diff --git a/README.md b/README.md
index 25e6008786f7ed95be7921573cbb613f2d306e01..f8088eee1f48c9e644cc9aa09f270f9c2cbb6d7c 100644
--- a/README.md
+++ b/README.md
@@ -19,7 +19,7 @@ v2.1 or higher, which is typically avaible in all major GNU/Linux distributions.
To install CAISAR via [opam](https://opam.ocaml.org/), do the following:
```
-$ opam install caisar
+opam install caisar
```
### Docker image
@@ -28,20 +28,20 @@ A ready-to-use [Docker](https://www.docker.com/) image of CAISAR is available on
[Docker Hub](https://hub.docker.com). To retrieve such an image, do the
following:
```
-$ docker pull laiser/caisar:pub
+docker pull laiser/caisar:pub
```
Alternatively, a [Docker](https://www.docker.com/) image for CAISAR can be
created locally by proceeding as follows:
```
-$ git clone https://git.frama-c.com/pub/caisar
-$ cd caisar
-$ make docker
+git clone https://git.frama-c.com/pub/caisar
+cd caisar
+make docker
```
To run the CAISAR [Docker](https://www.docker.com/) image, do the following:
```
-$ docker run -it laiser/caisar:pub sh
+docker run -it laiser/caisar:pub sh
```
### From source code
@@ -52,24 +52,24 @@ v2.1 or higher, which is typically avaible in all major GNU/Linux distributions.
To build and install CAISAR, do the following:
```
-$ git clone https://git.frama-c.com/pub/caisar
-$ cd caisar
-$ opam switch create --yes --no-install . 4.13.1
-$ opam install . --deps-only --with-test --yes
-$ make
-$ make install
+git clone https://git.frama-c.com/pub/caisar
+cd caisar
+opam switch create --yes --no-install . 4.13.1
+opam install . --deps-only --with-test --yes
+make
+make install
```
To run the tests:
```
-$ make test
+make test
```
## Usage
To start using CAISAR, please run the command:
```
-$ caisar --help
+caisar --help
```
### Property verification
@@ -79,7 +79,7 @@ machines (SVM).
The prototype command is:
```
-$ caisar verify --prover=PROVER FILE
+caisar verify --prover=PROVER FILE
```
`FILE` defines the property to verify, and it must be
@@ -97,7 +97,7 @@ path to the prover executables should appear in the environment variable
Run the following command to confirm that CAISAR detects the installed provers:
```
-$ caisar config --detect
+caisar config --detect
```
The following are the provers for which a support is provided in CAISAR:
diff --git a/config/caisar-detection-data.conf b/config/caisar-detection-data.conf
index 054b3d8ac73ff597af4afb95a406b080b65fb986..78e3f18272d8aedfe7910c5998b4d110d6bfece7 100644
--- a/config/caisar-detection-data.conf
+++ b/config/caisar-detection-data.conf
@@ -74,6 +74,17 @@ command = "%e -mp %{nnet-onnx} -pp %f --timeout %t --domain zono poly --split"
driver = "caisar_drivers/pyrat_vnnlib.drv"
use_at_auto_level = 1
+[ATP pyrat-acas]
+name = "PyRAT"
+alternative = "ACAS"
+exec = "pyrat.py"
+version_switch = "--version"
+version_regexp = "PyRAT \\([0-9.]+\\)"
+version_ok = "1.1"
+command = "%e -mp %{nnet-onnx} -pp %f --timeout %t --domain zono --split --scorer coef"
+driver = "caisar_drivers/pyrat.drv"
+use_at_auto_level = 1
+
[ATP nnenum]
name = "nnenum"
exec = "nnenum.sh"
diff --git a/doc/_static/media/acas_xu.png b/doc/_static/media/acas_xu.png
new file mode 100644
index 0000000000000000000000000000000000000000..d6de240a4a6fc470fc6a321de10ecc4a2a25ba17
Binary files /dev/null and b/doc/_static/media/acas_xu.png differ
diff --git a/doc/acas_xu.rst b/doc/acas_xu.rst
new file mode 100644
index 0000000000000000000000000000000000000000..1176bc4936c0ecbe50b80afc77a53f428a07a0c2
--- /dev/null
+++ b/doc/acas_xu.rst
@@ -0,0 +1,352 @@
+.. _acas_xu:
+
+Functional properties of ACAS-Xu
+********************************
+
+ACAS-Xu stands for Aircraft Collision Avoidance System. Introduced for instance
+in [Manfredi2016]_, it is a specification of a program which aim to output
+signals for an aircraft in a situation where there is a potential for collision.
+In the rest of this tutorial, we will use the flavour ACAS-Xu defined in
+[Katz2017]_, where the authors aim to verify a neural network implementing part
+of the ACAS-Xu specification. Its low dimensionality and well defined semantics
+make it a *de facto* benchmark for machine learning verification.
+
+Use case presentation
+=====================
+
+The system considers a 2D plane with two entities: the monitored airplane (the
+"ownship") and an incoming airplane (the "intruder").
+
+.. image:: _static/media/acas_xu.png
+ :alt: A picture with two aircrafts seen from above
+
+In the original implementation, the system state has seven inputs:
+
+* :math:`v_{own}`: speed of ownship
+* :math:`v_{int}`: speed of intruder
+* :math:`\rho`: distance from ownship to intruder
+* :math:`\theta`: angle to intruder relative to ownship heading direction
+* :math:`\psi`: heading angle of intruder relative to ownship heading direction
+* :math:`\tau`: time until vertical separation
+* :math:`a_{prev}`: previous advisory
+
+It has five outputs, that correspond to the different direction advisories the
+system can give:
+
+* :math:`COC`: Clear Of Conflict
+* :math:`WL`: Weak Left
+* :math:`SL`: Strong Left
+* :math:`WR`: Weak Right
+* :math:`SR`: Strong Right
+
+In the paper, the authors consider :math:`45` neural networks, for several
+values of :math:`\tau` and :math:`a_{prev}`, that operate on five inputs only
+while maintaining the same number of outputs. We will consider five-inputs
+networks in the future.
+
+Properties
+----------
+
+There are several functional properties one may want to verify on this system,
+for instance:
+
+* Guarantee that the system will never output COC advisory when the intruder is
+ nearby,
+* Guarantee that the system will never output an advisory that may result in a
+ collision,
+* Guarantee that the system will not output a strong advisory where a weak
+ variant would be enough.
+
+Authors of [Katz2017]_ propose ten properties to verify. We will reproduce the
+first and third properties here, and then show how to use CAISAR for verifying
+whether a given neural network respects them.
+
+**Property** :math:`\phi_1`
+
+* **Definition.**
+ If the intruder is distant and is significantly slower than
+ the ownship, the score of a COC advisory will always be below a certain fixed
+ threshold.
+
+* **Input constraints:**
+
+ * :math:`\rho \geq 55947.691`,
+ * :math:`v_{own} \geq 1145`,
+ * :math:`v_{int} \leq 60`.
+
+* **Desired output property:**
+
+ * :math:`COC \leq 1500`.
+
+**Property** :math:`\phi_3`
+
+* **Definition.**
+ If the intruder is directly ahead and is moving towards the
+ ownship, the score for COC will not be minimal.
+
+* **Input constraints:**
+
+ * :math:`1500 \leq \rho \leq 1800`,
+ * :math:`-0.06 \leq \theta \leq 0.06`,
+ * :math:`\psi \geq 3.10`,
+ * :math:`v_{own} \geq 980`,
+ * :math:`v_{int} \geq 960`.
+
+* **Desired output property:**
+
+ * :math:`COC` is not the minimal score.
