The most important requirements, for an operator to be viewed as a proposition, is that it must be hermitian and idempotent (which, in the Hilbert case corresponds to projectors). We interpret the above restrictions as follows. Hermitian operators have real eigenvalues. In particular, idempotent operators have eigenvalues 0 or 1, that is, they allow for asserting or negating in the classical way. When the operator is not hermitian, it is true that there is no way to interpret it directly as a logical proposition, because its eigenvalues are not real numbers, and the proposition cannot be asserted as usual.

This paper presents a case study in quantified multimodal logics. An interesting aspect of this case study is that off the shelf theorem provers and model generators for simple type theory, that is, classical higher-order logic, are employed to automate problems in quantified multimodal logics, that is, nonclassical logics. This is enabled by our recent embedding of normal quantified multimodal logics in simple type theory [8, 10], which is sound and complete [10]. Interestingly, not only reasoning within various nonclassical logics can be automated this way but also reasoning about them. For example, the equivalence between different properties of accessibility relations and their associated multimodal axioms can be proved automatically.

When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of detail (also called granularity). Often these presentations are neither intended nor suitable for human use. A challenge therefore is to develop user- and goal-adaptive proof presentation techniques that obey common mathematical practice. We present a flexible and adaptive approach to proof presentation that exploits machine learning techniques to extract a model of the specific granularity of proof examples and employs this model for the automated generation of further proofs at an adapted level of granularity.

Numerous logic frameworks have been proposed for modeling uncertainty and reasoning with such data. While different in syntax, the approaches of these frameworks have been classified into "annotation based" (AB) and "implication based" (IB). In this paper, we present a unified framework which allows evaluating programs in either approach. It extends existing query processing techniques to handle certainty constraints and uses heuristics to further improve the performance. Our experiments indicate that the proposed techniques yield useful tools for uncertainty reasoning.

The Multiple ANSwer EXtraction system is a framework for interpreting a conjecture with outermost existentially quantified variables as a question, and extracting multiple answers to the question by repetitive calls to a base system that can report the bindings for the variables in one proof of the conjecture. This paper describes the framework and demonstrates its use on an illustrative example.

Coinduction has recently been introduced into logic programming by Simon et al. The resulting paradigm, termed coinductive logic programming (co-LP), allows one to model and reason about infinite processes and objects. Co-LP extended with negation has many interesting applications: for instance in developing top-down, goaldirected evaluation strategies for Answer Set Programming. In this paper we show yet another application of co-LP, namely, elegantly realizing Boolean SAT solvers

Many important tasks can be cast as weighted relational satisfiability problems.  Propositionalizing relational theories and making inferences with them using SAT algorithms has proven effective in many cases.  However, these approaches require that all objects in a domain be known in advance.  Many domains, from language understanding to machine vision, involve reasoning about objects that are not known beforehand.  Theories with unknown objects can require models with infinite objects in their domain and thus lead to propositionalized SAT theories that existing algorithms cannot deal with.  To address these problems, we characterize a class of relational generative weighted satisfiability theories (GenSAT) over potentially infinite domains and propose an algorithm, GenDPLL, for finding models of these theories.  We introduce the notion of a relevant model and an increasing cost theory to identify conditions under which GenDPLL is complete, even when a theory has infinite models.

Graded modal logic is the formal language obtained from ordinary (propositional) modal logic by endowing its modal operators with cardinality constraints. Under the familiar possible-worlds semantics, these augmented modal operators receive interpretations such as "It is true at no fewer than 15 accessible worlds that...", or "It is true at no more than 2 accessible worlds that...". We investigate the complexity of satisfiability for this language over some familiar classes of frames. This problem is more challenging than its ordinary modal logic counterpart--especially in the case of transitive frames, where graded modal logic lacks the tree-model property. We obtain tight complexity bounds for the problem of determining the satisfiability of a given graded modal logic formula over the classes of frames characterized by any combination of reflexivity, seriality, symmetry, transitivity and the Euclidean property.

This paper further introduces and formalizes a novel concept of self-forensics for automotive vehicles, specified in the Forensic Lucid language. We argue that self-forensics, with the forensics taken out of the cybercrime domain, is applicable to "self-dissection" of intelligent vehicles and hardware systems for automated incident and anomaly analysis and event reconstruction by the software with or without the aid of the engineering teams in a variety of forensic scenarios. We propose a formal design, requirements, and specification of the self-forensic enabled units (similar to blackboxes) in vehicles that will help investigation of incidents and also automated reasoning and verification of theories along with the events reconstruction in a formal model. We argue such an analysis is beneficial to improve the safety of the passengers and their vehicles, like the airline industry does for planes.

We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple experiments, using existing higher-order theorem provers, to demonstrate that the embedding allows automated proofs of statements in these logics, as well as meta properties of them.