\emph{Canonical (logic) programs} (CP) refer to normal logic programs augmented with connective $not\ not$. In this paper we address the question of whether CP are \emph{succinctly incomparable} with \emph{propositional formulas} (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but \emph{only} has exponential representations in CP. In other words, PARITY \emph{separates} PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming $\mathsf{P}\nsubseteq \mathsf{NC^1/poly}$), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two \emph{models of computation}, i.e., we identify a language in $\mathsf{NC^1/poly}$ which is not in the set of languages computable by polynomial size CP programs.

In this paper we explore how machine learning techniques can be applied to the discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a grammar of math operators, we build trees that combine them in different ways, looking for compositions that are analytically equivalent to a target expression but of lower computational complexity. However, as the space of trees grows exponentially with the complexity of the target expression, brute force search is impractical for all but the simplest of expressions. Consequently, we introduce two novel learning approaches that are able to learn from simpler expressions to guide the tree search. The first of these is a simple n-gram model, the other being a recursive neural-network. We show how these approaches enable us to derive complex identities, beyond reach of brute-force search, or human derivation.

In this paper an ontological representation and reasoning paradigm has been proposed for interpretation of time-series signals. The signals come from sensors observing a smart environment. The signal chosen for the annotation process is a set of unintuitive and complex gas sensor data. The ontology of this paradigm is inspired form the SSN ontology (Semantic Sensor Network) and used for representation of both the sensor data and the contextual information. The interpretation process is mainly done by an incremental ASP solver which as input receives a logic program that is generated from the contents of the ontology. The contextual information together with high level domain knowledge given in the ontology are used to infer explanations (answer sets) for changes in the ambient air detected by the gas sensors.

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

An answer set is a plain set of literals which has no further structure that would explain why certain literals are part of it and why others are not. We show how argumentation theory can help to explain why a literal is or is not contained in a given answer set by defining two justification methods, both of which make use of the correspondence between answer sets of a logic program and stable extensions of the Assumption-Based Argumentation (ABA) framework constructed from the same logic program. Attack Trees justify a literal in argumentation-theoretic terms, i.e. using arguments and attacks between them, whereas ABA-Based Answer Set Justifications express the same justification structure in logic programming terms, that is using literals and their relationships. Interestingly, an ABA-Based Answer Set Justification corresponds to an admissible fragment of the answer set in question, and an Attack Tree corresponds to an admissible fragment of the stable extension corresponding to this answer set.

Linear logic programming languages have been identified in prior work as viable for specifying stories and analyzing their causal structure. We investigate the use of such a language for specifying story worlds, or settings where generalized narrative actions have uniform effects (not specific to a particular set of characters or setting elements), which may create emergent behavior through feedback loops. We show a sizable example of a story world specified in the language Celf and discuss its interpretation as a story-generating program, a simulation, and an interactive narrative. Further, we show that the causal analysis tools available by virtue of using a proof-theoretic language for specification can assist the author in reasoning about the structure and consequences of emergent stories.

Clinical practice guidelines (CPGs) were originally designed to help with evidence-based management of a single disease and such a single disease focus has impacted research on CPG computerization. This computerization is mostly concerned with supporting different representation formats and identifying potential inconsistencies in the definitions of CPGs. However, one of the biggest challenges facing physicians is the personalization of multiple CPGs to comorbid patients. Various research initiatives propose ways of mitigating adverse interactions in concurrently applied CPGs, however, there are no attempts to develop a generalized framework for mitigation that captures generic characteristics of the problem while handling nuances such as precedence relationships. In this paper we present our research towards developing a mitigation framework that relies on a first-order logic-based representation and related theorem proving and model finding techniques. The application of the proposed framework is illustrated with a simple clinical example.

The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the proofs, producing an AI system capable of answering a wide range of mathematical queries automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39% of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

We study the semantics of fuzzy if-then rules called fuzzy attribute implications parameterized by systems of isotone Galois connections. The rules express dependencies between fuzzy attributes in object-attribute incidence data. The proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches. We formalize the general parameterizations, propose bivalent and graded notions of semantic entailment of fuzzy attribute implications, show their characterization in terms of least models and complete axiomatization, and provide characterization of bases of fuzzy attribute implications derived from data.

For natural and artificial systems with some symmetry structure, computational understanding and manipulation can be achieved without learning by exploiting the algebraic structure. Here we describe this algebraic coordinatization method and apply it to permutation puzzles. Coordinatization yields a structural understanding, not just solutions for the puzzles.