+
+Modelling the problem using WhyML
+---------------------------------
+
+The first step for verifying anything with CAISAR is to write a specification
+file that describe the problem to verify as a so-called *theory*. A theory can
+be seen as a namespace inside which are defined logical terms, formulas and
+verification goals. In particular, being based on the `Why3
+<https://why3.lri.fr>`_ platform for deductive program verification, CAISAR
+supports the Why3 specification language
+`WhyML <https://why3.lri.fr/doc/syntaxref.html>`_, and inherits the Why3 standard
+library of logical theories (integer, float and real arithmetic, *etc.*) and
+basic programming data structures (arrays, queues, hash tables, *etc.*).
+
+Let us try to model the property :math:`\phi_1` defined earlier. We will call
+our theory ``ACASXU_P1``.
+
+.. code-block:: whyml
+
+ theory ACASXU_P1
+ end
+
+We will need to write down some numerical values. As of now, CAISAR allows
+writing values using floating-point arithmetic only. Why3 defines a float type
+and the relevant arithmetic operations according to the IEEE floating-point
+standard in a theory, astutely called ``ieee_float``. Specifically, we will
+import the ``Float64`` subtheory, that defines everything we need for 64-bit
+precision floating-point numbers. We thus import it in our theory using the
+``use`` keyword.
+
+Our file looks like this so far:
+
+.. code-block:: whyml
+
+ theory ACASXU_P1
+ use ieee_float.Float64
+ end
+
+We would like to verify our propety given a certain neural network. To do this,
+CAISAR extends the Why3 standard library for recognizing neural networks in ONNX
+and NNet formats. Given a file of such formats, CAISAR internally builds a
+theory named as the original neural network file, that contains the subtheories
+``AsTuple`` and ``AsArray`` that provide logical symbols for describing the
+input-output interface of a neural network as tuples and array, respectively. We
+will only consider the ``AsTuple`` subtheory for this tutorial.
+
+In particular, the theory built by CAISAR is equivalent to the following WhyML
+file:
+
+.. code-block:: whyml
+
+ theory NeuralNetworkFilename
+ theory AsTuple
+ type t
+ (* Tuple with as many elements as there are input *)
+ function nn_apply (t,_,...)
+ (* Tuple with as many elements as there are outputs *)
+ : (t,_,...)
+ end
+ (* Other stuff *)
+ end
+
+Note how the ``AsTuple`` theory defines the ``nn_apply`` function symbol that
+logically describes the input-output interface of a neural network using tuples.
+More importantly, CAISAR defines this function to take in input a tuple with as
+many elements as the inputs expected by the original neural network, and return
+a tuple with as many elements as the outputs provided by the original neural
+network.
+
+As our neural network takes five inputs and provides five outputs, adding ``use
+filename.AsTuple`` to our theory will provide a ``nn_apply`` function symbol
+that takes a 5-elements tuple as input, and provides a 5-elements tuple as
+output. Assuming we have a neural network named ``ACASXU_1_1.onnx``, the WhyML
+file looks like this:
+
+.. code-block:: whyml
+
+ theory ACASXU_P1
+ use ACASXU_1_1.AsTuple
+ use ieee_float.Float64
+ end
+
+Now is the time to define our verification goal, that will call ``P1_1_1`` for
+property :math:`\phi_1` on neural network :math:`N_{1,1}`.
+
+We first model the inputs of the neural network :math:`\rho, \theta, \psi,
+v_{own}, v_{int}` respectively as the floating-points constants :math:`x_i` for
+:math:`i \in [0..4]`. Moreover, we constraint these to the range of
+floating-point values each may take. According to the original authors, values
+were normalized during the training of the network, and so we adapt the values
+they provide in their `repository
+<https://github.com/NeuralNetworkVerification/Marabou/tree/master/resources/properties>`_.
+Then, we define the result of the application of ``net_apply`` on the inputs by
+taking advantage of the WhyML pattern-matching, and define the output constraint
+we want to enforce on the floating-point constant :math:`y_0` that we use to
+model the advisory *COC*.
+
+The final WhyML file looks like this:
+
+.. code-block:: whyml
+
+ theory ACASXU_P1
+ use ACASXU_1_1.AsTuple
+ use ieee_float.Float64
+
+ goal P1_1_1: forall x0 x1 x2 x3 x4.
+ (0.5999999999999999777955395074968691915273666381835937500000000000:t) .<= x0 .<= (0.6798577687000000313588543576770462095737457275390625000000000000:t) ->
+ (-0.5:t) .<= x1 .<= (0.5:t) ->
+ (-0.5:t) .<= x2 .<= (0.5:t) ->
+ (0.4500000000000000111022302462515654042363166809082031250000000000:t) .<= x3 .<= (0.5:t) ->
+ (-0.5:t) .<= x4 .<= (-0.4500000000000000111022302462515654042363166809082031250000000000:t) ->
+ let (y0, _, _, _, _) = nn_apply x0 x1 x2 x3 x4 in
+ y0 .<= (3.9911256458999999630066213285317644476890563964843750000000000000:t)
+ end
+
+This file is available, as is, under the ``/examples/acasxu/`` folder as
+`property_1.why
+<https://git.frama-c.com/pub/caisar/-/blob/master/examples/acasxu/property_1.why>`_.
+The corresponding neural network in ONNX format is available under the
+``/examples/acasxu/nets/onnx/`` folder as `ACASXU_1_1.onnx
+<https://git.frama-c.com/pub/caisar/-/blob/master/examples/acasxu/nets/onnx/ACASXU_1_1.onnx>`_.
+
+
+Verifying the property with CAISAR
+----------------------------------
+
+Once formalized, the specified property can be assessed by using CAISAR. We will
+use the *open-source* provers CAISAR supports for verifying properties of neural
+networks so to take advantage of the federating approach: whenever one prover
+cannot provide an answer, another may instead. In particular, we will use
+`Marabou <https://github.com/NeuralNetworkVerification/Marabou>`_ and `nnenum
+<https://github.com/stanleybak/nnenum>`_.
+
+Assuming the prover locations are available in ``PATH``, the following are the
+CAISAR verification invocations using Marabou first and nnenum afterwords, for
+verifying the ACAS-Xu property :math:`\phi_1`:
+
+.. code-block:: console
+
+ $ caisar verify --prover Marabou -L examples/acasxu/nets/onnx --format whyml examples/acasxu/property_1.why -t 10m
+ [caisar] Goal P1_1_1: Timeout
+
+.. code-block:: console
+
+ $ caisar verify --prover nnenum -L examples/acasxu/nets/onnx --format whyml examples/acasxu/property_1.why -t 10m
+ [caisar] Goal P1_1_1: Valid
+
+Note that the previous commands set the verification time limit to 10 minutes
+(*cf.* ``-t`` option), and the additional location ``examples/acasxu/nets/onnx``
+(*cf.* ``-L`` option) for letting CAISAR correctly locate the neural network
+file ``ACASXU_1_1.onnx`` that is used by the ``ACASXU_P1`` theory in
+``property_1.why``.
+
+Marabou is not able to prove the property valid in the specified time limit,
+while nnenum does. In general, the result of a CAISAR verification is typically
+either ``Valid``, ``Invalid``, ``Unknown`` or ``Timeout``. CAISAR may output
+``Failure`` whenever the verification process fails for whatever reason
+(typically, a prover internal failure).
+
+Using more advanced WhyML constructs
+------------------------------------
+
+Let us model the ACAS-Xu property :math:`\phi_3`, and verify it for the neural
+networks :math:`N_{1,1}` and :math:`N_{2,7}` [Katz2017]_.
+
+From the modelling standpoint, the main evident difference concerns the desired
+output property, meaining that *COC* should not be the minimal value. A
+straightforward way to express this property is that the corresponding
+floating-point constant :math:`y_0` is greater than or equal to at least one of
+the other five outputs. This can be formalized in first-order logic as a
+disjunction of clauses, that can be directly encoded into WhyML as follows:
+
+.. code-block:: whyml
+
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
+
+The delicate point is how to model the same property for two different neural
+networks. Of course, we could define a theory with two identical but distinct
+verification goals or two entirely distinct theories in a same WhyML file.
