Hans-Peter Störr

Approximating the Semantics of Logic Programs by Recurrent Neural Networks

In [1] we have shown how to construct a 3-layered recurrent neural network that computes the fixed point of the meaning function TP of a given propositional logic program P, which corresponds to the computation of the semantics of P. In this article we consider the first order case. We define a notion of approximation for interpretations and prove that there exists a 3-layered feed forward neural network that approximates the calculation of TP for a given first order acyclic logic program P with an injective level mapping arbitrarily well. Extending the feed forward network by recurrent connections we obtain a recurrent neural network whose iteration approximates the fixed point of TP. This result is proven by taking advantage of the fact that for acyclic logic programs the function TP is a contraction mapping on a complete metric space defined by the interpretations of the program. Mapping this space to the metric space R with Euclidean distance, a real valued function fP can be defined which corresponds to TP and is continuous as well as a contraction. Consequently it can be approximated by an appropriately chosen class of feed forward neural networks.

The Fuzzy Description Logic ALCFH with Hedge Algebras as Concept Modifiers

In this paper we present the fuzzy description logic ALCFH introduced, where primitive concepts are modified by means of hedges taken from hedge algebras. ALCFH is strictly more expressive than Fuzzy-ALC defined in [11]. We show that given a linearly ordered set of hedges primitive concepts can be modified to any desired degree by prefixing them with appropriate chains of hedges. Furthermore, we define a decision procedure for the unsatisfiability problem in ALCFH , and discuss knowledge base expansion when using terminologies, truth bounds, expressivity as well as complexity issues. We extend [8] by allowing modifiers on non-primitive concepts and extending the satisfiability procedure to handle concept definitions.

Incremental Fuzzy Decision Trees

We present a new classification algorithm that combines three properties: It generates decision trees, which proved a valuable and intelligible tool for classification and generalization of data; it utilizes fuzzy logic, that provides for a fine grained description of classified items adequate for human reasoning; and it is incremental, allowing rapid alternation of classification and learning of new data. The algorithm generalizes known non-incremental algorithms for top down induction of fuzzy decision trees, as well as known incremental algorithms for induction of decision trees in classical logic. The algorithm is shown to be terminating and to yield results equivalent to the non-incremental version.

Solving the Entailment Problem in the Fluent Calculus Using Binary Decision Diagrams

It is rigorously shown how planning problems encoded as a class of entailment problems in the fluent calculus can be mapped onto satisfiability problems for propositional formulas, which in turn can be mapped to the problem of finding models using binary decision diagrams (BDDs). The mapping is shown to be sound and complete. First experimental results of an implementation are presented and discussed.

Recurrent Neural Networks to Approximate the Semantics of Acceptable Logic Programs

In [9] we have shown how to construct a 3-layer recurrent neural network (RNN) that computes the iteration of the meaning function T p of a given propositional logic program, what corresponds to the computation of the semantics of the program.

A New Equational Foundation for the Fluent Calculus

A new equational foundation is presented for the Fluent Calculus, an established predicate calculus formalism for reasoning about actions. We discuss limitations of the existing axiomatizations of both equality of states and what it means for a fluent to hold in a state. Our new and conceptually even simpler theory is shown to overcome the restrictions of the existing approach. We prove that the correctness of the Fluent Calculus as a solution to the Frame Problem still holds under the new foundation. Furthermore, we extend our theory by an induction axiom needed for reasoning about integer-valued resources.

Complex Plans in the Fluent Calculus

Imagine an autonomous agent performing a task in the real world. Its performance is based on an internal plan. From a low level point of view, the world is its own representation and the actions of the aforementioned plan are simple commands controlling the effectors of the agent. At a higher level of abstraction, the world is internally represented by states and (primitive) actions are transformations on the space of states. In this article we do not want to discuss how actions on the higher level lead to commands on the lower level or whether the abstract level is needed or not, although these are very interesting and active open research problems. We also do not want to deal with another burning question of how the agent got hold of its plan. The plan may be given to it by a programmer, it may have (semi-)automatically generated the plan from the initial state, the goal state and the descriptions of the primitive actions it is able to perform or it may have learned it from examples. For the purpose of this article we just assume that a plan at the abstract level is given.