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535 lines
17 KiB
Plaintext
.. Copyright (C) 2001-2019 NLTK Project
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.. For license information, see LICENSE.TXT
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====================================
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Logical Inference and Model Building
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====================================
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>>> from nltk import *
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>>> from nltk.sem.drt import DrtParser
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>>> from nltk.sem import logic
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>>> logic._counter._value = 0
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------------
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Introduction
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------------
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Within the area of automated reasoning, first order theorem proving
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and model building (or model generation) have both received much
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attention, and have given rise to highly sophisticated techniques. We
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focus therefore on providing an NLTK interface to third party tools
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for these tasks. In particular, the module ``nltk.inference`` can be
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used to access both theorem provers and model builders.
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---------------------------------
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NLTK Interface to Theorem Provers
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---------------------------------
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The main class used to interface with a theorem prover is the ``Prover``
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class, found in ``nltk.api``. The ``prove()`` method takes three optional
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arguments: a goal, a list of assumptions, and a ``verbose`` boolean to
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indicate whether the proof should be printed to the console. The proof goal
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and any assumptions need to be instances of the ``Expression`` class
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specified by ``nltk.sem.logic``. There are currently three theorem provers
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included with NLTK: ``Prover9``, ``TableauProver``, and
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``ResolutionProver``. The first is an off-the-shelf prover, while the other
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two are written in Python and included in the ``nltk.inference`` package.
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>>> from nltk.sem import Expression
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>>> read_expr = Expression.fromstring
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>>> p1 = read_expr('man(socrates)')
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>>> p2 = read_expr('all x.(man(x) -> mortal(x))')
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>>> c = read_expr('mortal(socrates)')
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>>> Prover9().prove(c, [p1,p2])
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True
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>>> TableauProver().prove(c, [p1,p2])
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True
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>>> ResolutionProver().prove(c, [p1,p2], verbose=True)
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[1] {-mortal(socrates)} A
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[2] {man(socrates)} A
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[3] {-man(z2), mortal(z2)} A
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[4] {-man(socrates)} (1, 3)
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[5] {mortal(socrates)} (2, 3)
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[6] {} (1, 5)
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<BLANKLINE>
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True
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---------------------
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The ``ProverCommand``
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---------------------
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A ``ProverCommand`` is a stateful holder for a theorem
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prover. The command stores a theorem prover instance (of type ``Prover``),
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a goal, a list of assumptions, the result of the proof, and a string version
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of the entire proof. Corresponding to the three included ``Prover``
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implementations, there are three ``ProverCommand`` implementations:
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``Prover9Command``, ``TableauProverCommand``, and
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``ResolutionProverCommand``.
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The ``ProverCommand``'s constructor takes its goal and assumptions. The
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``prove()`` command executes the ``Prover`` and ``proof()``
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returns a String form of the proof
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If the ``prove()`` method has not been called,
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then the prover command will be unable to display a proof.
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>>> prover = ResolutionProverCommand(c, [p1,p2])
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>>> print(prover.proof()) # doctest: +ELLIPSIS
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Traceback (most recent call last):
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File "...", line 1212, in __run
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compileflags, 1) in test.globs
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File "<doctest nltk/test/inference.doctest[10]>", line 1, in <module>
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File "...", line ..., in proof
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raise LookupError("You have to call prove() first to get a proof!")
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LookupError: You have to call prove() first to get a proof!
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>>> prover.prove()
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True
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>>> print(prover.proof())
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[1] {-mortal(socrates)} A
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[2] {man(socrates)} A
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[3] {-man(z4), mortal(z4)} A
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[4] {-man(socrates)} (1, 3)
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[5] {mortal(socrates)} (2, 3)
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[6] {} (1, 5)
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<BLANKLINE>
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The prover command stores the result of proving so that if ``prove()`` is
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called again, then the command can return the result without executing the
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prover again. This allows the user to access the result of the proof without
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wasting time re-computing what it already knows.
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>>> prover.prove()
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True
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>>> prover.prove()
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True
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The assumptions and goal may be accessed using the ``assumptions()`` and
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``goal()`` methods, respectively.
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>>> prover.assumptions()
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[<ApplicationExpression man(socrates)>, <Alread_expression all x.(man(x) -> mortal(x))>]
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>>> prover.goal()
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<ApplicationExpression mortal(socrates)>
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The assumptions list may be modified using the ``add_assumptions()`` and
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``retract_assumptions()`` methods. Both methods take a list of ``Expression``
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objects. Since adding or removing assumptions may change the result of the
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proof, the stored result is cleared when either of these methods are called.
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That means that ``proof()`` will be unavailable until ``prove()`` is called and
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a call to ``prove()`` will execute the theorem prover.
