Distributed representations (such as those based on embeddings) and discrete representations (such as those based on logic) have complementary strengths. We explore one possible approach to combining these two kinds of representations. We present a model theory/semantics for first order logic based on vectors of reals. We describe the model theory, discuss some interesting properties of such a system and present a simple approach to query answering.

The interest in the combination of probability with logics for modeling the world has rapidly increased in the last few years. One of the most effective approaches is the Distribution Semantics which was adopted by many logic programming languages and in Descripion Logics. In this paper, we illustrate the work we have done in this research field by presenting a probabilistic semantics for description logics and reasoning and learning algorithms. In particular, we present in detail the system TRILL P, which computes the probability of queries w.r.t. probabilistic knowledge bases, which has been implemented in Prolog. Note: An extended abstract / full version of a paper accepted to be presented at the Doctoral Consortium of the 30th International Conference on Logic Programming (ICLP 2014), July 19-22, Vienna, Austria

Essentially, a connection graph is merely a data structure for a set of clauses indicating possible system. To use the graph, one has to introduce operations on the graph.

In AI we talk about expert systems.

The strategy has been implemented in a simple QLISP program.

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The first section discusses the importance of having systems that own M.S. thesis (Moore, 19)5), suggests that predicate calculus can understand the concept of knowledge, and how knowledge is be treated in a more natural manner than resolution and related to action. Section 2 points out some of the special problems combined with domain-dependent control information for greater that are involved in reasoning about knowledge, and section $ efficiency. Furthermore, the problems of reasoning about knowledge seem to require the full ability to handle quantifiers presents a logic of knowledge based on the idea of possible worlds. Section 4 integrates this with a logic of actions and gives an and logical connectives which only predicate calculus posseses.

The extensions are heuristically non-trivial, because the equivalent In (McCarthy and Hayes 1969), we proposed dividing the predicate calculus may be much longer and is usually much more artificial intelligence problem into two parts - an epistemological difficult to understand - for man or machine.

Circumscription formalizes such conjectural reasoning.

Program synthesis is the systematic derivation of a program from a given specification. A deductive approach to program synthesis is presented for the construction of recursive programs. This approach regards program synthesis as a theorem-proving task and relies on a theorem-proving method that combines the features of transformation rules, unification, and mathematical induction within a single framework. MOTIVATION The early work in program synthesis relied strongly on mechanical theoremproving techniques. The work of Green [5] and Waldinger and Lee [13], for example, depended on resolution-based theorem proving; however, the difficulty of representing the principle of mathematical induction in a resolution framework hampered these systems in the formation of programs with iterative or recursive loops. More recently, program synthesis and theorem proving have tended to go their separate ways. Newer theorem-proving systems are able to perform proofs by mathematical induction (e.g., Boyer and Moore [2]) but are useless for program synthesis because they have sacrificed the ability to prove theorems involving existential quantifiers. In this paper we describe a framework for program synthesis that again relies on a theorem-proving approach. This approach combines techniques of unification, mathematical induction, and transformation rules within a single deductive system. We outline the logical structure of this system without considering the strategic aspects of how deductions are directed. Although no implementation exists, the approach is machine oriented and ultimately intended for implementation in automatic synthesis systems. In the next section we give examples of specifications accepted by the system. In the succeeding sections we explain the relation between theorem proving and our approach to program synthesis. SPECIFICATION The specification of a program allows us to express the purpose of the desired program, without indicating an algorithm by which that purpose is to be achieved.