>>> from nltk import * >>> from nltk.sem.drt import DrtParser >>> from nltk.sem import logic >>> logic._counter._value = 0

Within the area of automated reasoning, first order theorem proving
and model building (or model generation) have both received much
attention, and have given rise to highly sophisticated techniques. We
focus therefore on providing an NLTK interface to third party tools
for these tasks. In particular, the module `nltk.inference` can be
used to access both theorem provers and model builders.

The main class used to interface with a theorem prover is the `Prover`
class, found in `nltk.api`. The `prove()` method takes three optional
arguments: a goal, a list of assumptions, and a `verbose` boolean to
indicate whether the proof should be printed to the console. The proof goal
and any assumptions need to be instances of the `Expression` class
specified by `nltk.sem.logic`. There are currently three theorem provers
included with NLTK: `Prover9`, `TableauProver`, and
`ResolutionProver`. The first is an off-the-shelf prover, while the other
two are written in Python and included in the `nltk.inference` package.

>>> from nltk.sem import Expression >>> read_expr = Expression.fromstring >>> p1 = read_expr('man(socrates)') >>> p2 = read_expr('all x.(man(x) -> mortal(x))') >>> c = read_expr('mortal(socrates)') >>> Prover9().prove(c, [p1,p2]) True >>> TableauProver().prove(c, [p1,p2]) True >>> ResolutionProver().prove(c, [p1,p2], verbose=True) [1] {-mortal(socrates)} A [2] {man(socrates)} A [3] {-man(z2), mortal(z2)} A [4] {-man(socrates)} (1, 3) [5] {mortal(socrates)} (2, 3) [6] {} (1, 5) <BLANKLINE> True

A `ProverCommand` is a stateful holder for a theorem
prover. The command stores a theorem prover instance (of type `Prover`),
a goal, a list of assumptions, the result of the proof, and a string version
of the entire proof. Corresponding to the three included `Prover`
implementations, there are three `ProverCommand` implementations:
`Prover9Command`, `TableauProverCommand`, and
`ResolutionProverCommand`.

The `ProverCommand`'s constructor takes its goal and assumptions. The
`prove()` command executes the `Prover` and `proof()`
returns a String form of the proof
If the `prove()` method has not been called,
then the prover command will be unable to display a proof.

>>> prover = ResolutionProverCommand(c, [p1,p2]) >>> print(prover.proof()) # doctest: +ELLIPSIS Traceback (most recent call last): File "...", line 1212, in __run compileflags, 1) in test.globs File "<doctest nltk/test/inference.doctest[10]>", line 1, in <module> File "...", line ..., in proof raise LookupError("You have to call prove() first to get a proof!") LookupError: You have to call prove() first to get a proof! >>> prover.prove() True >>> print(prover.proof()) [1] {-mortal(socrates)} A [2] {man(socrates)} A [3] {-man(z4), mortal(z4)} A [4] {-man(socrates)} (1, 3) [5] {mortal(socrates)} (2, 3) [6] {} (1, 5) <BLANKLINE>

The prover command stores the result of proving so that if `prove()` is
called again, then the command can return the result without executing the
prover again. This allows the user to access the result of the proof without
wasting time re-computing what it already knows.

>>> prover.prove() True >>> prover.prove() True

The assumptions and goal may be accessed using the `assumptions()` and
`goal()` methods, respectively.

>>> prover.assumptions() [<ApplicationExpression man(socrates)>, <Alread_expression all x.(man(x) -> mortal(x))>] >>> prover.goal() <ApplicationExpression mortal(socrates)>

The assumptions list may be modified using the `add_assumptions()` and
`retract_assumptions()` methods. Both methods take a list of `Expression`
objects. Since adding or removing assumptions may change the result of the
proof, the stored result is cleared when either of these methods are called.
That means that `proof()` will be unavailable until `prove()` is called and
a call to `prove()` will execute the theorem prover.

