Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Automation for interactive proof: First prototype

AbstractInteractive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, and clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality

Making IP=PSPACE\textsf{IP}=\textsf{PSPACE} Practical: Efficient Interactive Protocols for BDD Algorithms

We show that interactive protocols between a prover and a verifier, a well-known tool of complexity theory, can be used in practice to certify the correctness of automated reasoning tools. Theoretically, interactive protocols exist for all $\textsf{PSPACE}$ problems. The verifier of a protocol checks the prover's answer to a problem instance in polynomial time, with polynomially many bits of communication, and with exponentially small probability of error. (The prover may need exponential time.) Existing interactive protocols are not used in practice because their provers use naive algorithms, inefficient even for small instances, that are incompatible with practical implementations of automated reasoning. We bridge the gap between theory and practice by means of a novel interactive protocol whose prover uses BDDs. We consider the problem of counting the number of assignments to a QBF instance ($\#\textrm{CP}$), which has a natural BDD-based algorithm. We give an interactive protocol for $\#\textrm{CP}$ whose prover is implemented on top of an extended BDD library. The prover has only a linear overhead in computation time over the natural algorithm. We have implemented our protocol in $\textsf{blic}$, a certifying tool for $\#\textrm{CP}$. Experiments on standard QBF benchmarks show that \blic\ is competitive with state-of-the-art QBF-solvers. The run time of the verifier is negligible. While loss of absolute certainty can be concerning, the error probability in our experiments is at most $10^{-10}$ and reduces to $10^{-10k}$ by repeating the verification $k$ times

Investigation, Development, and Evaluation of Performance Proving for Fault-tolerant Computers

A number of methodologies for verifying systems and computer based tools that assist users in verifying their systems were developed. These tools were applied to verify in part the SIFT ultrareliable aircraft computer. Topics covered included: STP theorem prover; design verification of SIFT; high level language code verification; assembly language level verification; numerical algorithm verification; verification of flight control programs; and verification of hardware logic

An analysis and implementation of linear derivation strategies

This study examines the efficacy of six linear derivation strategies: (i) s-linear resolution, (ii) the ME procedure; (iii) t-linear resolution, (iv) SL -resolution, (v) the GC procedure, and (vi) SLM. The analysis is focused on the different restrictions and operations employed in each derivation strategy. The selection function, restrictive ancestor resolution, compulsory ancestor resolution on literals having atoms which are or become identical, compulsory merging operations, reuse of truncated literals, spreading of FALSE literals, no-tautologies resection, no two non-B-literals having identical atoms restriction, and the use of semantic information to trim irrelevant derivations from the search tree are the major features found In these six derivation strategies. Detecting loops and minimizing irrelevant derivations are the identified weak points of SLM. Two variations of SLM are suggested to rectify these problems. The ME procedure, SL-resolution, the GC procedure, SLM and one of the suggested variations of SLM were implemented using the Arity/Prolog compiler to produce the ME -TP, SL-TP, GC-TP, SLM-TP and SLM5-TP theorem provers respectively. In addition to the original features of each derivation strategy, the following search strategies were included in the implementations : the modified consecutively bounded depth-first search unit preference strategy, set of support strategy, pure literal elimination, tautologous clause elimination, selection function based on the computed weight of a literal, and a match check. The extension operation used by each theorem prover was extended to include subsumed unit extension and paramodulation. The performance of each theorem prover was determined. Experimental results were obtained using twenty four selected problems. The performance was measured in terms of the memory use and the execution time. A comparison of results between the five theorem provers using the, ME-TP as the basis was done. The results show that none of the theorem provers, consistently perform better than the others. Two of the selected problems were not proved by SL-TP and one problem was not proved by SLM-TP due to memory problems. The ME-TP, GC-TP and SLM5-TP proved all the selected problems. In some problems, the ME-TP and GC-TP performed better than SLM5-TP. However, the ME-TP and GC-TP had difficulties in some problems in which SLM5-TP performed well

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Formalizing Bachmair and Ganzinger’s Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter’s text, emphasizing the value of formal proofs in the field of automated reasoning