Tensor Theorem Prover

First-order logic theorem prover supporting unification with approximate vector similarity

Full docs: https://tensor-theorem-prover.readthedocs.io

Installation

pip install tensor-theorem-prover

Usage

tensor-theorem-prover can be used either as a standard symbolic first-order theorem prover, or it can be used with vector embeddings and fuzzy unification.

The basic setup requires listing out first-order formulae, and using the ResolutionProver class to generate proofs.

import numpy as np
from vector_theorem_prover import ResolutionProver, Constant, Predicate, Variable, Implies

X = Variable("X")
Y = Variable("Y")
Z = Variable("Z")
# predicates and constants can be given an embedding array for fuzzy unification
grandpa_of = Predicate("grandpa_of", np.array([1.0, 1.0, 0.0, 0.3, ...]))
grandfather_of = Predicate("grandfather_of", np.array([1.01, 0.95, 0.05, 0.33, ...]))
parent_of = Predicate("parent_of", np.array([ ... ]))
father_of = Predicate("father_of", np.array([ ... ]))
bart = Constant("bart", np.array([ ... ]))
homer = Constant("homer", np.array([ ... ]))
abe = Constant("abe", np.array([ ... ]))

knowledge = [
    parent_of(homer, bart),
    father_of(abe, homer),
    # father_of(X, Z) ^ parent_of(Z, Y) -> grandpa_of(X, Y)
    Implies(And(father_of(X, Z), parent_of(Z, Y)), grandpa_of(X, Y))
]

prover = ResolutionProver(knowledge=knowledge)

# query the prover to find who is bart's grandfather
proof = prover.prove(grandfather_of(X, bart))

# even though `grandpa_of` and `grandfather_of` are not identical symbols,
# their embedding is close enough that the prover can still find the answer
print(proof.substitutions[X]) # abe

# the prover will return `None` if a proof could not be found
failed_proof = prover.prove(grandfather_of(bart, homer))
print(failed_proof) # None

Working with proof results

The prover.prove() method will return a Proof object if a successful proof is found. This object contains a list of all the resolutions, substitutions, and similarity calculations that went into proving the goal.

proof = prover.prove(goal)

proof.substitutions # a map of all variables in the goal to their bound values
proof.similarity # the min similarity of all `unify` operations in this proof
proof.depth # the number of steps in this proof
proof.proof_steps # all the proof steps involved, including all resolutions and unifications along the way

The Proof object can be printed as a string to get a visual overview of the steps involved in the proof.

X = Variable("X")
Y = Variable("Y")
father_of = Predicate("father_of")
parent_of = Predicate("parent_of")
is_male = Predicate("is_male")
bart = Constant("bart")
homer = Constant("homer")

knowledge = [
    parent_of(homer, bart),
    is_male(homer),
    Implies(And(parent_of(X, Y), is_male(X)), father_of(X, Y)),
]

prover = ResolutionProver(knowledge=knowledge)

goal = father_of(homer, X)
proof = prover.prove(goal)

print(proof)
# Goal: [¬father_of(homer,X)]
# Subsitutions: {X -> bart}
# Similarity: 1.0
# Depth: 3
# Steps:
#   Similarity: 1.0
#   Source: [¬father_of(homer,X)]
#   Target: [father_of(X,Y) ∨ ¬is_male(X) ∨ ¬parent_of(X,Y)]
#   Unify: father_of(homer,X) = father_of(X,Y)
#   Subsitutions: {}, {X -> homer, Y -> X}
#   Resolvent: [¬is_male(homer) ∨ ¬parent_of(homer,X)]
#   ---
#   Similarity: 1.0
#   Source: [¬is_male(homer) ∨ ¬parent_of(homer,X)]
#   Target: [parent_of(homer,bart)]
#   Unify: parent_of(homer,X) = parent_of(homer,bart)
#   Subsitutions: {X -> bart}, {}
#   Resolvent: [¬is_male(homer)]
#   ---
#   Similarity: 1.0
#   Source: [¬is_male(homer)]
#   Target: [is_male(homer)]
#   Unify: is_male(homer) = is_male(homer)
#   Subsitutions: {}, {}
#   Resolvent: []

Finding all possible proofs

The prover.prove() method will return the proof with the highest similarity score among all possible proofs, if one exists. If you want to get a list of all the possible proofs in descending order of similarity score, you can call prover.prove_all() to return a list of all proofs.