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>>> prover.retract_assumptions([read_expr('man(socrates)')])
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>>> print(prover.proof()) # doctest: +ELLIPSIS
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Traceback (most recent call last):
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File "...", line 1212, in __run
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compileflags, 1) in test.globs
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File "<doctest nltk/test/inference.doctest[10]>", line 1, in <module>
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File "...", line ..., in proof
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raise LookupError("You have to call prove() first to get a proof!")
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LookupError: You have to call prove() first to get a proof!
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>>> prover.prove()
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False
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>>> print(prover.proof())
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[1] {-mortal(socrates)} A
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[2] {-man(z6), mortal(z6)} A
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[3] {-man(socrates)} (1, 2)
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<BLANKLINE>
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>>> prover.add_assumptions([read_expr('man(socrates)')])
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>>> prover.prove()
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True
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-------
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Prover9
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-------
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Prover9 Installation
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~~~~~~~~~~~~~~~~~~~~
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You can download Prover9 from http://www.cs.unm.edu/~mccune/prover9/.
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Extract the source code into a suitable directory and follow the
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instructions in the Prover9 ``README.make`` file to compile the executables.
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Install these into an appropriate location; the
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``prover9_search`` variable is currently configured to look in the
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following locations:
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>>> p = Prover9()
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>>> p.binary_locations() # doctest: +NORMALIZE_WHITESPACE
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['/usr/local/bin/prover9',
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'/usr/local/bin/prover9/bin',
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'/usr/local/bin',
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'/usr/bin',
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'/usr/local/prover9',
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'/usr/local/share/prover9']
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Alternatively, the environment variable ``PROVER9HOME`` may be configured with
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the binary's location.
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The path to the correct directory can be set manually in the following
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manner:
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>>> config_prover9(path='/usr/local/bin') # doctest: +SKIP
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[Found prover9: /usr/local/bin/prover9]
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If the executables cannot be found, ``Prover9`` will issue a warning message:
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>>> p.prove() # doctest: +SKIP
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Traceback (most recent call last):
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...
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LookupError:
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===========================================================================
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NLTK was unable to find the prover9 executable! Use config_prover9() or
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set the PROVER9HOME environment variable.
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<BLANKLINE>
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>> config_prover9('/path/to/prover9')
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<BLANKLINE>
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For more information, on prover9, see:
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<http://www.cs.unm.edu/~mccune/prover9/>
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===========================================================================
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Using Prover9
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~~~~~~~~~~~~~
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The general case in theorem proving is to determine whether ``S |- g``
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holds, where ``S`` is a possibly empty set of assumptions, and ``g``
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is a proof goal.
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As mentioned earlier, NLTK input to ``Prover9`` must be
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``Expression``\ s of ``nltk.sem.logic``. A ``Prover9`` instance is
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initialized with a proof goal and, possibly, some assumptions. The
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``prove()`` method attempts to find a proof of the goal, given the
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list of assumptions (in this case, none).
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>>> goal = read_expr('(man(x) <-> --man(x))')
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>>> prover = Prover9Command(goal)
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>>> prover.prove()
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True
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Given a ``ProverCommand`` instance ``prover``, the method
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``prover.proof()`` will return a String of the extensive proof information
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provided by Prover9, shown in abbreviated form here::
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============================== Prover9 ===============================
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Prover9 (32) version ...
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Process ... was started by ... on ...
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...
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The command was ".../prover9 -f ...".
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============================== end of head ===========================
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============================== INPUT =================================
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% Reading from file /var/...
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formulas(goals).
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(all x (man(x) -> man(x))).
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end_of_list.
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...
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============================== end of search =========================
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THEOREM PROVED
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Exiting with 1 proof.
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Process 6317 exit (max_proofs) Mon Jan 21 15:23:28 2008
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As mentioned earlier, we may want to list some assumptions for
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the proof, as shown here.
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>>> g = read_expr('mortal(socrates)')
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>>> a1 = read_expr('all x.(man(x) -> mortal(x))')
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>>> prover = Prover9Command(g, assumptions=[a1])
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>>> prover.print_assumptions()
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all x.(man(x) -> mortal(x))
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However, the assumptions are not sufficient to derive the goal:
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>>> print(prover.prove())
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False
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So let's add another assumption:
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>>> a2 = read_expr('man(socrates)')
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>>> prover.add_assumptions([a2])
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>>> prover.print_assumptions()
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all x.(man(x) -> mortal(x))
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man(socrates)
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>>> print(prover.prove())
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True
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We can also show the assumptions in ``Prover9`` format.
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>>> prover.print_assumptions(output_format='Prover9')
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all x (man(x) -> mortal(x))
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man(socrates)
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>>> prover.print_assumptions(output_format='Spass')
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Traceback (most recent call last):
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. . .