+However, these two solutions are not advisable in terms of clarity and
+maintainability.
+
+Reassuringly enough, WhyML provides all necessary features to come up with a
+better solution. First, WhyML allows for naming used (sub)theories in order to
+distinguish identical logic symbols coming from different theories. This is
+critical for identifying the correct ``nn_apply`` symbols in the two
+verification goals we will define. Second, WhyML allows for the hypotheses on
+the floating-point constants modelling the neural network inputs to be exported
+from the verification goal into the theory general context as axioms.
+
+In the end, the WhyML file looks like this:
+
+.. code-block:: whyml
+
+ theory ACASXU_P3
+ use ACASXU_1_1.AsTuple as N11
+ use ACASXU_2_7.AsTuple as N27
+ use ieee_float.Float64
+
+ constant x0:t
+ constant x1:t
+ constant x2:t
+ constant x3:t
+ constant x4:t
+
+ axiom H0:
+ (-0.3035311560999999769272506000561406835913658142089843750000000000:t) .<= x0 .<= (-0.2985528118999999924731980627257144078612327575683593750000000000:t)
+
+ axiom H1:
+ (-0.0095492965999999998572000947660853853449225425720214843750000000:t) .<= x1 .<= (0.0095492965999999998572000947660853853449225425720214843750000000:t)
+
+ axiom H2:
+ (0.4933803236000000036476365039561642333865165710449218750000000000:t) .<= x2 .<= (0.5:t)
+
+ axiom H3:
+ (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x3 .<= (0.5:t)
+
+ axiom H4:
+ (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x4 .<= (0.5:t)
+
+ goal P3_1_1:
+ let (y0, y1, y2, y3, y4) = N11.nn_apply x0 x1 x2 x3 x4 in
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
+
+ goal P3_2_7:
+ let (y0, y1, y2, y3, y4) = N27.nn_apply x0 x1 x2 x3 x4 in
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
+ end
+
+Note how the two verification goals ``P3_1_1`` and ``P3_2_7`` are clearly almost
+identical, but for the ``nn_apply`` logic symbol used, identifying respectively
+the ``ACASXU_1_1.onnx`` and ``ACASXU_2_7.onnx`` neural networks.
+
+We can then verify the resulting verification problem as before:
+
+.. code-block:: console
+
+ $ caisar verify --prover Marabou -L examples/acasxu/nets/onnx --format whyml examples/acasxu/property_3.why -t 10m
+ [caisar] Goal P3_1_1: Timeout
+ [caisar] Goal P3_2_7: Valid
+
+.. code-block:: console
+
+ $ caisar verify --prover nnenum -L examples/acasxu/nets/onnx --format whyml examples/acasxu/property_3.why -t 10m
+ [caisar] Goal P3_1_1: Valid
+ [caisar] Goal P3_2_7: Valid
+
+
+.. [Manfredi2016] G. Manfredi and Y. Jestin, *An introduction to ACAS Xu and the
+ challenges ahead*, 2016 IEEE/AIAA 35th Digital Avionics Systems Conference
+ (DASC), 2016, pp. 1-9, doi: 10.1109/DASC.2016.7778055
+
+.. [Katz2017] Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.
+ (2017). *Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks.*
+ CAV 2017, doi: 10.1007/978-3-319-63387-9_5
diff --git a/doc/conf.py b/doc/conf.py
index 1eff397760cee828038f42fcb8bb3c778f106bac..42a6c97be5114c72b61c4f133852410c8dfac530 100644
--- a/doc/conf.py
+++ b/doc/conf.py
@@ -14,10 +14,13 @@
# import sys
# sys.path.insert(0, os.path.abspath('.'))
+import os
+import sys
+sys.path.append(os.path.abspath('./ext'))
# -- Project information -----------------------------------------------------
-project = 'Caisar'
+project = 'CAISAR'
copyright = '2022, Michele Alberti, Julien Girard, François Bobot'
author = 'Michele Alberti, Julien Girard, François Bobot'
@@ -28,6 +31,10 @@ author = 'Michele Alberti, Julien Girard, François Bobot'
# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom
# ones.
extensions = [
+ 'sphinx.ext.mathjax',
+ 'sphinx.ext.graphviz',
+ # 'sphinxcontrib.bibtex',
+ 'why3'
]
# Add any paths that contain templates here, relative to this directory.
@@ -49,4 +56,4 @@ html_theme = 'alabaster'
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
-html_static_path = ['_static']
\ No newline at end of file
+html_static_path = ['_static']
diff --git a/doc/contents.rst b/doc/contents.rst
deleted file mode 100644
index daa46db69729e3f5b7d11b30c01926ca8b00c326..0000000000000000000000000000000000000000
--- a/doc/contents.rst
+++ /dev/null
@@ -1,10 +0,0 @@
-Table of content
-================
-
-.. toctree::
- :maxdepth: 2
- :caption: Contents:
-
- installation
- usage
- modules
diff --git a/doc/examples.rst b/doc/examples.rst
new file mode 100644
index 0000000000000000000000000000000000000000..62eb15e90e224d26c656510cdae85c4d234bb4e6
--- /dev/null
+++ b/doc/examples.rst
@@ -0,0 +1,15 @@
+.. _examples:
+
+CAISAR by Examples
+==================
+
+This page regroups some example use cases for CAISAR. All examples will describe
+the use case, the formalization using the WhyML specification language, and the
+CAISAR execution.
