The Proof Certifier Checkers

Abstract

International audienceDifferent theorem provers work within different formalisms and paradigms, and therefore produce various incompatible proof objects. Currently there is a big effort to establish foundational proof certificates (FPC), which would serve as a common " specification language " for all these formats. Such framework enables the uniform checking of proof objects from many different theorem provers while relying on a small and trusted kernel to do so. Checkers is an implementation of a proof checker using foundational proof certificates. By trusting a small kernel based on (focused) sequent calculus on the one hand and by supporting FPC specifications in a prolog-like language on the other hand, it can be used for checking proofs of a wide range of theorem provers. The focus of this paper is on the output of equational resolution theorem provers and for this end, we specify the paramodulation rule. We describe the architecture of Checkers and demonstrate how it can be used to check proof objects by supplying the FPC specification for a subset of the inferences used by E-prover and checking proofs using these inferences

Greedy pebbling for proof space compression

“…Secondly, whereas the main motivation for deletion information in the DRUP format was to speed up proof checking, the main concern here is memory consumption. And finally, whereas DRUP proofs are typically processed in a bottom-up way, here it is assumed that the proof consumer will do a top-down traversal of the proof, as usual for resolution [2,19,[24][25][26]36,44,55].…”

“…This is the common way to process all kinds of proofs in proof checking systems such as CERes [25], GAPT [24,26,36,44,55], the Checkers tool [19] for foundational proof certificates [20] and Dedukti [2,17]. Moreover, when the conclusion clauses of resolution inferences (or chains of inferences) are omitted in propositional resolution proofs in the TraceCheck format, only top-down checking is possible, because the conclusion clauses need to be recomputed based on their premises.…”

Defining the meaning of TPTP formatted proofs

“…In particular, nondeterminism is allowed in the Foundational Proof Certificates (FPC) framework [7,19] where client-side inference rules (i.e., rules implemented in theorem provers) are translated into low-level rules of sequent calculus. The checkers proof certifier [5], based on the FPC framework, used the λProlog logic programming language [20] to provide for a backtracking search approach to exploring any nondeterminism in such translations. The basic idea is to program a set of predicates which will guide the search in the target calculus.…”

“…Currently, the normal process for certifying the output of a certain theorem prover is for a dedicated team on the certifier side to try to understand the semantics of each inference rule of the object calculus. This approach suffers many times from missing documentation, different names and versions of actual software, and insufficient information in the proofs themselves [5]. This gap is enlarged by the fact that teams of implementers and certifiers can reside in different locations or even work in different periods, thus making the communication between them difficult or even impossible.…”

“…The predicate logic form of the problems, their solutions, and now, also their semantics, will allow proof certifiers to easily access the semantics and will further diminish the gap between the theorem provers and their certifiers. For example, the checkers proof certifier [5] (written using logic programming), will only require minor computations to be applied to the input files, if any. Such simplicity helps to improve the trust of the certification process.…”