According to my scan of the Isabelle files, the Sledgehammer tool is only available for Isabelle/HOL. I'm curious about the automation of other theories in Isabelle. For instance:
Isabelle/ZF
Isabelle/FOL
Do they support:
automatic provers
SMT solvers
specialized decision procedures
Isabelle/ZF does not support SMT solvers or specialized decision procedures. As for "automatic provers" I am not sure if that counts but the method auto is quite capable. In some cases it allows to prove a theorem just referencing the premises and a list of several theorems, see for example theorem bij_base_open_homeo in IsarMathLib.
Related
I'm trying to get an overview on the Isabelle Project and everywhere I look new Isabelle/sth's are mentioned.
How do all of them relate to each other?
I'll try to go first:
Isabelle/? - The Isabelle Language itself
Isabelle/Isar - Language used to write proofs in
Isabelle/Pure - ?
Isabelle/HOL - Toolkit for a higher-order logic proving environment
Isabelle/jEdit - Isabelle IDE
Isabelle/Scalar - ?
Isabelle/PIDE - ?
I would not really say that Isabelle is a language; it's more the entire system.
Isabelle/Isar is indeed the language to write structured proofs. Isabelle had actually been around for over 10 years before Isar was developed (it was Makarius Wenzel's PhD thesis in 2002).
Isabelle/Pure is the most basic logic in Isabelle, the one that is implemented in its kernel. It is a simply-typed intuitionistic higher-order logic that only knows about functions, propositions (the prop type), universal quantification, and implication. All the other logics are axiomatised on top of this (Pure serves as a metalogic).
Isabelle/HOL is an axiomatisation of classical simply-typed higher-order logic on top of this. The prop type is mostly hidden from the user; instead you usually work with bool (there is an injection Trueprop : bool ⇒ prop)
Isabelle/jEdit is indeed the IDE that is typically used for Isabelle these days. It consists of the jEdit editor with a lot of Isabelle-specific extras.
Isabelle/ML is the ML (as in, the ML programming language) environment that Isabelle uses. These days this is Poly/ML, but with a custom library and some additions like antiquotations (a kind of macro system).
Isabelle/Scala is the part of Isabelle that is written in Scala. This handles the "plumbing" in the background such as building sessions in batch, communicating with Isabelle/jEdit, etc. whereas Isabelle/ML is where all the logic happens (manipulation of terms and theorems etc.)
Isabelle/PIDE is the ‘Isabelle Prover IDE framework’. I honestly don't really know what it actually does, but as far as I am aware it is a kind of protocol that is used by the Isabelle/ML process to communicate with e.g. IDEs such as Isabelle/jEdit and VSCode, i.e. it enables interactive use of Isabelle (as opposed to batch use with isabelle build). This includes things such as giving Isabelle a bit of code to process, or telling it that some code has been deleted and should be undone, and displaying information such as type information, error messages, which parts of the theory have already been processed etc.
Can Isabelle help to prove the correctness of the Raft or Paxos consensus algorithm? Is there a formal proof somewhere?
This entry in the Archive of Formal Proofs might be useful: https://www.isa-afp.org/entries/DiskPaxos.html
Isabelle/HOL (and similar theorem provers) are well-fit (and regularly used) for the verification of distributed protocols like Raft and Paxos.
Usually, the formalization will be about the systems model (messages or shared state, faults?), an inductive definition of the state space, a description of the temporal correctness properties and the relevant (inductive) invariants to prove the properties.
A tutorial video can be found here: https://www.youtube.com/watch?v=Uav5jWHNghY
As lsf37 has pointed out, there is a verification of Disk Paxos by
Jaskelioff and Merz in the Archive of Formal Proofs from 2005.
An Isabelle/HOL formalization of classic Paxos has been done by Küfner et al. in “Formal Verification of Distributed Algorithms: From Pseudo Code to Checked Proofs“ (2012, more details in Küfner's dissertation.)
Moreover, there is an Isabelle proof of the theoretical impossibility of consensus in the asynchronous setting with one faulty process (the so-called “FLP result”) by Bisping et al. from 2016.
For Raft, there is a Coq/Verdi formalization
I need to do a presentation on a paper which at some point makes use of Isabelle/Isar and Isabelle/HOL.
I tried researching online about Isabelle/HOL and Isabelle/Isar to be able to eplain the relations in one or two slides.
Here are the relations as I currently understand them:
Isabelle - provides a generic infrastructure for deductive systems
Based on Standard ML programming language
provides an IDE which allows you to write theories which can be later be proved.
