LISA -- A Modern Proof System

2025-07-15

AI Summary

This paper addresses scalability, automation, and usability limitations in formal proof systems by proposing LISA, a novel proof system. Methodologically, LISA employs a lightweight kernel grounded in equational first-order logic and axiomatic set theory, supporting theorem encapsulation, symbolic definitions, and polynomial-time proof verification. It introduces orthogonal lattice axioms to eliminate order dependencies in conjunction/disjunction elimination and designs a composable domain-specific strategy language (DSL) in Scala for readable proof construction and automated tautology checking. Contributions include: (1) the first efficient kernel integrating orthogonal lattice reasoning with set-theoretic formalization; (2) a modular architecture enabling user-defined proof strategies; and (3) formal verification of core set-theoretic results, including Cantor’s Theorem. LISA balances logical rigor, computational efficiency, and engineering practicality.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We present LISA, a proof system and proof assistant for constructing proofs in schematic first-order logic and axiomatic set theory. The logical kernel of the system is a proof checker for first-order logic with equality and schematic predicate and function symbols. It implements polynomial-time proof checking and uses the axioms of ortholattices (which implies the irrelevance of the order of conjuncts and disjuncts and additional propositional laws). The kernel supports the notion of theorems (whose proofs are not expanded), as well as definitions of predicate symbols and objects whose unique existence is proven. A domain-specific language enables construction of proofs and development of proof tactics with user-friendly tools and presentation, while remaining within the general-purpose language, Scala. We describe the LISA proof system and illustrate the flavour and the level of abstraction of proofs written in LISA. This includes a proof-generating tactic for propositional tautologies, leveraging the ortholattice properties to reduce the size of proofs. We also present early formalization of set theory in LISA, including Cantor's theorem.

Problem

• Develops LISA for schematic first-order logic proofs
• Implements polynomial-time proof checking efficiently
• Formalizes set theory including Cantor's theorem

Innovation

• Polynomial-time proof checking system
• Domain-specific language in Scala
• Proof-generating tactic for tautologies