>>> prover.retract_assumptions([read_expr('man(socrates)')]) >>> print(prover.proof()) # doctest: +ELLIPSIS Traceback (most recent call last): File "...", line 1212, in __run compileflags, 1) in test.globs File "<doctest nltk/test/inference.doctest[10]>", line 1, in <module> File "...", line ..., in proof raise LookupError("You have to call prove() first to get a proof!") LookupError: You have to call prove() first to get a proof! >>> prover.prove() False >>> print(prover.proof()) [1] {-mortal(socrates)} A [2] {-man(z6), mortal(z6)} A [3] {-man(socrates)} (1, 2) <BLANKLINE> >>> prover.add_assumptions([read_expr('man(socrates)')]) >>> prover.prove() True

You can download Prover9 from http://www.cs.unm.edu/~mccune/prover9/.

Extract the source code into a suitable directory and follow the
instructions in the Prover9 `README.make` file to compile the executables.
Install these into an appropriate location; the
`prover9_search` variable is currently configured to look in the
following locations:

>>> p = Prover9() >>> p.binary_locations() # doctest: +NORMALIZE_WHITESPACE ['/usr/local/bin/prover9', '/usr/local/bin/prover9/bin', '/usr/local/bin', '/usr/bin', '/usr/local/prover9', '/usr/local/share/prover9']

Alternatively, the environment variable `PROVER9HOME` may be configured with
the binary's location.

The path to the correct directory can be set manually in the following manner:

>>> config_prover9(path='/usr/local/bin') # doctest: +SKIP [Found prover9: /usr/local/bin/prover9]

If the executables cannot be found, `Prover9` will issue a warning message:

>>> p.prove() # doctest: +SKIP Traceback (most recent call last): ... LookupError: =========================================================================== NLTK was unable to find the prover9 executable! Use config_prover9() or set the PROVER9HOME environment variable. <BLANKLINE> >> config_prover9('/path/to/prover9') <BLANKLINE> For more information, on prover9, see: <http://www.cs.unm.edu/~mccune/prover9/> ===========================================================================

The general case in theorem proving is to determine whether `S |- g`
holds, where `S` is a possibly empty set of assumptions, and `g`
is a proof goal.

As mentioned earlier, NLTK input to `Prover9` must be
`Expression`s of `nltk.sem.logic`. A `Prover9` instance is
initialized with a proof goal and, possibly, some assumptions. The
`prove()` method attempts to find a proof of the goal, given the
list of assumptions (in this case, none).

>>> goal = read_expr('(man(x) <-> --man(x))') >>> prover = Prover9Command(goal) >>> prover.prove() True

Given a `ProverCommand` instance `prover`, the method
`prover.proof()` will return a String of the extensive proof information
provided by Prover9, shown in abbreviated form here:

============================== Prover9 =============================== Prover9 (32) version ... Process ... was started by ... on ... ... The command was ".../prover9 -f ...". ============================== end of head =========================== ============================== INPUT ================================= % Reading from file /var/... formulas(goals). (all x (man(x) -> man(x))). end_of_list. ... ============================== end of search ========================= THEOREM PROVED Exiting with 1 proof. Process 6317 exit (max_proofs) Mon Jan 21 15:23:28 2008

As mentioned earlier, we may want to list some assumptions for the proof, as shown here.

>>> g = read_expr('mortal(socrates)') >>> a1 = read_expr('all x.(man(x) -> mortal(x))') >>> prover = Prover9Command(g, assumptions=[a1]) >>> prover.print_assumptions() all x.(man(x) -> mortal(x))

However, the assumptions are not sufficient to derive the goal:

>>> print(prover.prove()) False

So let's add another assumption:

>>> a2 = read_expr('man(socrates)') >>> prover.add_assumptions([a2]) >>> prover.print_assumptions() all x.(man(x) -> mortal(x)) man(socrates) >>> print(prover.prove()) True

We can also show the assumptions in `Prover9` format.

>>> prover.print_assumptions(output_format='Prover9') all x (man(x) -> mortal(x)) man(socrates)>>> prover.print_assumptions(output_format='Spass') Traceback (most recent call last): . . . NameError: Unrecognized value for 'output_format': Spass

Assumptions can be retracted from the list of assumptions.

