package logtk
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md5=e72de75e9f0f87da9e6e8c0a4d4c89f9
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Description
README
Zipperposition
Automated theorem prover for firstorder logic with equality and theories.
Logic toolkit (
logtk
), designed primarily for firstorder automated reasoning. It aims at providing basic types and algorithms (terms, unification, orderings, indexing, etc.) that can be factored out of several applications.
Short summary
Zipperposition is intended to be a superposition prover for full first order logic, plus some extensions (datatypes, recursive functions, lambdafree higher order). The accent is on flexibility, modularity and simplicity rather than performance, to allow quick experimenting on automated theorem proving. It generates TSTP traces or graphviz files for nice graphical display.
Zipperposition supports several input formats:
TPTP (fof, cnf, tff)
its own native input, extension
.zf
(see directoryexamples/
and section below)
Zipperposition is written in the functional and imperative language OCaml. The name is a bad play on the words "zipper" (a functional data structure) and "superposition" (the calculus used by the prover), although the current implementation is written in quite an imperative style. Superpositionbased theorem proving is an active field of research, so there is a lot of literature about it; for this implementation the main references for the base calculus are:
the chapter Paramodulationbased theorem proving of the Handbook of Automated Reasoning,
the paper E: a brainiac theorem prover that describes the E prover by S. Schulz,
the paper Superposition with equivalence reasoning and delayed clause normal form transformation by H. Ganzinger and J. Stuber
Disclaimer: Note that the prover is a research project. Please don't use it to drive your personal nuclear power plant, nor as a trusted tool for critical applications.
License
This project is licensed under the BSD2 license. See the LICENSE
file.
Build
Zipperposition requires OCaml >= 4.03.0, and some libraries that are available on opam.
Via opam
The recommended way to install Zipperposition is through opam. You need to have GMP (with headers) installed (it's not handled by opam). Once you have installed GMP and opam, type:
$ opam install zipperposition
To upgrade to more recent versions:
$ opam update
$ opam upgrade
If you want to try the development (unstable) version, which has more dependencies (in particular dune
for the build), try:
$ opam pin k git https://github.com/sneeuwballen/zipperposition.git#master
NOTE: do not install logtk
. It now ships with zipperposition itself. NOTE: if installation fails, you might want to try to opam update
and opam upgrade
: it might be because some of the dependencies are too old.
Manually
If you really need to, you can download a release on the following github page for releases.
Look in the file opam
to see which dependencies you need to install. They include menhir
, zarith
, containers
, msat and sequence
, but maybe also other libraries. Consider using opam directly if possible.
$ make install
Additional sublibraries can be built if their respective dependencies are met.
If menhir is installed, the parsers library Logtk_parsers
will automatically be built.
If you have installed qcheck and alcotest, for instance via opam install qcheck alcotest
, you can enable the propertybased testing and random term generators with
$ make test
NOTE: in case of build errors, it might be because of outdated dependencies (see via opam for more details), or stale build files. Try rm _build rf
to try to build from scratch.
Documentation
See this page.
There are some examples of how to use the libraries in src/tools/
and src/demo/
.
Use
Typical usage:
$ zipperposition help
$ zipperposition problem_file [options]
$ zipperposition dot /tmp/foo.dot examples/ind/nat1.zf
to run the prover. Help is available with the option help
. For instance,
$ zipperposition examples/pelletier_problems/pb47.p timeout 30
To build the library, documentation, and tools, type in a terminal located in the root directory of the project:
$ make
If you use ocamlfind
(which is strongly recommended), installation/uninstallation are just:
$ make install
$ make uninstall
