Algebraic anti-unification

Abstraction is key to human and artificial intelligence as it allows one to see common structure in otherwise distinct objects or situations and as such it is a key element for generality in AI. Anti-unification (or generalization) is \textit{the} part of theoretical computer science and AI studying abstraction. It has been successfully applied to various AI-related problems, most importantly inductive logic programming. Up to this date, anti-unification is studied only from a syntactic perspective in the literature. The purpose of this paper is to initiate an algebraic (i.e. semantic) theory of anti-unification within general algebras. This is motivated by recent applications to similarity and analogical proportions.

Neural logic programs and neural nets

Neural-symbolic integration aims to combine the connectionist subsymbolic with the logical symbolic approach to artificial intelligence. In this paper, we first define the answer set semantics of (boolean) neural nets and then introduce from first principles a class of neural logic programs and show that nets and programs are equivalent.

Analogical proportions II

Analogical reasoning is the ability to detect parallels between two seemingly distant objects or situations, a fundamental human capacity used for example in commonsense reasoning, learning, and creativity which is believed by many researchers to be at the core of human and artificial general intelligence. Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. It is the purpose of this paper to further develop the mathematical theory of analogical proportions within that framework as motivated by the fact that it has already been successfully applied to logic program synthesis in artificial intelligence.

Similarity-based analogical proportions

The author has recently introduced abstract algebraic frameworks of analogical proportions and similarity within the general setting of universal algebra. The purpose of this paper is to build a bridge from similarity to analogical proportions by formulating the latter in terms of the former. The benefit of this similarity-based approach is that the connection between proportions and similarity is built into the framework and therefore evident which is appealing since proportions and similarity are both at the center of analogy; moreover, future results on similarity can directly be applied to analogical proportions.

Bilingual analogical proportions

Analogical proportions are expressions of the form ``$a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning which itself is at the core of human and artificial intelligence. The author has recently introduced {\em from first principles} an abstract algebro-logical framework of analogical proportions within the general setting of universal algebra and first-order logic. In that framework, the source and target algebras have the {\em same} underlying language. The purpose of this paper is to generalize his unilingual framework to a bilingual one where the underlying languages may differ. This is achieved by using hedges in justifications of proportions. The outcome is a major generalization vastly extending the applicability of the underlying framework. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.

Generalization-based similarity

Detecting and exploiting similarities between seemingly distant objects is at the core of analogical reasoning which itself is at the core of artificial intelligence. This paper develops {\em from the ground up} an abstract algebraic and {\em qualitative} notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can be naturally embedded into first-order logic via model-theoretic types. In a broader sense, this paper is a further step towards a mathematical theory of analogical reasoning.

Analogical proportions in monounary algebras and difference proportions

This paper studies analogical proportions in monounary algebras consisting only of a universe and a single unary function. We show that the analogical proportion relation is characterized in the infinite monounary algebra formed by the natural numbers together with the successor function via difference proportions.

Boolean proportions

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper studies analogical proportions between booleans of the form `$a$ is to $b$ what $c$ is to $d$' called boolean proportions. Technically, we instantiate an abstract algebraic framework of analogical proportions -- recently introduced by the author -- in the boolean domain consisting of the truth values true and false together with boolean functions. It turns out that our notion of boolean proportions has appealing mathematical properties and that it coincides with a prominent model of boolean proportions in the general case. In a broader sense, this paper is a further step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.

Analogical Proportions

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper contributes to the foundations of artificial general intelligence by introducing an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. The main idea is to define solutions to analogical proportions in terms of generalizations and to derive abstract terms of concrete elements from a `known' source domain which can then be instantiated in an `unknown' target domain to obtain analogous elements. We extensively compare our framework with two prominent and recently introduced frameworks of analogical proportions from the literature in the concrete domains of sets, numbers, and words and show that our framework is strictly more general in all of these cases which provides strong evidence for the applicability of our framework. In a broader sense, this paper is a first step towards an algebraic theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.

Fixed Point Semantics for Stream Reasoning

Reasoning over streams of input data is an essential part of human intelligence. During the last decade {\em stream reasoning} has emerged as a research area within the AI-community with many potential applications. In fact, the increased availability of streaming data via services like Google and Facebook has raised the need for reasoning engines coping with data that changes at high rate. Recently, the rule-based formalism {\em LARS} for non-monotonic stream reasoning under the answer set semantics has been introduced. Syntactically, LARS programs are logic programs with negation incorporating operators for temporal reasoning, most notably {\em window operators} for selecting relevant time points. Unfortunately, by preselecting {\em fixed} intervals for the semantic evaluation of programs, the rigid semantics of LARS programs is not flexible enough to {\em constructively} cope with rapidly changing data dependencies. Moreover, we show that defining the answer set semantics of LARS in terms of FLP reducts leads to undesirable circular justifications similar to other ASP extensions. This paper fixes all of the aforementioned shortcomings of LARS. More precisely, we contribute to the foundations of stream reasoning by providing an operational fixed point semantics for a fully flexible variant of LARS and we show that our semantics is sound and constructive in the sense that answer sets are derivable bottom-up and free of circular justifications.