+
+.. toctree::
+ :maxdepth: 2
+ :caption: CAISAR by Examples:
+
+ acas_xu
+ mnist
diff --git a/doc/ext/why3.py b/doc/ext/why3.py
new file mode 100644
index 0000000000000000000000000000000000000000..29a8fd889715a39a543d0e40d454616247e870c0
--- /dev/null
+++ b/doc/ext/why3.py
@@ -0,0 +1,199 @@
+from pygments.lexer import RegexLexer, words
+from pygments.token import Text, Comment, Operator, Keyword, Name, String, Number, Punctuation, Error
+
+class WhyMLLexer(RegexLexer):
+ name = 'WhyML'
+ aliases = 'whyml'
+
+ keywords = (
+ 'abstract', 'absurd', 'alias', 'any', 'as', 'assert', 'assume', 'at', 'axiom',
+ 'begin', 'break', 'by', 'check', 'clone', 'coinductive', 'constant', 'continue',
+ 'diverges', 'do', 'done', 'downto',
+ 'else', 'end', 'ensures', 'epsilon', 'exception', 'exists', 'export',
+ 'false', 'float', 'for', 'forall', 'fun', 'function', 'ghost', 'goal',
+ 'if', 'import', 'in', 'inductive', 'invariant', 'label', 'lemma', 'let',
+ 'match', 'meta', 'module', 'mutable', 'not', 'old',
+ 'partial', 'predicate', 'private', 'pure',
+ 'raise', 'raises', 'range', 'reads', 'rec', 'ref', 'requires', 'return', 'returns',
+ 'scope', 'so', 'then', 'theory', 'to', 'true', 'try', 'type', 'use', 'val', 'variant',
+ 'while', 'with', 'writes',
+ )
+
+ tokens = {
+ 'root': [
+ (r'\s+', Text),
+ (r'\(\*\)', Operator),
+ (r'\(\*', Comment, 'comment'),
+ (r'\[@[^]]*\]', Comment),
+ (words(keywords, suffix=r'\b'), Keyword),
+ (r'[-~!%^&*+=|?<>/\\]', Operator),
+ (r'[][{};:.,()]', Punctuation),
+ (r"[^\W\d][\w']*", Name),
+ (r'\bresult\b', Name.Builtin.Pseudo),
+
+ (r'-?\d\d*([.]\d*)?([eE][+-]?\d\d*)', Number.Float),
+ (r'-?0[xX][\da-fA-F][\da-fA-F]*([.][\da-fA-F]*)?([pP][+-]?\d\d*)', Number.Float),
+ (r'0[xX][\da-fA-F][\da-fA-F_]*', Number.Hex),
+ (r'0[oO][0-7][0-7_]*', Number.Oct),
+ (r'0[bB][01][01_]*', Number.Bin),
+ (r'\d[\d_]*', Number.Integer),
+
+ (r"'", Keyword),
+ (r'"', String.Double, 'string'),
+ ],
+ 'comment': [
+ (r'[^(*)]+', Comment),
+ (r'\(\*', Comment, '#push'),
+ (r'\*\)', Comment, '#pop'),
+ (r'[(*)]', Comment),
+ ],
+ 'string': [
+ (r'[^\\"]+', String.Double),
+ (r'\\[\\"\'ntbr]', String.Escape),
+ (r'\\[0-9]{3}', String.Escape),
+ (r'\\x[0-9a-fA-F]{2}', String.Escape),
+ (r'\\\n', String.Double),
+ (r'"', String.Double, '#pop'),
+ ],
+ }
+
+from sphinx.highlighting import lexers
+
+lexers['whyml'] = WhyMLLexer(startinline=True)
+
+import re
+from docutils import nodes
+from sphinx import addnodes
+from sphinx.directives import ObjectDescription
+from sphinx.domains import Domain
+from sphinx.roles import XRefRole
+from sphinx.util.docutils import SphinxDirective
+from sphinx.util.nodes import make_refnode
+
+ws_re = re.compile(r'\s+')
+
+class Why3ToolRole(XRefRole):
+ def process_link(self, env, refnode, has_explicit_title, title, target):
+ target = ws_re.sub(' ', target)
+ if target.startswith('why3 '):
+ target = target[5:]
+ return title, target
+
+attr_re = re.compile(r'\[@(.*)\]')
+
+class Why3AttributeRole(XRefRole):
+ def process_link(self, env, refnode, has_explicit_title, title, target):
+ target = ws_re.sub(' ', target)
+ m = attr_re.match(target)
+ if m:
+ target = m.group(1)
+ return title, target
+
+class Why3ToolDirective(SphinxDirective):
+ has_content = False
+ required_arguments = 1
+ optional_arguments = 0
+ final_argument_whitespace = True
+ option_spec = {}
+
+ def run(self):
+ fullname = ws_re.sub(' ', self.arguments[0].strip())
+ targetname = '%s-%s' % (self.name, fullname)
+ node = nodes.target('', '', ids = [targetname])
+ self.state.document.note_explicit_target(node)
+ indexentry = '%s; command' % (fullname,)
+ inode = addnodes.index(entries = [('pair', indexentry, targetname, '', None)])
+ domain = self.env.get_domain('why3')
+ domain.add_object('tool', fullname, targetname)
+ return [inode, node]
+
+class Why3Thing(ObjectDescription):
+ has_content = True
+ required_arguments = 1
+ optional_arguments = 0
+ final_argument_whitespace = True
+ option_spec = {}
+ thing_index = ''
+ thing_kind = ''
+
+ def handle_signature(self, sig, signode):
+ signode += addnodes.desc_name(text = sig)
+ return sig
+
+ def add_target_and_index(self, name, sig, signode):
+ targetname = '%s-%s' % (self.name, name)
+ signode['ids'].append(targetname)
+ indexentry = '%s; %s' % (name, self.thing_index)
+ self.indexnode['entries'].append(('pair', indexentry, targetname, '', None))
+ domain = self.env.get_domain('why3')
+ domain.add_object(self.thing_kind, name, targetname)
+
+class Why3Attribute(Why3Thing):
+ thing_index = 'attribute'
+ thing_kind = 'attribute'
+
+class Why3Debug(Why3Thing):
+ thing_index = 'debug flag'
+ thing_kind = 'debug'
+
+class Why3Meta(Why3Thing):
+ thing_index = 'meta'
+ thing_kind = 'meta'
+
+class Why3Transform(Why3Thing):
+ thing_index = 'transformation'
+ thing_kind = 'transform'
+
+class Why3Domain(Domain):
+ name = 'why3'
+ label = 'Why3'
+ roles = {
+ 'attribute': Why3AttributeRole(),
+ 'debug': XRefRole(),
+ 'meta': XRefRole(),
+ 'tool': Why3ToolRole(),
+ 'transform': XRefRole(),
+ }
+ directives = {
+ 'attribute': Why3Attribute,
+ 'debug': Why3Debug,
+ 'meta': Why3Meta,
+ 'tool': Why3ToolDirective,
+ 'transform': Why3Transform,
+ }
+ indices = {
+ }
+ initial_data = {}
+ initial_data['objects'] = {
+ 'attribute': {},
+ 'debug': {},
+ 'meta': {},
+ 'tool': {},
+ 'transform': {},
+ }
+
+ def get_objects(self):
+ for role, objects in self.data['objects'].items():
+ prio = 0 # self.object_types[role].attrs['searchprio']
+ for name, (docname, targetname) in objects.items():
+ yield (name, name, role, docname, targetname, prio)
+
+ def resolve_xref(self, env, fromdocname, builder, role, targetname, node, contnode):
+ resolved = self.data['objects'][role].get(targetname)
+ if resolved:
+ (todocname, targetname) = resolved
+ return make_refnode(builder, fromdocname, todocname, targetname, contnode, targetname)
+ return None
+
+ def add_object(self, role, name, targetname):
+ self.data['objects'][role][name] = (self.env.docname, targetname)
+
+
+def setup(app):
+ app.add_domain(Why3Domain)
+
+ return {
+ 'version': '0.1',
+ 'parallel_read_safe': True,
+ 'parallel_write_safe': True,
+ }
diff --git a/doc/index.rst b/doc/index.rst
index 7a26e02ada6f54db0dce86889e8a1ab5b1a7fa87..f042f054973abf8bb06115866019e767ecaefccd 100644
--- a/doc/index.rst
+++ b/doc/index.rst
@@ -21,16 +21,15 @@ Content
=======
.. toctree::
- :maxdepth: 2
- :caption: Contents:
+ :maxdepth: 3
+ :caption: Contents
installation
usage
- modules
+ examples
-
-Indices and tables
-==================
-
-* :ref:`search`
+..
+ Indices and tables
+ ==================
+ * :ref:`search`
diff --git a/doc/installation.rst b/doc/installation.rst
index be23b34be19e86657fca6d1fa7516221d758ffd1..ca8c5117a6a9a0d7d125a1fdb33e5450cb1c8269 100644
--- a/doc/installation.rst
+++ b/doc/installation.rst
@@ -18,7 +18,9 @@ GNU/Linux distributions.
