We propose a new linear algebraic approach to the computation of Tarskian semantics in logic. We embed a finite model M in first-order logic with N entities in N-dimensional Euclidean space R^N by mapping entities of M to N dimensional one-hot vectors and k-ary relations to order-k adjacency tensors (multi-way arrays). Second given a logical formula F in prenex normal form, we compile F into a set Sigma_F of algebraic formulas in multi-linear algebra with a nonlinear operation. In this compilation, existential quantifiers are compiled into a specific type of tensors, e.g., identity matrices in the case of quantifying two occurrences of a variable. It is shown that a systematic evaluation of Sigma_F in R N gives the truth value, 1(true) or 0(false), of F in M. Based on this framework, we also propose an unprecedented way of computing the least models defined by Datalog programs in linear spaces via matrix equations and empirically show its effectiveness compared to state-of-the-art approaches.

Epistemic Specifications allow for the correct representation of incomplete information in the presence of multiple belief sets by expanding Answer Set Programming with modal operators $K$ and M. The meaning of M in the existing work does not correspond well to the principle of justifiedness accepted by the community. It is, however, challenging to characterize the justfiedness of each belief, due to the complexity introduced by M. We address this issue by identifying a belief set with a program which uniquely decides the belief set. This idea leads to a novel definition of the semantics of Epistemic Specifications which assures that each belief in any belief set is well justified.  We also show that conformant planning problems can be naturally represented by Epistemic Specification under our semantics.

Free variables occur frequently in mathematics and computer science with ad hoc and altering semantics. We present the most recent version of our free-variable framework for two-valued logics with properly improved functionality, but only two kinds of free variables left (instead of three): implicitly universally and implicitly existentially quantified ones, now simply called "free atoms" and "free variables", respectively. The quantificational expressiveness and the problem-solving facilities of our framework exceed standard first-order and even higher-order modal logics, and directly support Fermat's descente infinie. With the improved version of our framework, we can now model also Henkin quantification, neither using quantifiers (binders) nor raising (Skolemization). We propose a new semantics for Hilbert's epsilon as a choice operator with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the epsilon supports reductive proof search optimally.

We study the effectiveness of neural sequence models for premise selection in automated theorem proving, one of the main bottlenecks in the formalization of mathematics. We propose a two stage approach for this task that yields good results for the premise selection task on the Mizar corpus while avoiding the hand-engineered features of existing state-of-the-art models. To our knowledge, this is the first time deep learning has been applied to theorem proving on a large scale.

In the past twenty years, various large corpora of computer-understandable reasoning knowledge have been developed (Harrison et al., 2014). Apart from axioms, definitions, and conjectures, such corpora include proofs derived in the selected logical foundation with sufficient detail to be machine-checkable. This is either given in the form of premises-conclusion pairs (Sutcliffe, 2009) or as procedures and intermediate steps (Wenzel, 1999). The development of many of these formal proofs required dozens of person-years, their sizes are measured in tens of thousands of human-named theorems and the complete proofs contain billions of low-level inference steps. These formal proof libraries are also interesting for AIbased methods, with tasks such as concept matching, theory exploration, and structure formation (Autexier & Hutter, 2015). Furthermore, the AI methods can be augmented by automated reasoning: progress in the development of efficient first-order automated theorem provers (ATPs) (Kovács & Voronkov, 2013) allows applying them not only as tools that redo the formal proofs, but also to find the missing steps (Urban, 2006). Together with proof translations from the richer logics of the interactive systems to the simpler logics of the ATPs this becomes a commonly used tool in certain interactive provers (Blanchette et al., 2016). Many significant proof developments covering both mathematics and computer science have been created using such technologies. Examples include the formal proof of the Kepler conjecture (Hales et al., 2015), or the proof of correctness of the seL4 operating system kernel (Klein et al., 2010).

Artificial Intelligence (which I'll refer to hereafter by its nickname, "AI") is the subfield of Computer Science devoted to developing programs that enable computers to display behavior that can (broadly) be characterized as intelligent.[1] Most research in AI is devoted to fairly narrow applications, such as planning or speech-to-speech translation in limited, well defined task domains. But substantial interest remains in the long-range goal of building generally intelligent, autonomous agents,[2] even if the goal of fully human-like intelligence is elusive and is seldom pursued explicitly and as such. Throughout its relatively short history, AI has been heavily influenced by logical ideas. AI has drawn on many research methodologies: the value and relative importance of logical formalisms is questioned by some leading practitioners, and has been debated in the literature from time to time.[3]

In April 2008 I wrote "Where Have All the Great Programmers Gone?" . In trying to answer the question, I contrasted the contemporary introduction to programming with the way it was learned in the 1950s. The first was that one of the prerequisites for being a promising oddball was that of having a nontrivial college degree. Examples were philosophy (J. A. Robinson), English literature (Mark Halpern), Classics (C. A. R. Hoare), Physics (E.W. Dijkstra). Being thrown into the deep end in this way was educative, something that cannot be said of the typical first-year programming text, dumbed down in the way that only Educators have the secret of. A Programmer's Place (APP) has yet to snag Hoare or Halpern, but was fortunate to find J.A. Robinson available for an interview.

Predicate logic in which predicates take only individuals as arguments and quantifiers only bind individual variables. Predicate logic in which predicates take other predicates as arguments and quantifiers bind predicate variables. For example, second-order predicates take first-order predicates as arguments. Order n predicates take order n-1 predicates as arguments (n 1). Predicate logic that does not exclude interpretations with empty domains.

To indicate that a compound is to be taken as a whole or single statement, we put it in parentheses. For example, p q is a compound. Its negation is (p q). When the negation sign is outside the parentheses, it affects the entire compound, not just the first component, p. Its conjunction with the compound, q r, would be expressed, (p q)·(q r). If we wanted to say that the whole latter statement was false, we would write, [(p q)·(q r)].

Presently successful applications areas for ILP systems include the learning of structure-activity rules for drug design, finite-element mesh analysis design rules, primary-secondary prediction of protein structure and fault diagnosis rules for satellites.