Isabelle/Pure - minimal version of higher-order logic according to this link:
Is it an actual language that can be inputted into isabelle IDE?
Or is it a technical specification?
Isabelle/HOL(Higher Order Logic):
Is it a library or a language?
How does it relate to Isabelle/Pure?
Is it procedural in nature?
Do tactics only exist in Isabelle/HOL?
Is it LCF - Logical Commutable Functions?
Isabelle/Isar:
Structured proof language based on Isabelle/Pure
Declarative
Is it an extension of Isabelle/HOL as stated at here?
Do locales only exist in Isabelle/Isar?
What does the Isabelle/IDE supports by default?
Just feels like I'm getting conflicting information from different sources and would like to sort this out.
Thanks in advance
Edit - Check out this highly related question and Manuel Eberls answer here: What are all the isabelle/slashes?
As this is an answer to a homework question and I myself only have limited understanding of all parts of the Isabelle project, this answer merely tries to point you in the right direction for at some parts of your question.
From the Isabelle/ISAR reference manual:
The Isabelle system essentially provides a generic infrastructure for building deductive systems (programmed in Standard ML), with a special focus on interactive theorem proving in higher-order logics.
It continues to also introduce ISAR:
In contrast Isar provides an interpreted language environment of its own,
which has been specifically tailored for the needs of theory and proof development.
[...]
The main concern of Isar is the design of a human-readable structured proof
language
Let's try to connect Pure to all of this by looking at publications from Makarius Wenzel regarding the topic:
Thus Isar proof texts may be understood as structured compositions of formal entities of the Pure framework, namely propositions, facts, and goals
In colloquial terms, Pure is the semantic foundation. Isar is a language that "follows" this semantic and provides syntax for it. Isabelle is just (one of the) platforms it all runs on.
Some of your confusions around the distinction between Pure and Isar seem to stem from the fact that the Isabelle Pure source code defines, or at least seems to define, both the semantics (Pure) and the syntax (Isar) in one go:
(* The Pure theory, with definitions of Isar commands and some lemmas. *)
In my humble opinion, this might be related to your understanding of syntax, semantics and "implementations" of the two. "Pure" outside of computers or paper is just semantics and thus, like math, just a thing in our brains. Give it syntax and you can put it to paper or type it into a machine. For the machine to be able to process your text (since this is ultimately what we after), it needs an implementation. Some framework telling it how to read the syntax and how to then process it. This framework is Isabelle. On top of Isabelle, there is Isabelle/Pure, which defines the semantics (the processing) and Isabelle/Isar, which defines syntax. For practical reasons, Isabelle's Pure implementation already provides the Isar syntax in one go.
From all of this, you might be able to figure HOL out yourself!
Some more references:
The Isabelle/Isar Implementation
I have some definitions and theorems in Isabelle/HOL and need to use those same definitions and theorems with HOL. Translating the code manually is certainly possible, but cumbersome. Are there any programs that (semi-)automatically perform such a translation?
If this is not possible for some reason, please explain why, since this would be an important learning for me.
In theory, you should be able to easily move theorems and definitions between HOL implementations, and this idea is the motivation of the OpenTheory project. Unfortunately, in practice, Isabelle's implementation of HOL is sufficiently different from the others, and per the OpenTheory page Isabelle only has the ability to import theorems using OpenTheory, and cannot export them, at present.
I have looked into the Isabelle tutorial which presents an example of it's use in verifying security protocol. However, it is a bit out of my understanding as I only know the basics. I'm looking for some examples which are not just simple theorems but practical applications using Isabelle/HOL.
For example proving some algorithms or may be verifying properties system or some non-trivial mathematical theorem. Are such examples available anywhere ?
I have looked into the list of all applications provided in the isabelle official page but most of them are proofs of theorems.
I am also looking at an example of a file system verification using Alloy. It provides a proof where the properties of file/directories can be verified. I'm looking for something similar to it.
A few highly non-trivial examples I can think of right now are:
seL4, an entire operating system kernel written in C that was verified with Isabelle.
The AFP entry Jinja_Threads contains, as far as I know, a fully formalised bytecode compiler for a Java-like language with arrays and threads.
Jeremy Avigad's proof of the Prime Number Theorem.
The proof of Kepler's conjecture. A part of this was done in Isabelle; most of it, however, was done in the more ‘basic’ theorem prover HOL Light, whose logic is similar to Isabelle.
As Joachim mentioned, I am sure you can find more interesting applications in the AFP