First-order logical neural networks with bidirectional inference
A system for configuring and using a logical neural network including a graph syntax tree of formulae in a represented knowledgebase connected to each other via nodes representing each proposition. One neuron exists for each logical connective occurring in each formula and, additionally, one neuron for each unique proposition occurring in any formula. All neurons return pairs of values representing upper and lower bounds on truth values of their corresponding subformulae and propositions. Neurons corresponding to logical connectives accept as input the output of neurons corresponding to their operands and have activation functions configured to match the connectives' truth functions. Neurons corresponding to propositions accept as input the output of neurons established as proofs of bounds on the propositions' truth values and have activation functions configured to aggregate the tightest such bounds. Bidirectional inference permits every occurrence of each proposition in each formula to be used as a potential proof.
1. A computer-implemented method comprising:
receiving, at one or more hardware processors, an input query and input propositions used to infer a logical truth based on said input query;
configuring, using said one or more hardware processors, a logical neural network (LNN) structure comprising one or more propositional neurons representing said input propositions and one or more neurons representing a logical connective according to a system of logical formulae corresponding to inference rules;
configuring at each logical connective neuron, using said one or more hardware processors, a respective activation function used to compute a truth of a logical connective of said logical formulae;
evaluating, using said LNN structure, a logical inference based on said input query; and
responsive to said evaluating, outputting an output truth value either proving or not the logical inference within an aggregated bounds.
2. The computer-implemented method of claim 1 , wherein said system of logical formula for said LNN are obtained from an ontology knowledge base, said configuring a LNN comprising:
choosing neural inputs and parameters for said one or more propositional and logical connective neurons to match a syntax and behavior of connectives making up the system of logical formulae from the knowledge base and choosing initial bounds on formula and proposition truth values chosen to match input facts.
3. The computer-implemented method of claim 1 , wherein said evaluating a logical inference comprises:
performing a forward inference pass using said one or more hardware processors to evaluate expressions of said LNN structure using an activation function to compute said truth values at each neuron, said truth values indicating an upper and lower bound on a logical formula or subformula of the system formulae.
4. The computer-implemented method of claim 3 , wherein said evaluating a logical inference comprises:
performing a backwards (downward) inference pass using said one or more hardware processors, to evaluate inverse inference rules during said downward inference pass; and
aggregating said truth values at propositional neurons and logical connective neurons during said backward inference pass.
5. The computer-implemented method of claim 4 , wherein said evaluating inverse inference rules during said downward inference path comprises:
computing an inverse activation function at a logical connective neuron to update said truth values and aggregating said truth values obtained in the downward inference pass.
6. The computer-implemented method of claim 5 , wherein said LNN structure is configured as a syntax tree, said backwards (downward) inference pass is performed using a recursive depth-first traversal of the syntax tree performed from one or more root nodes to each of the roots' leaves.
7. The computer-implemented method of claim 4 , wherein said aggregating said truth values at propositional neurons and logical connective neuron comprises: obtaining a tightest upper and lower bounds by taking a maximum lower bound value and a minimum upper bound value of said obtained truth values.
8. The computer-implemented method of claim 4 , wherein said one or more propositional neurons represent an input predicate comprising an n-dimensional table or tensor representation, where n is a non-negative number, the method further comprising:
providing for a neuron a table of truth value bounds uniquely keyed by a group of columns pertaining to an unbound logical variable occurring in a corresponding sub-formula or predicate; and
modifying a neural activation function to perform joins over columns pertaining to shared variables while computing truth value bounds at the associated rows.
9. The computer-implemented method of claim 8 , wherein said evaluating a logical inference to obtain truth values comprises:
performing joins between table representations of known bounds for a neuron's output and for the neuron's other inputs,
computing truth value bounds at the associated rows of said tensor representation; and
reducing over any columns pertaining to logical variables absent from the target input's corresponding subformula so as to aggregate the tightest such bounds.
10. The computer-implemented method of claim 1 , further comprising: initializing parameters of each neuron's respective activation function to effect an importance weighting over the neuron's inputs, output, or both inputs and outputs.
11. The computer-implemented method of claim 1 , wherein said logical connectives are configured according to a weighted Lukasiewicz logic scheme.