Native Syntax
The native syntax, with file extension .zf
, resembles a simple fragment of ML with explicit polymorphism. Many examples in examples/
are written using this syntax. A vim syntax coloring file can be found in utils/vim
(see the readme for instructions on how to install it).
Description of the native format `.zf`
Basics
Comments start with #
and continue to the end of the line. Every symbol must be declared, using the builtin type prop
for propositions. A type is declared like this: val i : type.
and a parametrized type: val array: type > type.
val i : type.
val a : i.
val f : i > i. # a function
val p : i > i > prop. # a binary predicate
Then, axioms and the goal:
assert forall x y. p x y => p y x.
assert p a (f a).
goal exists (x:i). p (f x) x.
We can run the prover on a file containing these declarations. It will display a proof very quickly:
$ ./zipperposition.native example.zf
% done 3 iterations
% SZS status Theorem for 'example.zf'
% SZS output start Refutation
* ⊥/7 by simp simplify with [⊥]/5
* [⊥]/5 by
inf s_sup with {X2[1] → a[0]}
with [p (f a) a]/4, forall (X2:i). [¬p (f X2) X2]/2
* forall (X2:i). [¬p (f X2) X2]/2 by
esa cnf with ¬ (∃ x/13:i. (p (f x/13) x/13))
* [p (f a) a]/4 by simp simplify with [p (f a) a ∨ ⊥]/3
* [p (f a) a ∨ ⊥]/3 by
inf s_sup with {X0[0] → f a[1], X1[0] → a[1]}
with [p a (f a)]/1, forall (X0:i) (X1:i). [p X0 X1 ∨ ¬p X1 X0]/0
* ¬ (∃ x/13:i. (p (f x/13) x/13)) by
esa neg_goal negate goal to find a refutation
with ∃ x/13:i. (p (f x/13) x/13)
* ∃ x/13:i. (p (f x/13) x/13) by goal 'example.zf'
* forall (X0:i) (X1:i). [p X0 X1 ∨ ¬p X1 X0]/0 by
esa cnf with ∀ x/9:i y/11:i. ((p x/9 y/11) ⇒ (p y/11 x/9))
* [p a (f a)]/1 by esa cnf with p a (f a)
* p a (f a) by 'example.zf'
* ∀ x/9:i y/11:i. ((p x/9 y/11) ⇒ (p y/11 x/9)) by 'example.zf'
% SZS output end Refutation
Each *
prefixed item in the list is an inference step. The top step is the empty clause: zipperposition works by negating the goal before looking for proving false
. Indeed, proving a ⇒ b
is equivalent to deducing false
from a ∧ ¬b
.
Connectives and Quantifiers
The connectives are:
true:
true
false:
false
conjunction:
a && b
disjunction:
a  b
negation:
~ a
equality:
a = b
disequality:
a != b
(synonym for~ (a = b)
)implication:
a => b
equivalence:
a <=> b
Implication and equivalence have the same priority as disjunction. Conjunction binds tighter, meaning that a && b  c
is actually parsed as (a && b)  c
. Negation is even stronger: ~ a && b
means (~ a) && b
.
Binders extend as far as possible to their right, and are typed, although the type constraint can be omitted if it can be inferred:
universal quantification:
forall x. F
or in its typed form:forall (x:ty). F
existential quantification:
exists x. F
Polymorphic symbols can be declare using pi <var>. type
, for instance val f : pi a b. a > array a b > b
is a polymorphic function that takes 2 type arguments, then 2 term arguments. An application of f
will look like f nat (list bool) (Succ Z) empty
. Type arguments might be omitted if they can be inferred.
Inclusion
It can be convenient to put commonly used axioms in a separate file. The statement
include "foo.zf".
will include the corresponding file (whose path is relative to the current file).
Advanced Syntax
There are more advanced concepts that are mostly related to induction:
datatypes: (here, Peano numbers and polymorphic lists)
data nat := Zero  Succ nat.
data list a := nil  cons a (list a).
simple definitions:
def four : nat := Succ (Succ (Succ (Succ Zero))).
rewrite rules:
A rewrite rule is similar to an assert
statement, except it is much more efficient. Zipperposition assumes that the set of rewrite rules in its input is confluent and terminating (otherwise, no guarantee applies). Rewriting can be done on terms and on atomic formulas:
val set : type > type.
val member : pi a. a > set a > prop.
val union : pi a. set a > set a > set a.
rewrite forall a (x:a)(s1:set a)(s2:set a).
member x (union s1 s2) <=> (member x s1  member x s2).
val subset : pi a. set a > set a > prop.
rewrite forall a (s1:set a)(s2:set a).
subset s1 s2 <=> (forall x. member x s1 => member x s2).
val equal_set : pi a. set a > set a > prop.
rewrite forall a (s1:set a) s2.
equal_set s1 s2 <=> subset s1 s2 && subset s2 s1.