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NameError: Unrecognized value for 'output_format': Spass
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Assumptions can be retracted from the list of assumptions.
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>>> prover.retract_assumptions([a1])
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>>> prover.print_assumptions()
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man(socrates)
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>>> prover.retract_assumptions([a1])
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Statements can be loaded from a file and parsed. We can then add these
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statements as new assumptions.
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>>> g = read_expr('all x.(boxer(x) -> -boxerdog(x))')
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>>> prover = Prover9Command(g)
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>>> prover.prove()
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False
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>>> import nltk.data
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>>> new = nltk.data.load('grammars/sample_grammars/background0.fol')
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>>> for a in new:
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... print(a)
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all x.(boxerdog(x) -> dog(x))
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all x.(boxer(x) -> person(x))
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all x.-(dog(x) & person(x))
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exists x.boxer(x)
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exists x.boxerdog(x)
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>>> prover.add_assumptions(new)
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>>> print(prover.prove())
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True
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>>> print(prover.proof()) # doctest: +ELLIPSIS
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============================== prooftrans ============================
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Prover9 (...) version ...
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Process ... was started by ... on ...
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...
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The command was ".../prover9".
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============================== end of head ===========================
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<BLANKLINE>
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============================== end of input ==========================
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<BLANKLINE>
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============================== PROOF =================================
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<BLANKLINE>
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% -------- Comments from original proof --------
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% Proof 1 at ... seconds.
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% Length of proof is 13.
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% Level of proof is 4.
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% Maximum clause weight is 0.000.
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% Given clauses 0.
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<BLANKLINE>
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<BLANKLINE>
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1 (all x (boxerdog(x) -> dog(x))). [assumption].
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2 (all x (boxer(x) -> person(x))). [assumption].
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3 (all x -(dog(x) & person(x))). [assumption].
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6 (all x (boxer(x) -> -boxerdog(x))). [goal].
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8 -boxerdog(x) | dog(x). [clausify(1)].
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9 boxerdog(c3). [deny(6)].
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11 -boxer(x) | person(x). [clausify(2)].
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12 boxer(c3). [deny(6)].
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14 -dog(x) | -person(x). [clausify(3)].
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15 dog(c3). [resolve(9,a,8,a)].
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18 person(c3). [resolve(12,a,11,a)].
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19 -person(c3). [resolve(15,a,14,a)].
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20 $F. [resolve(19,a,18,a)].
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<BLANKLINE>
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============================== end of proof ==========================
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----------------------
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The equiv() method
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----------------------
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One application of the theorem prover functionality is to check if
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two Expressions have the same meaning.
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The ``equiv()`` method calls a theorem prover to determine whether two
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Expressions are logically equivalent.
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>>> a = read_expr(r'exists x.(man(x) & walks(x))')
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>>> b = read_expr(r'exists x.(walks(x) & man(x))')
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>>> print(a.equiv(b))
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True
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The same method can be used on Discourse Representation Structures (DRSs).
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In this case, each DRS is converted to a first order logic form, and then
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passed to the theorem prover.
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>>> dp = DrtParser()
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>>> a = dp.parse(r'([x],[man(x), walks(x)])')
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>>> b = dp.parse(r'([x],[walks(x), man(x)])')
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>>> print(a.equiv(b))
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True
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--------------------------------
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NLTK Interface to Model Builders
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--------------------------------
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The top-level to model builders is parallel to that for
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theorem-provers. The ``ModelBuilder`` interface is located
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in ``nltk.inference.api``. It is currently only implemented by
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``Mace``, which interfaces with the Mace4 model builder.
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Typically we use a model builder to show that some set of formulas has
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a model, and is therefore consistent. One way of doing this is by
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treating our candidate set of sentences as assumptions, and leaving
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the goal unspecified.
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Thus, the following interaction shows how both ``{a, c1}`` and ``{a, c2}``
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are consistent sets, since Mace succeeds in a building a
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model for each of them, while ``{c1, c2}`` is inconsistent.
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>>> a3 = read_expr('exists x.(man(x) and walks(x))')
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>>> c1 = read_expr('mortal(socrates)')
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>>> c2 = read_expr('-mortal(socrates)')
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>>> mace = Mace()
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>>> print(mace.build_model(None, [a3, c1]))
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True
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>>> print(mace.build_model(None, [a3, c2]))
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True
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We can also use the model builder as an adjunct to theorem prover.
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Let's suppose we are trying to prove ``S |- g``, i.e. that ``g``
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is logically entailed by assumptions ``S = {s1, s2, ..., sn}``.
