An automatic theorem prover in OCaml for typed logic with equality, datatypes and arithmetic, based on superposition+rewriting
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README.adoc

Zipperposition

  • Automated theorem prover for first-order logic with equality and theories.

  • Logic toolkit (logtk), designed primarily for first-order 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, arithmetic). 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)

  • TIP

  • its own native input, extension .zf (see directory examples/)

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. Superposition-based theorem proving is an active field of research, so there is a lot of literature about it; for this implementation my main references are:

  • the chapter paramodulation-based 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 currently a prototype and is likely not complete. 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

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 Oasis for the build), try:

$ opam pin add zipperposition -k git https://github.com/c-cube/zipperposition.git#dev

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, oclock, msat and sequence, but maybe also other libraries. Consider using opam directly if possible.

$ ./configure

$ make install

Additional sub-libraries can be built if their respective dependencies are met, and the appropriate ./configure --enable-foobar flag was set.

If menhir is installed, the parsers library Logtk_parsers can be built with

$ ./configure --enable-parsers

If you have installed qcheck, for instance via opam install qcheck, you can enable the property-based testing and random term generators with

$ ./configure --enable-qcheck --enable-tests
$ make tests

Use

Typical usage:

$ zipperposition --help
$ zipperposition problem_file [options]
$ zipperposition --arith examples/ARI114=1.p
$ 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 --ord rpo6 --timeout 30

Several tools are shipped with Zipperposition, including a CNF converter, a type-checker, etc. They are built if the flag --enable-tools is set. Documentation will be built provided --enable-docs is set.

After the configuration is done, to build the library, documentation and tools (given the appropriate flags are set), 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 is available.

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).
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.

pattern-matching

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.

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
simple proof graph

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 logic-related 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 (type-checking, reduction to CNF, etc.);

Documentation

See this page.

There are some examples of how to use the code in src/tools/ and src/demo/.