To install CAISAR via opam, do the following:
-``$ opam install caisar``
+.. code-block:: console
+
+ $ opam install caisar
Install through Docker
----------------------
@@ -28,33 +30,41 @@ A ready-to-use Docker image of CAISAR is available on
`Docker Hub <https://hub.docker.com>`_. To retrieve it, do the
following:
-``$ docker pull laiser/caisar:pub``
+.. code-block:: console
+
+ $ docker pull laiser/caisar:pub
Alternatively, a Docker image for CAISAR can be
created locally by proceeding as follows:
-::
+.. code-block:: console
-$ git clone https://git.frama-c.com/pub/caisar
-$ cd caisar
-$ make docker
+ $ git clone https://git.frama-c.com/pub/caisar
+ $ cd caisar
+ $ make docker
To run the CAISAR Docker image, do the following:
-``$ docker run -it laiser/caisar:pub sh``
+
+.. code-block:: console
+
+ $ docker run -it laiser/caisar:pub sh
Compile from source
-------------------
To build and install CAISAR, do the following:
-::
+.. code-block:: console
-$ git clone https://git.frama-c.com/pub/caisar
-$ cd caisar
-$ opam switch create --yes --no-install . 4.13.1
-$ opam install . --deps-only --with-test --yes
-$ make
-$ make install
+ $ git clone https://git.frama-c.com/pub/caisar
+ $ cd caisar
+ $ opam switch create --yes --no-install . 4.13.1
+ $ opam install . --deps-only --with-test --yes
+ $ make
+ $ make install
To run the tests:
-``$ make test``
+
+.. code-block:: console
+
+ $ make test
diff --git a/doc/mnist.rst b/doc/mnist.rst
new file mode 100644
index 0000000000000000000000000000000000000000..00d62cfb410c77725c22739c3dec54ad84b056c6
--- /dev/null
+++ b/doc/mnist.rst
@@ -0,0 +1,159 @@
+.. _mnist:
+
+(Local) Robustness of MNIST dataset
+***********************************
+
+CAISAR provides a convenient way for verifying (local) robustness properties of
+neural networks on datasets, for classification problems only, in a specific
+``CSV`` format. In particular, each of the ``CSV`` lines is interpreted as
+providing the classification label in the first column, and the dataset element
+features in the other columns.
+
+We recall that a neural network is deemed robust on a dataset element whenever
+it classify with a same label all other elements being at an
+:math:`l_\infty`-distance of at most :math:`\epsilon \geq 0` from it. More in
+general, a neural network is deemed (locally) robust on a dataset whenever the
+former property is valid on all the dataset elements. The CAISAR standard
+library specifies such a property in terms of the predicate ``robust``, which
+CAISAR implements as a builtin.
+
+In the following, we will describe how to use CAISAR for verifying a neural
+network robust on (a fragment of) the MNIST dataset.
+
+Use case presentation
+=====================
+
+MNIST is a dataset of handwritten digits normalized and centered to fit into
+grayscale images of :math:`28 \times 28` pixels, along with the classification
+labels [LiDeng2012]_. Although it is mostly irrelevant as dataset for
+benchmarking machine learning models for computer vision tasks, MNIST is still
+valuable for assessing robustness properties by means of formal method tools.
+
+CAISAR provides in `mnist_test.csv
+<https://git.frama-c.com/pub/caisar/-/blob/master/examples/mnist/csv/mnist_test.csv>`_
+a fragment (:math:`100` images) of the MNIST dataset under the
+``examples/mnist/csv`` folder. Each line in this file represents an MNIST image:
+in particular, the first column represents the classification label, and the
+remaining :math:`784` columns represent the greyscale value of the respective
+pixels.
+
+Properties
+----------
+
+Generally speaking, the property we are interested in verifying is the local
+robustness of a machine learning model on the elements of a set. That is, the
+model classifies all elements of a set being at an :math:`l_\infty`-distance of
+at most :math:`\epsilon \geq 0` with a same label. A general formulation of this
+latter states that, given a classifier :math:`C`, a set :math:`X`, and some
+perturbation :math:`\epsilon \geq 0`, it must hold that :math:`\forall x,x'
+\in X. \ \lVert x - x' \rVert_\infty \leq \epsilon \Rightarrow C(x) = C(x')`.
+
+Since we actually deal with a *dataset* of finite elements for which we also
+know the expected labels, we will instead verify a slightly different property:
+given a classifier :math:`C`, an element :math:`x \in X` such that :math:`C(x) =
+y`, and some perturbation :math:`\epsilon \geq 0`, it must hold that
+:math:`\forall x'. \ \lVert x - x' \rVert_\infty \leq \epsilon \Rightarrow C(x)
+= y = C(x')`. Obviously, such a property must be verified for all elements of a
+dataset.
+
+Modelling the problem using WhyML
+---------------------------------
+
+As described for the example on :ref:`acas_xu`, we first need to write a
+specification file containing a WhyML theory to describe the verification
+problem. In principle, we need to formalize the local robustness property as
+well as the notions of classifier and dataset.
+
+The CAISAR standard library `caisar.mlw
+<https://git.frama-c.com/pub/caisar/-/blob/master/stdlib/caisar.mlw>`_ provides
+some utilities for dealing with verification problems about classification
+datasets. Of particular interest for us is the ``robust`` predicate, defined in
+the theory ``DataSetProps`` as follows:
+
+.. code-block:: whyml
+
+ predicate robust (m: model) (d: dataset) (eps: t) =
+ forall i: int. 0 <= i < d.data.length -> robust_at m d.data[i] eps
+
+Note that the predicate is defined over a ``model``, a ``dataset`` and a
+floating-point value ``eps``. The latter determines the perturbation
+:math:`\epsilon`. The other two are custom WhyML types that respectively
+formalize the notions of classifier and dataset in CAISAR. These types are
+respectively defined in the ``Model`` and ``DataSetClassification`` theories.
+
+Moreover, it is defined in terms of the predicate ``robust_at`` that formalizes
+the local robustness property:
+
+.. code-block:: whyml
+
+ predicate robust_at (m: model) (d: datum) (eps: t) =
+ forall x': features.
+ let (x, _) = d in
+ linfty_distance x x' eps ->
+ predict m x = predict m x'
+
+Note that a ``datum`` is a dataset element given as a pair of *features* and
+(classification) *label*. Morever, ``linfty_distance`` is a predicate that
+describes how two arrays of floating-point values (*i.e.* ``features``) are
+considered close up-to a pertubation ``eps``, while ``predict`` is a function
+that formalizes the execution of a model on some features to obtain the
+corresponding classification label.
+
+In order to use the ``robust`` predicate in our WhyML specification, we need
+values of types ``model`` and ``dataset``. For the former, CAISAR makes
+available the constant ``model`` upon interpreting the ``AsArray`` subtheory
+that is built by the extension of the Why3 standard library for recognizing
+neural network ONNX and NNet formats. For the latter, the CAISAR standard
+library provides the constant ``dataset`` in ``DataSetClassification`` that will
+be interpreted as the actual dataset the user needs to provide via the
+command-line interface when launching a CAISAR verification.
+
+Assuming we have a neural network named ``MNIST_256_2.onnx`` for MNIST
+classification, the final WhyML file for specifying its local robustness on a
+dataset, with each element's feature pertubed by :math:`1 \%`, looks like this:
+
+.. code-block:: whyml
+
+ theory MNIST
+ use MNIST_256_2.AsArray
+ use ieee_float.Float64
+ use caisar.DataSetClassification
+ use caisar.DataSetProps
+
+ goal robustness:
+ let normalized_dataset = min_max_scale true (0.0:t) (1.0:t) dataset in
+ robust model normalized_dataset (0.0100000000000000002081668171172168513294309377670288085937500000:t)
+ end
+
+Note the presence of the ``min_max_scale`` function defined in
+``DataSetClassification`` for normalizing all feature values in :math:`[0,1]`.
+Besides classic *Min-Max scaling*, CAISAR also provides ``z_norm`` function for
+*Z-normalization*.
+
+This file is available, as is, under the ``/examples/mnist/`` folder as
+`property.why
+<https://git.frama-c.com/pub/caisar/-/blob/master/examples/mnist/property.why>`_.
+The corresponding neural network in ONNX format is available under the
+``/examples/mnist/nets/`` folder as `MNIST_256_2.onnx
+<https://git.frama-c.com/pub/caisar/-/blob/master/examples/mnist/nets/MNIST_256_2.onnx>`_.