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We can this same input to Mace4, and the model builder will try to
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find a counterexample, that is, to show that ``g`` does *not* follow
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from ``S``. So, given this input, Mace4 will try to find a model for
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the set ``S' = {s1, s2, ..., sn, (not g)}``. If ``g`` fails to follow
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from ``S``, then Mace4 may well return with a counterexample faster
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than Prover9 concludes that it cannot find the required proof.
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Conversely, if ``g`` *is* provable from ``S``, Mace4 may take a long
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time unsuccessfully trying to find a counter model, and will eventually give up.
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In the following example, we see that the model builder does succeed
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in building a model of the assumptions together with the negation of
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the goal. That is, it succeeds in finding a model
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where there is a woman that every man loves; Adam is a man; Eve is a
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woman; but Adam does not love Eve.
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>>> a4 = read_expr('exists y. (woman(y) & all x. (man(x) -> love(x,y)))')
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>>> a5 = read_expr('man(adam)')
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>>> a6 = read_expr('woman(eve)')
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>>> g = read_expr('love(adam,eve)')
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>>> print(mace.build_model(g, [a4, a5, a6]))
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True
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The Model Builder will fail to find a model if the assumptions do entail
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the goal. Mace will continue to look for models of ever-increasing sizes
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until the end_size number is reached. By default, end_size is 500,
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but it can be set manually for quicker response time.
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>>> a7 = read_expr('all x.(man(x) -> mortal(x))')
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>>> a8 = read_expr('man(socrates)')
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>>> g2 = read_expr('mortal(socrates)')
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>>> print(Mace(end_size=50).build_model(g2, [a7, a8]))
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False
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There is also a ``ModelBuilderCommand`` class that, like ``ProverCommand``,
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stores a ``ModelBuilder``, a goal, assumptions, a result, and a model. The
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only implementation in NLTK is ``MaceCommand``.
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-----
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Mace4
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-----
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Mace4 Installation
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~~~~~~~~~~~~~~~~~~
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Mace4 is packaged with Prover9, and can be downloaded from the same
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source, namely http://www.cs.unm.edu/~mccune/prover9/. It is installed
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in the same manner as Prover9.
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Using Mace4
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~~~~~~~~~~~
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Check whether Mace4 can find a model.
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>>> a = read_expr('(see(mary,john) & -(mary = john))')
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>>> mb = MaceCommand(assumptions=[a])
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>>> mb.build_model()
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True
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Show the model in 'tabular' format.
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>>> print(mb.model(format='tabular'))
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% number = 1
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% seconds = 0
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<BLANKLINE>
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% Interpretation of size 2
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<BLANKLINE>
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john : 0
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<BLANKLINE>
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mary : 1
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<BLANKLINE>
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see :
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| 0 1
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---+----
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0 | 0 0
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1 | 1 0
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<BLANKLINE>
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Show the model in 'tabular' format.
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>>> print(mb.model(format='cooked'))
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% number = 1
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% seconds = 0
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<BLANKLINE>
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% Interpretation of size 2
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<BLANKLINE>
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john = 0.
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<BLANKLINE>
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mary = 1.
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<BLANKLINE>
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- see(0,0).
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- see(0,1).
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see(1,0).
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- see(1,1).
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<BLANKLINE>
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The property ``valuation`` accesses the stored ``Valuation``.
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>>> print(mb.valuation)
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{'john': 'a', 'mary': 'b', 'see': {('b', 'a')}}
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We can return to our earlier example and inspect the model:
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>>> mb = MaceCommand(g, assumptions=[a4, a5, a6])
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>>> m = mb.build_model()
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>>> print(mb.model(format='cooked'))
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% number = 1
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% seconds = 0
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<BLANKLINE>
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% Interpretation of size 2
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<BLANKLINE>
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adam = 0.
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<BLANKLINE>
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eve = 0.
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<BLANKLINE>
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c1 = 1.
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<BLANKLINE>
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man(0).
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- man(1).
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<BLANKLINE>
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woman(0).
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woman(1).
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<BLANKLINE>
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- love(0,0).
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love(0,1).
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- love(1,0).
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- love(1,1).
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<BLANKLINE>
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|
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Here, we can see that ``adam`` and ``eve`` have been assigned the same
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individual, namely ``0`` as value; ``0`` is both a man and a woman; a second
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individual ``1`` is also a woman; and ``0`` loves ``1``. Thus, this is
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|
an interpretation in which there is a woman that every man loves but
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Adam doesn't love Eve.
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Mace can also be used with propositional logic.
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|
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>>> p = read_expr('P')
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>>> q = read_expr('Q')
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>>> mb = MaceCommand(q, [p, p>-q])
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>>> mb.build_model()
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True
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|
>>> mb.valuation['P']
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True
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|
>>> mb.valuation['Q']
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|
False
|