>>> prover.retract_assumptions([a1]) >>> prover.print_assumptions() man(socrates) >>> prover.retract_assumptions([a1])

Statements can be loaded from a file and parsed. We can then add these statements as new assumptions.

>>> g = read_expr('all x.(boxer(x) -> -boxerdog(x))') >>> prover = Prover9Command(g) >>> prover.prove() False >>> import nltk.data >>> new = nltk.data.load('grammars/sample_grammars/background0.fol') >>> for a in new: ... print(a) all x.(boxerdog(x) -> dog(x)) all x.(boxer(x) -> person(x)) all x.-(dog(x) & person(x)) exists x.boxer(x) exists x.boxerdog(x) >>> prover.add_assumptions(new) >>> print(prover.prove()) True >>> print(prover.proof()) # doctest: +ELLIPSIS ============================== prooftrans ============================ Prover9 (...) version ... Process ... was started by ... on ... ... The command was ".../prover9". ============================== end of head =========================== <BLANKLINE> ============================== end of input ========================== <BLANKLINE> ============================== PROOF ================================= <BLANKLINE> % -------- Comments from original proof -------- % Proof 1 at ... seconds. % Length of proof is 13. % Level of proof is 4. % Maximum clause weight is 0.000. % Given clauses 0. <BLANKLINE> <BLANKLINE> 1 (all x (boxerdog(x) -> dog(x))). [assumption]. 2 (all x (boxer(x) -> person(x))). [assumption]. 3 (all x -(dog(x) & person(x))). [assumption]. 6 (all x (boxer(x) -> -boxerdog(x))). [goal]. 8 -boxerdog(x) | dog(x). [clausify(1)]. 9 boxerdog(c3). [deny(6)]. 11 -boxer(x) | person(x). [clausify(2)]. 12 boxer(c3). [deny(6)]. 14 -dog(x) | -person(x). [clausify(3)]. 15 dog(c3). [resolve(9,a,8,a)]. 18 person(c3). [resolve(12,a,11,a)]. 19 -person(c3). [resolve(15,a,14,a)]. 20 $F. [resolve(19,a,18,a)]. <BLANKLINE> ============================== end of proof ==========================

One application of the theorem prover functionality is to check if
two Expressions have the same meaning.
The `equiv()` method calls a theorem prover to determine whether two
Expressions are logically equivalent.

>>> a = read_expr(r'exists x.(man(x) & walks(x))') >>> b = read_expr(r'exists x.(walks(x) & man(x))') >>> print(a.equiv(b)) True

The same method can be used on Discourse Representation Structures (DRSs). In this case, each DRS is converted to a first order logic form, and then passed to the theorem prover.

>>> dp = DrtParser() >>> a = dp.parse(r'([x],[man(x), walks(x)])') >>> b = dp.parse(r'([x],[walks(x), man(x)])') >>> print(a.equiv(b)) True

The top-level to model builders is parallel to that for
theorem-provers. The `ModelBuilder` interface is located
in `nltk.inference.api`. It is currently only implemented by
`Mace`, which interfaces with the Mace4 model builder.

Typically we use a model builder to show that some set of formulas has
a model, and is therefore consistent. One way of doing this is by
treating our candidate set of sentences as assumptions, and leaving
the goal unspecified.
Thus, the following interaction shows how both `{a, c1}` and `{a, c2}`
are consistent sets, since Mace succeeds in a building a
model for each of them, while `{c1, c2}` is inconsistent.

>>> a3 = read_expr('exists x.(man(x) and walks(x))') >>> c1 = read_expr('mortal(socrates)') >>> c2 = read_expr('-mortal(socrates)') >>> mace = Mace() >>> print(mace.build_model(None, [a3, c1])) True >>> print(mace.build_model(None, [a3, c2])) True

We can also use the model builder as an adjunct to theorem prover.
Let's suppose we are trying to prove `S |- g`, i.e. that `g`
is logically entailed by assumptions `S = {s1, s2, ..., sn}`.
We can this same input to Mace4, and the model builder will try to
find a counterexample, that is, to show that `g` does *not* follow
from `S`. So, given this input, Mace4 will try to find a model for
the set `S' = {s1, s2, ..., sn, (not g)}`. If `g` fails to follow
from `S`, then Mace4 may well return with a counterexample faster
than Prover9 concludes that it cannot find the required proof.
Conversely, if `g` *is* provable from `S`, Mace4 may take a long
time unsuccessfully trying to find a counter model, and will eventually give up.

