary
The paper investigates proof‑theoretic foundations for PC(ID), the propositional fragment of the logic FO(ID) that enriches classical propositional logic with non‑monotonic inductive definitions. The authors introduce LPC(ID), a Gentzen‑style sequent calculus that integrates the usual logical inference rules with two new families of definition rules. The logical rules (∧‑left/right, ∨‑left/right, ¬‑left/right, structural rules, cut, weakening, contraction) are exactly those of the standard sequent calculus for propositional logic. The definition rules are designed to reflect the semantics of inductive definitions, which are interpreted as least fixpoints of rule sets.
The first definition rule, called Definition Expansion (Def‑Exp), allows a defined atom P occurring in the antecedent to be replaced by the body φ of the rule that defines P (P ← φ). This mirrors one step of the fixpoint construction: if P is assumed true, then its defining condition must also hold. The second rule, Definition Reduction (Def‑Red), deals with a negated defined atom ¬P in the succedent. It replaces ¬P by ¬φ, thereby using the fact that if P is false in the least model, then its defining condition must be false. Both rules are applied only when the underlying definition is stratified (i.e., there is a well‑founded ordering of defined atoms that prevents circular negative dependencies).
The authors prove soundness by showing that each inference rule preserves truth under the least‑fixpoint semantics of PC(ID). For Def‑Exp, they argue that assuming P in a model forces φ to be true in the same model; for Def‑Red, they use the contrapositive property of the least fixpoint: if P is false, then the body φ cannot be satisfied. Consequently, any sequent derivable in LPC(ID) is semantically valid.
For completeness, the paper restricts attention to a fragment where all definitions are stratified and can be transformed into a Definition Normal Form (DNF). The transformation rewrites arbitrary PC(ID) formulas into an equivalent set of clauses where each clause contains at most one defined atom, and the definitions are ordered according to their strata. Using this normal form, the authors construct a proof by induction on the strata: for the lowest stratum the required sequents are provable by ordinary propositional reasoning; for higher strata, the induction hypothesis supplies proofs for the bodies of the defining rules, and the Def‑Exp and Def‑Red rules lift these to proofs about the defined atoms themselves. The cut rule is employed to connect different strata, yielding a full derivation for any valid formula in the fragment.
The paper also analyses the computational complexity of PC(ID). The satisfiability problem (does a given PC(ID) formula have a model?) is shown to be NP‑complete. The reduction to NP uses the fact that, for stratified definitions, the least fixpoint can be computed in polynomial time, so the remaining propositional part reduces to ordinary SAT. The validity problem (is a formula true in all models?) is co‑NP‑complete, being the complement of SAT. The authors note that if definitions are allowed to be non‑stratified, the complexity can rise to EXPTIME or beyond, reflecting the difficulty of evaluating arbitrary non‑monotonic fixpoints.
In the related work section, the authors position their contribution relative to earlier efforts on FO(ID), logic programming, abductive reasoning, and Datalog. While previous studies focused on model theory, translation to other logics, or implementation of solvers, this work is the first to provide a sequent‑calculus proof system that directly incorporates inductive definitions as inference rules. This integration offers a clean, modular proof theory that can be the basis for future automated theorem provers for FO(ID) and its extensions.
The conclusion emphasizes that LPC(ID) supplies a solid proof‑theoretic foundation for PC(ID) and, by extension, for FO(ID). It demonstrates that inductive definitions can coexist with classical logical reasoning in a single, well‑behaved deductive system. The authors outline future directions: extending completeness to non‑stratified definitions, lifting the calculus to the full first‑order level (FO(ID)), investigating cut‑elimination, and developing efficient proof search strategies that exploit the structure of definitions.
Overall, the paper makes a significant contribution by bridging the gap between the declarative semantics of inductive definitions and the syntactic machinery of sequent calculi, thereby opening new avenues for both theoretical investigation and practical reasoning tools in knowledge representation and logic programming.