This paper addresses the current lack of a unified formal framework in machine learning theory, as well as the absence of robust theoretical foundations for interpretability and ethical safety assurance. We first construct a formal information model, employing sets of well-formed formulas (WFFs) to explicitly define the ontological states and carrier mappings for the core components of machine learning. By introducing learnable and processable predicates, as well as learning and processing functions, we analyze the logical inference and constraint rules underlying causal chains in models, thereby establishing the Machine Learning Theory Meta-Framework (MLT-MF). Building upon this framework, we propose universal definitions for model interpretability and ethical safety, and rigorously prove and validate four key theorems: the equivalence between model interpretability and information existence, the constructive formulation of ethical safety assurance and two types of total variation distance (TVD) upper bounds. This work overcomes the limitations of previous fragmented approaches, providing a unified theoretical foundation from an information science perspective to systematically address the critical challenges currently facing machine learning.

There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we present MATR, a new framework for automated theorem proving explicitly designed to easily adapt to unusual logics or integrate new reasoning processes. MATR is formalism-agnostic, highly modular, and programmer-friendly. We explain the high-level design of MATR as well as some details of its implementation. To demonstrate MATR's utility, we then describe a formalized metalogic suitable for proofs of G\"odel's Incompleteness Theorems, and report on our progress using our metalogic in MATR to semi-autonomously generate proofs of both the First and Second Incompleteness Theorems.

I used to teach the calculus in a private high school where I was headmaster and would occasionally play a game called "WFF'N Proof" with my students. They had to roll a set of six dice that had sides marked with the following: p, q, r, C, A, K, E or N. After a throw they had to see if they could construct a WFF, a well-formed-formula, by arranging the outcomes in various combinations. A WFF was defined as any of the following: p, q or r. (Addendum 1, 2/7/16: Note that there is no indication of p, q, or r having common features.) Also WFF's were combinations of the letters C, A, K or E followed by two WFF's. Or N followed by a single WFF. So beginning with p, q, or r, one could possibly construct the new WFF's, Np or Nq or Nr. Continuing in this vein, one might construct Apq, or ANpp, or CAqKqrr, … (Addendum 2, 2/7/16: Note also that no two complex WFF's need have common features, e.g.

Justification logic originated from the study of the logic of proofs.  However, in a more general setting, it may be regarded as a kind of explicit epistemic logic. In such logic, the reasons why a fact is believed are explicitly represented as justification terms.  Traditionally, the modeling of uncertain beliefs is crucially important for epistemic reasoning. While graded modal logics interpreted with possibility theory semantics have been successfully applied to the representation and reasoning of uncertain beliefs, they cannot keep track of the reasons why an agent believes a fact. The objective of this paper is to extend the graded modal logics with explicit justifications. We introduce a possibilistic justification logic,  present its syntax and semantics, and investigate its meta-properties, such as soundness, completeness, and realizability.

In AI we talk about expert systems.

RUM (Reasoning with Uncertainty Module), is an integrated software tool based on a KEE, a frame system implemented in an object oriented language. RUM's architecture is composed of three layers: representation, inference, and control. The representation layer is based on frame-like data structures that capture the uncertainty information used in the inference layer and the uncertainty meta-information used in the control layer. The inference layer provides a selection of five T-norm based uncertainty calculi with which to perform the intersection, detachment, union, and pooling of information. The control layer uses the meta-information to select the appropriate calculus for each context and to resolve eventual ignorance or conflict in the information. This layer also provides a context mechanism that allows the system to focus on the relevant portion of the knowledge base, and an uncertain-belief revision system that incrementally updates the certainty values of well-formed formulae (wffs) in an acyclic directed deduction graph. RUM has been tested and validated in a sequence of experiments in both naval and aerial situation assessment (SA), consisting of correlating reports and tracks, locating and classifying platforms, and identifying intents and threats. An example of naval situation assessment is illustrated. The testbed environment for developing these experiments has been provided by LOTTA, a symbolic simulator implemented in Flavors. This simulator maintains time-varying situations in a multi-player antagonistic game where players must make decisions in light of uncertain and incomplete data. RUM has been used to assist one of the LOTTA players to perform the SA task.

The paper considers the class of information systems capable of solving heuristic problems on basis of formal theory that was termed modal and vector theory of formal intelligent systems (FIS). The paper justifies the construction of FIS resolution algorithm, defines the main features of these systems and proves theorems that underlie the theory. The principle of representation diversity of FIS construction is formulated. The paper deals with the main principles of constructing and functioning formal intelligent system (FIS) on basis of FIS modal and vector theory. The following phenomena are considered: modular architecture of FIS presentation sub-system, algorithms of data processing at every step of the stage of creating presentations. Besides the paper suggests the structure of neural elements, i.e. zone detectors and processors that are the basis for FIS construction.

Of the common commutative binary logical connectives, only and and or may be used as operators that take arbitrary numbers of arguments with order and multiplicity being irrelevant, that is, as connectives that take sets of arguments. This is especially evident in the Common Logic Interchange Format, in which it is easy for operators to be given arbitrary numbers of arguments. The reason is that and and or are associative and idempotent, as well as commutative. We extend the ability of taking sets of arguments to the other common commutative connectives by defining generalized versions of nand , nor , xor ,and iff , as well as the additional, parameterized connectives andor and thresh . We prove that andor is expressively complete — all the other connectives may be considered abbreviations of it.

These algorithms are part of the "folk culture",

An initial version of the program has been implemented in LISP on a PDP-10 and is being used in conjunction with robot research at SRI. STRIPS is a member of the class of problem solvers that search a space of "world models" to ind one in w hich a given goal is achieved. For any world model, we assume that there exists a set of appllcable ope rators, each of w hi eh transforms the world model to some other world model. The task of the problem solver is to find some composl11on of ope rat ors that trans forms a given initial worId mode] into one t hat satisfies some stated goa1 condltion. This f rarnewo rk for probl em so 1 v i ng has l een cen t ra 1 to much of t he research I n artificial Intel licence (1). Ou r p nmary interest he re is in the class of p robJ ems faced by a robot in rea rranging ob]ec t s and in navigatlng, l.e.