In the following example, we see that the model builder does succeed in building a model of the assumptions together with the negation of the goal. That is, it succeeds in finding a model where there is a woman that every man loves; Adam is a man; Eve is a woman; but Adam does not love Eve.

>>> a4 = read_expr('exists y. (woman(y) & all x. (man(x) -> love(x,y)))') >>> a5 = read_expr('man(adam)') >>> a6 = read_expr('woman(eve)') >>> g = read_expr('love(adam,eve)') >>> print(mace.build_model(g, [a4, a5, a6])) True

The Model Builder will fail to find a model if the assumptions do entail the goal. Mace will continue to look for models of ever-increasing sizes until the end_size number is reached. By default, end_size is 500, but it can be set manually for quicker response time.

>>> a7 = read_expr('all x.(man(x) -> mortal(x))') >>> a8 = read_expr('man(socrates)') >>> g2 = read_expr('mortal(socrates)') >>> print(Mace(end_size=50).build_model(g2, [a7, a8])) False

There is also a `ModelBuilderCommand` class that, like `ProverCommand`,
stores a `ModelBuilder`, a goal, assumptions, a result, and a model. The
only implementation in NLTK is `MaceCommand`.

Mace4 is packaged with Prover9, and can be downloaded from the same source, namely http://www.cs.unm.edu/~mccune/prover9/. It is installed in the same manner as Prover9.

Check whether Mace4 can find a model.

>>> a = read_expr('(see(mary,john) & -(mary = john))') >>> mb = MaceCommand(assumptions=[a]) >>> mb.build_model() True

Show the model in 'tabular' format.

>>> print(mb.model(format='tabular')) % number = 1 % seconds = 0 <BLANKLINE> % Interpretation of size 2 <BLANKLINE> john : 0 <BLANKLINE> mary : 1 <BLANKLINE> see : | 0 1 ---+---- 0 | 0 0 1 | 1 0 <BLANKLINE>

Show the model in 'tabular' format.

>>> print(mb.model(format='cooked')) % number = 1 % seconds = 0 <BLANKLINE> % Interpretation of size 2 <BLANKLINE> john = 0. <BLANKLINE> mary = 1. <BLANKLINE> - see(0,0). - see(0,1). see(1,0). - see(1,1). <BLANKLINE>

The property `valuation` accesses the stored `Valuation`.

>>> print(mb.valuation) {'john': 'a', 'mary': 'b', 'see': {('b', 'a')}}

We can return to our earlier example and inspect the model:

>>> mb = MaceCommand(g, assumptions=[a4, a5, a6]) >>> m = mb.build_model() >>> print(mb.model(format='cooked')) % number = 1 % seconds = 0 <BLANKLINE> % Interpretation of size 2 <BLANKLINE> adam = 0. <BLANKLINE> eve = 0. <BLANKLINE> c1 = 1. <BLANKLINE> man(0). - man(1). <BLANKLINE> woman(0). woman(1). <BLANKLINE> - love(0,0). love(0,1). - love(1,0). - love(1,1). <BLANKLINE>

Here, we can see that `adam` and `eve` have been assigned the same
individual, namely `0` as value; `0` is both a man and a woman; a second
individual `1` is also a woman; and `0` loves `1`. Thus, this is
an interpretation in which there is a woman that every man loves but
Adam doesn't love Eve.

Mace can also be used with propositional logic.

>>> p = read_expr('P') >>> q = read_expr('Q') >>> mb = MaceCommand(q, [p, p>-q]) >>> mb.build_model() True >>> mb.valuation['P'] True >>> mb.valuation['Q'] False