+
+Verifying the property with CAISAR
+----------------------------------
+
+Now we may verify whether the previous robustness specification holds on the
+MNIST fragment ``mnist_test.csv`` by means of the nnenum prover. This can be
+done via CAISAR as follows:
+
+.. code-block:: console
+
+ $ caisar verify --prover nnenum -L examples/mnist/nets --format whyml --dataset=examples/mnist/csv/mnist_test.csv examples/mnist/property.why
+ [caisar] Goal robustness: Invalid
+
+The result tells us that there exists at least one image in ``mnist_test.csv``
+for which nnenum is sure that the model ``MNIST_256_2.onnx`` is not robust with
+respect to :math:`1 \%` perturbation. At the moment, CAISAR is not able to tell
+which are the images in the dataset that cause such result.
+
+.. [LiDeng2012] Li Deng, *The MNIST Database of Handwritten Digit Images for
+ Machine Learning Research*, IEEE Signal Process. Mag., 2012, pp.
+ 141-142, doi: 10.1109/MSP.2012.2211477
diff --git a/doc/usage.rst b/doc/usage.rst
index 07606ae4920c770072607e49a5038451dd342845..70cd0bbd3cbdde766435344e9e4343596a2bb551 100644
--- a/doc/usage.rst
+++ b/doc/usage.rst
@@ -7,10 +7,10 @@ Property verification
---------------------
CAISAR can be used to verify properties on neural networks.
-The prototype command is ``caisar verify --solver=SOLVER property.mlw``
-File property.mlw defines the goal you want to verify,
+The prototype command is ``caisar verify --prover=PROVER property.mlw``.
+File ``property.mlw`` defines the goal you want to verify,
written in the Whyml language.
-Example of .mlw files can be seen in tests folder.
+Example of ``.mlw`` files can be seen in tests folder.
External verifiers detection
----------------------------
diff --git a/examples/acasxu/nets/ACASXU_1_1.nnet b/examples/acasxu/nets/nnet/ACASXU_1_1.nnet
similarity index 100%
rename from examples/acasxu/nets/ACASXU_1_1.nnet
rename to examples/acasxu/nets/nnet/ACASXU_1_1.nnet
diff --git a/examples/acasxu/nets/onnx/ACASXU_1_1.onnx b/examples/acasxu/nets/onnx/ACASXU_1_1.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..107622f165db728e6cc2984afb42ee2964f5ebf1
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_1_1.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_1_9.onnx b/examples/acasxu/nets/onnx/ACASXU_1_9.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..5b5dfa714321acd013216d4157e9f325b4970361
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_1_9.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_2_2.onnx b/examples/acasxu/nets/onnx/ACASXU_2_2.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..8d62b272de75115e883af912db3473cf7c10cb42
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_2_2.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_2_7.onnx b/examples/acasxu/nets/onnx/ACASXU_2_7.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..eb98b1523490f617c75349ce3c956ccf38b732a6
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_2_7.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_2_9.onnx b/examples/acasxu/nets/onnx/ACASXU_2_9.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..99bb5fd878cb1761afeb5d40b34b7c260074819a
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_2_9.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_3_3.onnx b/examples/acasxu/nets/onnx/ACASXU_3_3.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..9da9dd7ec66e489539748094e152166d3f287833
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_3_3.onnx differ
diff --git a/examples/acasxu/nets/onnx/ACASXU_4_5.onnx b/examples/acasxu/nets/onnx/ACASXU_4_5.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..73e71fc91eb788ef13f953057ffcb50cf5973ed8
Binary files /dev/null and b/examples/acasxu/nets/onnx/ACASXU_4_5.onnx differ
diff --git a/examples/acasxu/property_1.why b/examples/acasxu/property_1.why
index 360958cfe896e2c9386a1c7e588b75ee66850de4..6e51d7b8cb79d41f4bda8a80f8fb5456225cabb0 100644
--- a/examples/acasxu/property_1.why
+++ b/examples/acasxu/property_1.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P1
+theory ACASXU_P1
use ACASXU_1_1.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P1_1_1: forall x0 x1 x2 x3 x4.
(0.5999999999999999777955395074968691915273666381835937500000000000:t) .<= x0 .<= (0.6798577687000000313588543576770462095737457275390625000000000000:t) ->
(-0.5:t) .<= x1 .<= (0.5:t) ->
(-0.5:t) .<= x2 .<= (0.5:t) ->
(0.4500000000000000111022302462515654042363166809082031250000000000:t) .<= x3 .<= (0.5:t) ->
(-0.5:t) .<= x4 .<= (-0.4500000000000000111022302462515654042363166809082031250000000000:t) ->
let (y0, _, _, _, _) = nn_apply x0 x1 x2 x3 x4 in
- y0 .< (3.9911256458999999630066213285317644476890563964843750000000000000:t)
+ y0 .<= (3.9911256458999999630066213285317644476890563964843750000000000000:t)
end
diff --git a/examples/acasxu/property_10.why b/examples/acasxu/property_10.why
index 79149ea499e8ca0b8df5b6eea24ce97b1e03d100..6e260fabebd5e52a51fa5640be659b426eccdec2 100644
--- a/examples/acasxu/property_10.why
+++ b/examples/acasxu/property_10.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P10
+theory ACASXU_P10
use ACASXU_4_5.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P10_4_5: forall x0 x1 x2 x3 x4.
(0.2689779999999999948734341614908771589398384094238281250000000000:t) .<= x0 .<= (0.6798579999999999623483404320722911506891250610351562500000000000:t) ->
(0.1114079999999999931459271351741335820406675338745117187500000000:t) .<= x1 .<= (0.5:t) ->
(-0.5:t) .<= x2 .<= (-0.4933803236000000036476365039561642333865165710449218750000000000:t) ->
(0.2272730000000000027959856652159942314028739929199218750000000000:t) .<= x3 .<= (0.5:t) ->
(0.0:t) .<= x4 .<= (0.5:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y0 .< y1 /\ y0 .< y2 /\ y0 .< y3 /\ y0 .< y4
+ y0 .<= y1 /\ y0 .<= y2 /\ y0 .<= y3 /\ y0 .<= y4
end
diff --git a/examples/acasxu/property_2.why b/examples/acasxu/property_2.why
index fd8fe0a498e5d670d5a0b7b8fb6089c6138ef626..bac22019613a4915f4e30d1cfd2a89992c6eed5a 100644
--- a/examples/acasxu/property_2.why
+++ b/examples/acasxu/property_2.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P2
+theory ACASXU_P2
use ACASXU_2_2.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P2_2_2: forall x0 x1 x2 x3 x4.