Custom matching functions and similarity thresholds

By default, the prover will use cosine similarity for unification. If you'd like to use a different similarity function, you can pass in a function to the prover to perform the similarity calculation however you wish.

def fancy_similarity(item1, item2):
    norm = np.linalg.norm(item1.embedding) + np.linalg.norm(item2.embedding)
    return np.linalg.norm(item1.embedding - item2.embedding) / norm

prover = ResolutionProver(knowledge=knowledge, similarity_func=fancy_similarity)

By default, there is a minimum similarity threshold of 0.5 for a unification to success. You can customize this as well when creating a ResolutionProver instance

prover = ResolutionProver(knowledge=knowledge, min_similarity_threshold=0.9)

Working with Tensors (Pytorch, Tensorflow, etc...)

By default, the similarity calculation assumes that the embeddings supplied for constants and predicates are numpy arrays. If you want to use tensors instead, this will work as long as you provide a similarity_func which can work with the tensor types you're using and return a float.

For example, if you're using Pytorch, it might look like the following:

import torch

def torch_cosine_similarity(item1, item2):
    similarity = torch.nn.functional.cosine_similarity(
        item1.embedding,
        item2.embedding,
        0
    )
    return similarity.item()

prover = ResolutionProver(knowledge=knowledge, similarity_func=torch_cosine_similarity)

# for pytorch you may want to wrap the proving in torch.no_grad()
with torch.no_grad():
    proof = prover.prove(goal)

Max proof depth

By default, the ResolutionProver will abort proofs after a depth of 10. You can customize this behavior by passing max_proof_depth when creating the prover

prover = ResolutionProver(knowledge=knowledge, max_proof_depth=10)

Max resolvent width

By default, the ResolutionProver has no limit on how wide resolvents can get during the proving process. If the proofs are running too slowly, you can try to set max_resolvent_width to limit how many literals intermediate resolvents are allowed to contain. This should narrow the search tree, but has the trade-off of not finding proofs if the proof requires unifying together a lot of very wide clauses.

prover = ResolutionProver(knowledge=knowledge, max_resolvent_width=10)

Skipping seen resolvents

A major performance improvement when searching through a large proof space is to stop searching any branches that encounter a resolvent that's already been seen. Doing this is still guaranteed to find the proof with the highest similarity score, but it means the prover is no longer guaranteed to find every possible proof when running prover.prove_all(). Although, when dealing with anything beyond very small knowledge bases, finding every possible proof is likely not going to be computationally feasible anyway.

Searching for a proof using prover.prove() always enables this optimization, but you can enable it when using prover.prove_all() as well by passing the option skip_seen_resolvents=True when creating the ResolutionProver, like below:

prover = ResolutionProver(knowledge=knowledge, skip_seen_resolvents=True)

Max resolution attempts

As a final backstop against the search tree getting too large, you can set a maximum resolution attempts parameter to force the prover to give up after a finite amount of attempts. You can set this parameter when creating a ResolutionProver as shown below:

prover = ResolutionProver(knowledge=knowledge, max_resolution_attempts=100_000_000)

Multithreading

By default, the ResolutionProver will try to use available CPU cores up to a max of 6, though this may change in future releases. If you want to explicitly control the number of worker threads used for solving, pass num_workers when creating the ResolutionProver, like below:

prover = ResolutionProver(knowledge=knowledge, num_workers=1)

Acknowledgements

This library borrows code and ideas from the earier library fuzzy-reasoner. The main difference between these libraries is that tensor-theorem-prover supports full first-order logic using Resolution, whereas fuzzy-reasoner is restricted to Horn clauses and uses backwards chaining. This library is also much more optimized than the fuzzy-reasoner, as the core of tensor-theorem-prover is written in rust and supports multithreading, while fuzzy-reasoner is pure Python.

Like fuzzy-reasoner, this library also takes inspiration from the following papers for the idea of using vector similarity in theorem proving:

• End-to-End Differentiable Proving by Rocktäschel et al.
• Braid - Weaving Symbolic and Neural Knowledge into Coherent Logical Explanations by Kalyanpur et al.

Contributing

Contributions are welcome! Please leave an issue in the Github repo if you find any bugs, and open a pull request with and fixes or improvements that you'd like to contribute.