Tensor Theorem Prover
First-order logic theorem prover supporting unification with approximate vector similarity
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) # NoneWorking 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 wayThe 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)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 restricting to Horn clauses and uses backwards chaining.
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.