(0.5999999999999999777955395074968691915273666381835937500000000000:t) .<= x0 .<= (0.6798577687000000313588543576770462095737457275390625000000000000:t) ->
(-0.5:t) .<= x1 .<= (0.5:t) ->
(-0.5:t) .<= x2 .<= (0.5:t) ->
(0.4500000000000000111022302462515654042363166809082031250000000000:t) .<= x3 .<= (0.5:t) ->
(-0.5:t) .<= x4 .<= (-0.4500000000000000111022302462515654042363166809082031250000000000:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y0 .< y1 \/ y0 .< y2 \/ y0 .< y3 \/ y0 .< y4
+ y0 .<= y1 \/ y0 .<= y2 \/ y0 .<= y3 \/ y0 .<= y4
end
diff --git a/examples/acasxu/property_3.why b/examples/acasxu/property_3.why
index 6cbbd4f575599a714ed8266fb3a4a270188118a2..84f816bf108f543074a6b27cf065b04d6e0c6cdc 100644
--- a/examples/acasxu/property_3.why
+++ b/examples/acasxu/property_3.why
@@ -1,13 +1,34 @@
-theory ACAS_XU_P3
- use ACASXU_1_1.AsTuple
+theory ACASXU_P3
+ use ACASXU_1_1.AsTuple as N11
+ use ACASXU_2_7.AsTuple as N27
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
- (-0.3035311560999999769272506000561406835913658142089843750000000000:t) .<= x0 .<= (-0.2985528118999999924731980627257144078612327575683593750000000000:t) ->
- (-0.0095492965999999998572000947660853853449225425720214843750000000:t) .<= x1 .<= (0.0095492965999999998572000947660853853449225425720214843750000000:t) ->
- (0.4933803236000000036476365039561642333865165710449218750000000000:t) .<= x2 .<= (0.5:t) ->
- (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x3 .<= (0.5:t) ->
- (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x4 .<= (0.5:t) ->
- let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y0 .> y1 \/ y0 .> y2 \/ y0 .> y3 \/ y0 .> y4
+ constant x0:t
+ constant x1:t
+ constant x2:t
+ constant x3:t
+ constant x4:t
+
+ axiom H0:
+ (-0.3035311560999999769272506000561406835913658142089843750000000000:t) .<= x0 .<= (-0.2985528118999999924731980627257144078612327575683593750000000000:t)
+
+ axiom H1:
+ (-0.0095492965999999998572000947660853853449225425720214843750000000:t) .<= x1 .<= (0.0095492965999999998572000947660853853449225425720214843750000000:t)
+
+ axiom H2:
+ (0.4933803236000000036476365039561642333865165710449218750000000000:t) .<= x2 .<= (0.5:t)
+
+ axiom H3:
+ (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x3 .<= (0.5:t)
+
+ axiom H4:
+ (0.2999999999999999888977697537484345957636833190917968750000000000:t) .<= x4 .<= (0.5:t)
+
+ goal P3_1_1:
+ let (y0, y1, y2, y3, y4) = N11.nn_apply x0 x1 x2 x3 x4 in
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
+
+ goal P3_2_7:
+ let (y0, y1, y2, y3, y4) = N27.nn_apply x0 x1 x2 x3 x4 in
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
end
diff --git a/examples/acasxu/property_4.why b/examples/acasxu/property_4.why
index 40551f23c8fa06ce7a2a4b85961e0bbe58d5b504..cfc7df2af37117109cfb4c544497dfe297188be1 100644
--- a/examples/acasxu/property_4.why
+++ b/examples/acasxu/property_4.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P4
+theory ACASXU_P4
use ACASXU_1_1.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P4_1_1: forall x0 x1 x2 x3 x4.
(-0.3035311560999999769272506000561406835913658142089843750000000000:t) .<= x0 .<= (-0.2985528118999999924731980627257144078612327575683593750000000000:t) ->
(-0.0095492965999999998572000947660853853449225425720214843750000000:t) .<= x1 .<= (0.0095492965999999998572000947660853853449225425720214843750000000:t) ->
(0.0:t) .<= x2 .<= (0.0:t) ->
(0.3181818182000000216902435568044893443584442138671875000000000000:t) .<= x3 .<= (0.5:t) ->
(0.0833333333000000064938461719066253863275051116943359375000000000:t) .<= x4 .<= (0.1666666666999999935061538280933746136724948883056640625000000000:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y0 .> y1 \/ y0 .> y2 \/ y0 .> y3 \/ y0 .> y4
+ y0 .>= y1 \/ y0 .>= y2 \/ y0 .>= y3 \/ y0 .>= y4
end
diff --git a/examples/acasxu/property_5.why b/examples/acasxu/property_5.why
index 02f97b01878ce3e79619d52577618bbe60408a18..df38218dfffe142a11f8bb46828124817e6bdefc 100644
--- a/examples/acasxu/property_5.why
+++ b/examples/acasxu/property_5.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P5
+theory ACASXU_P5
use ACASXU_1_1.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P5_1_1: forall x0 x1 x2 x3 x4.
(-0.3242740000000000066826544298237422481179237365722656250000000000:t) .<= x0 .<= (-0.3217849999999999877076106713502667844295501708984375000000000000:t) ->
(0.0318309999999999981845633101329440250992774963378906250000000000:t) .<= x1 .<= (0.0636619999999999963691266202658880501985549926757812500000000000:t) ->
(-0.5:t) .<= x2 .<= (-0.4992039999999999810853523740661330521106719970703125000000000000:t) ->
(-0.5:t) .<= x3 .<= (-0.2272730000000000027959856652159942314028739929199218750000000000:t) ->
(-0.5:t) .<= x4 .<= (-0.1666670000000000095852215054037515074014663696289062500000000000:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y4 .< y1 \/ y4 .< y2 \/ y4 .< y3 \/ y4 .< y0
+ y4 .<= y1 \/ y4 .<= y2 \/ y4 .<= y3 \/ y4 .<= y0
end
diff --git a/examples/acasxu/property_6.why b/examples/acasxu/property_6.why
index 34b30b3d120d5f4c272f20af9b6534391331b425..aea0e63aba341e4fb66a1b34dce7bfdf0595501f 100644
--- a/examples/acasxu/property_6.why
+++ b/examples/acasxu/property_6.why
@@ -1,8 +1,8 @@
-theory ACAS_XU_P6
+theory ACASXU_P6
use ACASXU_1_1.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P6_1_1: forall x0 x1 x2 x3 x4.
(-0.1292889999999999872670741751790046691894531250000000000000000000:t) .<= x0 .<= (0.7004349250000000415283807342348154634237289428710937500000000000:t) ->
(-0.5:t) .<= x1 .<= (-0.1114079999999999931459271351741335820406675338745117187500000000:t)
\/
@@ -11,5 +11,5 @@ theory ACAS_XU_P6
(-0.5:t) .<= x3 .<= (0.5:t) ->
(-0.5:t) .<= x4 .<= (0.5:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y0 .< y1 /\ y0 .< y2 /\ y0 .< y3 /\ y0 .< y4
+ y0 .<= y1 /\ y0 .<= y2 /\ y0 .<= y3 /\ y0 .<= y4
end
diff --git a/examples/acasxu/property_7.why b/examples/acasxu/property_7.why
index 96fa7c3868e095e256a4f21f5b55ff4cd9925f26..f33ea31617d538d777347d2436945ce2d07645b6 100644
--- a/examples/acasxu/property_7.why
+++ b/examples/acasxu/property_7.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P7
+theory ACASXU_P7
use ACASXU_1_9.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P7_1_9: forall x0 x1 x2 x3 x4.
(-0.3284230000000000204707362172484863549470901489257812500000000000:t) .<= x0 .<= (0.6798579999999999623483404320722911506891250610351562500000000000:t) ->
(-0.5:t) .<= x1 .<= (0.5:t) ->
(-0.5:t) .<= x2 .<= (0.5:t) ->
(-0.5:t) .<= x3 .<= (0.5:t) ->
(-0.5:t) .<= x4 .<= (0.5:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- (y4 .> y1 \/ y4 .> y2 \/ y4 .> y3 \/ y4 .> y0) /\ (y3 .> y1 \/ y3 .> y2 \/ y3 .> y0 \/ y3 .> y4)
+ (y4 .>= y1 \/ y4 .>= y2 \/ y4 .>= y3 \/ y4 .>= y0) /\ (y3 .>= y1 \/ y3 .>= y2 \/ y3 .>= y0 \/ y3 .>= y4)
end
diff --git a/examples/acasxu/property_8.why b/examples/acasxu/property_8.why
index c30de4c19538bda2b8c2325f6095cefbb25cdf82..1e4703d2c08908ca1dc765486b1826135e55fdec 100644
--- a/examples/acasxu/property_8.why
+++ b/examples/acasxu/property_8.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P8
+theory ACASXU_P8
use ACASXU_2_9.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P8_2_9: forall x0 x1 x2 x3 x4.
