Proof Checking Technology for Satisfiability Modulo Theories

Abstract

A common proof format for solvers for Satisfiability Modulo Theories (SMT) is proposed, based on the Edinburgh Logical Framework (LF). Two problems arise: checking very large proofs, and keeping proofs compact in the presence of complex side conditions on rules. Incremental checking combines parsing and proof checking in a single step, to avoid building in-memory representations of proof subterms. LF with Side Conditions (LFSC) extends LF to allow side conditions to be expressed using a simple first-order functional programming language. Experimental data with an implementation show very good proof checking times and memory usage on benchmarks including the important example of resolution inferences.

Fast and flexible proof checking for SMT

“…As described in previous work [11], lfsc interleaves parsing and checking of proofs. This results in significant performance improvements over parsing the whole proof into memory and then checking it, and opens the way for checking proofs too large to fit into main memory.…”

Foundational Proof Certificates in First-Order Logic

“…A more interesting approach might be to find ways to handle huge proofs efficiently, which can be done both in a practical way using, for instance, incremental checking as proposed in [24], and in a theoretical way by designing certificates that can be recognized in deterministic log space (e.g., the RUP format for proofs of unsatisfiability [27]). …”

“…Also the use of axioms leads one away from truly analytic proof theory in which subformulas of a conjectured sequent are needed for consideration within (cut-free) proofs. Also in the general area of enhancing intuitionistic proof representations for checking, there is also recent work on extending the λΠ-calculus with side conditions [24] and with external predicates [11].…”

Computer-Aided Security Proofs for the Working Cryptographer

“…As there is currently no SMT solver that generates Coq proofs, the verification conditions are admitted in order to make the output derivations checkable by the Coq proof assistant. Making SMT solvers proof-producing is an active subject of research [21], and advances towards this goal shall benefit immediately to EasyCrypt; -Computation of probability: EasyCrypt generates proof skeletons for claims about probability rather than fully machine-checked proofs. While it is entirely feasible to extend the compiler for justifying more reasonings, a more principled solution would require a tool that can symbolically compute the probability of an event in a distribution.…”