# now show that union is associative:
goal forall a (s1:set a) s2 s3.
equal_set
(union s1 (union s2 s3))
(union (union s1 s2) s3).
there are several variations on literal rewrite rules:
rewrite forall x. p x
(short forp x <=> true
)rewrite forall x. ~ p x
(short forp x <=> false
)rewrite forall x. p x => q x
(one way rule, will rewritep x
but not~ p x
; also called polarized rewriting)rewrite forall x. ~ p x => q x
(negative polarized rule)
recursive definitions:
one can write recursive functions (assuming they terminate), they will be desugared to a declaration + a set of rewrite rules:
def plus : nat > nat > nat where
forall y. plus Zero y = y;
forall x y. plus (Succ x) y = Succ (plus x y).
Mutually recursive definitions are separated by and
:
def even : nat > prop where
even Zero;
forall x. even (Succ x) = odd x
and odd : nat > prop where
forall x. odd (Succ x) = even x.
Zipperposition is able to do simple inductive proofs using these recursive functions and datatypes:
$ cat doc/plus_assoc.zf
data nat := Zero  Succ nat.
def plus : nat > nat > nat where
forall y. plus Zero y = y;
forall x y. plus (Succ x) y = Succ (plus x y).
goal forall (x:nat) y z. plus x (plus y z) = plus (plus x y) z.
$ zipperposition doc/plus_assoc.zf o none
% done 17 iterations
% SZS status Theorem for 'doc/plus_assoc.zf'
conditionals:
tests on boolean formulas are written if a then b else c
, where a:prop
, b
, and c
, are terms. b
and c
must have the same type.
patternmatching:
shallow pattern matching is written match <term> with [case]+ end
where each case is  <constructor> [var]* > <term>
.
AC symbols:
Some symbols can be declared "associative commutative": they satisfy
forall x y z. f x (f y z) = f (f x y) z
forall x y. f x y = f y x
.
the following statement is a bit more efficient than writing the corresponding axioms:
val[AC] f : foo > foo > foo.
Axioms in Set of Support:
Some axioms (introduced using assert [sos] <formula>.
) will be considered as part of the socalled "set of support" strategy. No saturation among SOS axioms is done. They are only used for inferences (and simplifications) with nonSOS axioms and goals. Typically this is useful for introducing general lemmas while preventing them from interacting in ways not related to the current goal.
Named Axioms:
An axiom can be given a name, as in TPTP, to retrieve it easily in proofs. The syntax is:
assert[name "foo"] bar.
Graphical Display of Proofs
A handy way of displaying the proof is to use graphviz:
$ ./zipperposition.native dot /tmp/example.dot example.zf
$ dot Txlib /tmp/example.dot
One can generate an image from the .dot
file:
$ dot Tsvg /tmp/example.dot > some_picture.svg
Proof Format
It is possible to avoid displaying the proof at all, by using o none
. A TSTP derivation can be obtained with o tstp
.
Library
Zipperposition's library provides several useful parts for logicrelated implementations:
a library packed in a module
Logtk
, with terms, formulas, etc.;a library packed in a module
Logtk_parsers
, with parsers for input formats;small tools (see directory
src/tools/
) to illustrate how to use the library and provide basic services (typechecking, reduction to CNF, etc.);
Hacking
Some advices if you want to hack on the code:
debug 5
prints everything the prover doesdebug.foo <n>
changes the verbosity only forfoo
(seehelp
for a list of such flags)backtrace
is very useful to get stack traces when a wild uncaught exception appearsstats
prints some statistics, and you can add your own easily withUtil.mk_stat
dot <somefile>.dot
dumps the proof in the given file in graphviz. This is very useful for reading proofs, e.g. usingdot Txlib <somefile>.dot
. See Graphical display of proofs for more details.many flags control the behavior of the prover; to dumb heuristics down a bit you can try:
cq bfs
(BFS traversal of the search space, instead of weightbased clause selection rules)ord none
for disabling term orderings
Profiling
a script using
perf
can be found inutils/profile.sh
profiling probes are inserted into the code, but they're disabled by default (see
src/core/ZProf.ml
,let __prof=false
). By setting__prof=true
the probes will become active, and the command line optionprofile
will be available.
StarExec
StarExec is a service for experimental evaluation of logic solvers like Zipperposition.