(-0.3284230000000000204707362172484863549470901489257812500000000000:t) .<= x0 .<= (0.6798579999999999623483404320722911506891250610351562500000000000:t) ->
(-0.5:t) .<= x1 .<= (0.375:t) ->
(-0.0159149999999999985922372047753015067428350448608398437500000000:t) .<= x2 .<= (0.0159149999999999985922372047753015067428350448608398437500000000:t) ->
(-0.0454550000000000023470114740575809264555573463439941406250000000:t) .<= x3 .<= (0.5:t) ->
(0.0:t) .<= x4 .<= (0.5:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- (y1 .< y0 /\ y1 .< y2 /\ y1 .< y3 /\ y1 .< y4) \/ (y0 .< y1 /\ y0 .< y2 /\ y0 .< y3 /\ y0 .< y4)
+ (y1 .<= y0 /\ y1 .<= y2 /\ y1 .<= y3 /\ y1 .<= y4) \/ (y0 .<= y1 /\ y0 .<= y2 /\ y0 .<= y3 /\ y0 .<= y4)
end
diff --git a/examples/acasxu/property_9.why b/examples/acasxu/property_9.why
index 7e67075f6c4207271f5d949f4d2771374333e9d0..34d9b007c38c845df84cea528e9150f12b01d99a 100644
--- a/examples/acasxu/property_9.why
+++ b/examples/acasxu/property_9.why
@@ -1,13 +1,13 @@
-theory ACAS_XU_P9
+theory ACASXU_P9
use ACASXU_3_3.AsTuple
use ieee_float.Float64
- goal G: forall x0 x1 x2 x3 x4.
+ goal P9_3_3: forall x0 x1 x2 x3 x4.
(-0.2952339158000000240988924815610516816377639770507812500000000000:t) .<= x0 .<= (-0.2122615123999999908743774312824825756251811981201171875000000000:t) ->
(-0.0636619771999999972678097037714906036853790283203125000000000000:t) .<= x1 .<= (-0.0222816919999999987767047571196599164977669715881347656250000000:t) ->
(-0.4999998959999999992298569395643426105380058288574218750000000000:t) .<= x2 .<= (-0.4984083464999999879552206039079464972019195556640625000000000000:t) ->
(-0.5:t) .<= x3 .<= (-0.4545454545000000012855423392466036602854728698730468750000000000:t) ->
(-0.5:t) .<= x4 .<= (-0.375:t) ->
let (y0, y1, y2, y3, y4) = nn_apply x0 x1 x2 x3 x4 in
- y3 .< y1 /\ y3 .< y2 /\ y3 .< y0 /\ y3 .< y4
+ y3 .<= y1 /\ y3 .<= y2 /\ y3 .<= y0 /\ y3 .<= y4
end
diff --git a/examples/mnist/csv/mnist_test.csv b/examples/mnist/csv/mnist_test.csv
new file mode 100644
index 0000000000000000000000000000000000000000..3ea4d6f03d8dcbc7f83e4270e116ad83a43bd1f3
--- /dev/null
+++ b/examples/mnist/csv/mnist_test.csv
@@ -0,0 +1,100 @@
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diff --git a/examples/mnist/csv/single_image.csv b/examples/mnist/csv/single_image.csv
new file mode 100644
index 0000000000000000000000000000000000000000..879a80118068720293b896c49ce4c8f6149f80d0
--- /dev/null
+++ b/examples/mnist/csv/single_image.csv
@@ -0,0 +1 @@
+5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,18,18,18,126,136,175,26,166,255,247,127,0,0,0,0,0,0,0,0,0,0,0,0,30,36,94,154,170,253,253,253,253,253,225,172,253,242,195,64,0,0,0,0,0,0,0,0,0,0,0,49,238,253,253,253,253,253,253,253,253,251,93,82,82,56,39,0,0,0,0,0,0,0,0,0,0,0,0,18,219,253,253,253,253,253,198,182,247,241,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,80,156,107,253,253,205,11,0,43,154,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,1,154,253,90,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,139,253,190,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,190,253,70,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,35,241,225,160,108,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,81,240,253,253,119,25,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,45,186,253,253,150,27,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,93,252,253,187,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,249,253,249,64,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,46,130,183,253,253,207,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,39,148,229,253,253,253,250,182,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,114,221,253,253,253,253,201,78,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,66,213,253,253,253,253,198,81,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,171,219,253,253,253,253,195,80,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,55,172,226,253,253,253,253,244,133,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,136,253,253,253,212,135,132,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
diff --git a/examples/mnist/nets/MNIST_256_2.onnx b/examples/mnist/nets/MNIST_256_2.onnx
new file mode 100644
index 0000000000000000000000000000000000000000..462b098c3a37faa518a9a3bcfaad8403ba60ccad
Binary files /dev/null and b/examples/mnist/nets/MNIST_256_2.onnx differ
diff --git a/examples/mnist/property.why b/examples/mnist/property.why
new file mode 100644
index 0000000000000000000000000000000000000000..9f47a84b1a8f52b0c694a3e33f800eac10fc3a28
--- /dev/null
+++ b/examples/mnist/property.why
@@ -0,0 +1,10 @@
+theory MNIST
+ use MNIST_256_2.AsArray
+ use ieee_float.Float64
+ use caisar.DataSetClassification
+ use caisar.DataSetProps
+
+ goal robustness:
+ let normalized_dataset = min_max_scale true (0.0:t) (1.0:t) dataset in
+ robust model normalized_dataset (0.0100000000000000002081668171172168513294309377670288085937500000:t)
+end
diff --git a/stdlib/caisar.mlw b/stdlib/caisar.mlw
index 1a2a793c390aa8d4a9a23cf8663e68fbb82a8144..889f92be052088c797fe12645f1582341b6ed76e 100644
--- a/stdlib/caisar.mlw
+++ b/stdlib/caisar.mlw
@@ -26,13 +26,13 @@ theory DataSetClassification
use array.Array
type features = array t
- type class = int
+ type label_ = int
- type datum = (features, class)
+ type datum = (features, label_)
type dataset = {
nb_features: int;
- nb_classes: int;
+ nb_label_s: int;
data: array datum
}
@@ -51,7 +51,7 @@ theory Model
nb_outputs: int;
}
- function predict_class: model -> features -> class
+ function predict: model -> features -> label_
end
theory DataSetProps
@@ -67,13 +67,13 @@ theory DataSetProps
predicate correct_at (m: model) (d: datum) =
let (x, y) = d in
- y = predict_class m x
+ y = predict m x
predicate robust_at (m: model) (d: datum) (eps: t) =
forall x': features.
let (x, _) = d in
linfty_distance x x' eps ->
- predict_class m x = predict_class m x'
+ predict m x = predict m x'
predicate cond_robust_at (m: model) (d: datum) (eps: t) =
correct_at m d /\ robust_at m d eps
diff --git a/tests/autodetect.t b/tests/autodetect.t
index dc9de889d81ac056b9673c29290a6a351f7dfdcc..36d66f12046ac5df419dc6a0e4444422522eeb9c 100644
--- a/tests/autodetect.t
+++ b/tests/autodetect.t
@@ -42,13 +42,15 @@ Test autodetect
<autodetect>Found prover Marabou version 1.0.+, OK.
<autodetect>Found prover PyRAT version 1.1, OK.
<autodetect>Found prover PyRAT version 1.1 (alternative: VNNLIB)
+ <autodetect>Found prover PyRAT version 1.1 (alternative: ACAS)
<autodetect>Found prover nnenum version dummy, OK.
<autodetect>Found prover SAVer version v1.0, OK.
- <autodetect>7 prover(s) added
+ <autodetect>8 prover(s) added
[caisar] Alt-Ergo 2.4.0
CVC5 1.0.2
Marabou 1.0.+
PyRAT 1.1
+ PyRAT 1.1 (ACAS)
PyRAT 1.1 (VNNLIB)
SAVer v1.0
nnenum dummy