Automated theorem proving

Summary

Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science. Logical foundations While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics. Frege's Begriffsschrift (1879) introduced both a complete propositional calculus and what is essentially modern predicate logic. His Foundations of Arithmetic, published in 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Russell and Whitehead in their influential Principia Mathematica, first published 1910–1913, and with a revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules

Official source

https://en.wikipedia.org/wiki/Automated_theorem_proving

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Supporting Loop Proofs in KeY by using BLAST

Mathias Krebs

In contrast to beta-testing, formal verification can guarantee correctness of a program against a specification. Two basic verification techniques are theorem proving and model checking. Both have strengths and weaknesses. Theorem proving is powerful, but difficult to use for a software engineer. Model checking is fully automatic, but less powerful and hard to extend. This paper shows a possibility, how to combine both approaches, in order to surround the weaknesses. Concretely, we automatize the proof of while loops in the theorem prover KeY, by using the model checker BLAST.

2006

We present a trustworthy connection between the Leon verification system and the Isabelle proof assistant. Leon is a system for verifying functional Scala programs. It uses a variety of automated theorem provers (ATPs) to check verification conditions (VCs) stemming from the input program. Isabelle, on the other hand, is an interactive theorem prover used to verify mathematical specifications using its own input language Isabelle/Isar. Users specify (inductive) definitions and write proofs about them manually, albeit with the help of semi-automated tactics. The integration of these two systems allows us to exploit Isabelle’s rich standard library and give greater confidence guarantees in the correctness of analysed programs.

2016

Almost-Invariants: From Bugs in Distributed Systems to Invariants

Abhishek Anand, Marco Canini, Dejan Kostic, Maysam Yabandeh

It is notoriously hard to develop dependable distributed systems. This is partly due to the difficulties in foreseeing various corner cases and failure scenarios while implementing a system that will be deployed over an asynchronous network. In contrast, reasoning about the desired distributed system behavior and the corresponding invariants is easier than reasoning about the code itself. Further, the invariants can be used for testing, theorem proving, and runtime enforcement. In this paper, we propose an approach to observe the system behavior and automatically infer invariants which reveal implementation bugs. Using our tool, Avenger, we automatically generate a large number of potentially relevant properties, check them within the time and spatial domains using traces of system executions, and filter out all but a few properties before reporting them to the developer. Our key insight in filtering is that a good candidate for an invariant is the one that holds in all but a few cases, i.e., an "almost-invariant". Our experimental results with the BGP, RandTree, and Chord implementations demonstrate Avenger's ability to identify the almost-invariants that lead the developer to programming errors.

2009