How to build Zipperposition for StarExec
The easiest way to import Zipperposition as a solver is to precompile Zipperposition on the StarExec virtual machine. Download the VM image and open it in VirtualBox.
Open the settings of the VM. Set "Network > Adapter 1 > Attached to" to NAT to have internet access from inside the VM. To allow SSH access into the VM open "Network > Adapter 1 > Advanced > Port Forwarding" and create a new rule:
Name: ssh
Protocol: TCP
Host Port: 3022
Guest Port: 22
Leave the two IP fields empty.
Start the VM. Log in as root using the password "St@rexec".
starclone login: root
Password: St@rexec
Install the openssh server to get a more convenient access to the machine and to copy the compiled binary later.
$ yum install opensshserver
Now open a terminal on the host machine while the VM is still running. Tunnel into the VM via SSH:
$ ssh p 3022 root@127.0.0.1
root@127.0.0.1's password: St@rexec
Install OPAM:
$ wget https://raw.github.com/ocaml/opam/master/shell/opam_installer.sh O   sh s /usr/local/bin
So far we have used the superuser root. To download and compile Zipperposition we will use a regular user that we create as follows:
$ useradd m bob
$ passwd bob
New password: bob
BAD PASSWORD: The password is a palindrome
Retype new password: bob
passwd: all authentication tokens updated successfully.
Close the SSH connection and reopen it as the new user:
$ exit
$ ssh p 3022 bob@127.0.0.1
bob@127.0.0.1's password: bob
Initialize OPAM. Install OCaml 4.05 and the dependencies of Zipperposition (Look in the file opam
to see which dependencies you need to install).
$ opam init
$ opam switch 4.05.0+flambda
$ eval `opam config env`
$ opam install dune zarith containers sequence msat menhir
Clone Zipperposition and compile it:
$ git clone https://github.com/sneeuwballen/zipperposition.git branch dev
$ cd zipperposition
$ make
Close the SSH connection and copy the binary from the VM onto your host machine.
$ exit
$ scp P 3022 bob@127.0.0.1:~/zipperposition/zipperposition.native /some/path/on/the/host/machine
bob@127.0.0.1's password: bob
As described in the StarExec documentation you need a script whose filename has the prefix starexec_run_
to execute your solver. For Zipperposition this script could look like this:
#!/bin/sh
./zipperposition.native o tptp "$1" \
timeout "$STAREXEC_WALLCLOCK_LIMIT" \
memlimit "$STAREXEC_MAX_MEM"
Put this script and the file zipperposition.native
into a folder called bin
. Create a ZIP archive containing that folder. Now Zipperposition is ready to be uploaded to StarExec!
Docker
(experimental)
to build an image:
docker build t zipper .
to use the image:
docker run i zipper < examples/pelletier_problems/pb47.zf
Howto (for devs)
Make a release
edit
*.opam
files to update the version number (fieldversion
).git commit a m "prepare for <version>"
(to save the changes on the stable branch)make clean all
(to check everything builds properly)git tag <version>
(e.g.git tag 1.4
)git push origin <version>
(origin
being the name of the github remote)opam publish prepare zipperposition.1.4 https://github.com/sneeuwballen/zipperposition/archive/1.4.tar.gz
(using the actual version number). This might require toopam install opampublish
first, it's a handy opam plugin for managing releases.if that works properly, then it will create a directory
zipperposition.<version>
. Just runopam publish submit zipperposition.1.4
to open a PR against opamrepository.
If something is wrong with the release, it's possible to change it. This is a bit brutal, never do it for older releases that have been merged into opamrepo, only for the next release while no one has seen it yet.
git tag f <version>; git push origin :<version> ; git push origin <version>
to change the tagrerun the two
opam publish
commands to update the directory and the PR.
Dependencies (12)

ocaml
>= "4.07"

dune
>= "1.11"

menhir
build & < "20211215"

iter
>= "1.2"

mtime
>= "1.0" & < "2.0"

containersdata
>= "3.0" & < "4.0"

containers
>= "3.0" & < "4.0"

oseq
>= "0.3" & < "0.4"
 num
 zarith
 baseunix
 basebytes
Dev Dependencies (3)

qcheckalcotest
withtest & >= "0.9"

qcheckcore
withtest & >= "0.9"

alcotest
withtest
Used by (3)

libzipperposition
>= "2.1"

zipperposition
>= "2.1"

zipperpositiontools
>= "2.1"