# Logic & Formal Reasoning

### Stratified Negation in Limit Datalog Programs

There has recently been an increasing interest in declarative data analysis, where analytic tasks are specified using a logical language, and their implementation and optimisation are delegated to a general-purpose query engine. Existing declarative languages for data analysis can be formalised as variants of logic programming equipped with arithmetic function symbols and/or aggregation, and are typically undecidable. In prior work, the language of $\mathit{limit\ programs}$ was proposed, which is sufficiently powerful to capture many analysis tasks and has decidable entailment problem. Rules in this language, however, do not allow for negation. In this paper, we study an extension of limit programs with stratified negation-as-failure. We show that the additional expressive power makes reasoning computationally more demanding, and provide tight data complexity bounds. We also identify a fragment with tractable data complexity and sufficient expressivity to capture many relevant tasks.

### A Stronger Foundation for Computer Science and P=NP

This article describes a Turing machine which can solve for $\beta^{'}$ which is RE-complete. RE-complete problems are proven to be undecidable by Turing's accepted proof on the Entscheidungsproblem. Thus, constructing a machine which decides over $\beta^{'}$ implies inconsistency in ZFC. We then discover that unrestricted use of the axiom of substitution can lead to hidden assumptions in a certain class of proofs by contradiction. These hidden assumptions create an implied axiom of incompleteness for ZFC. Later, we offer a restriction on the axiom of substitution by introducing a new axiom which prevents impredicative tautologies from producing theorems. Our discovery in regards to these foundational arguments, disproves the SPACE hierarchy theorem which allows us to solve the P vs NP problem using a TIME-SPACE equivalence oracle.

### The Higher-Order Prover Leo-III (Extended Version)

The automated theorem prover Leo-III for classical higher-order logic with Henkin semantics and choice is presented. Leo-III is based on extensional higher-order paramodulation and accepts every common TPTP dialect (FOF, TFF, THF), including their recent extensions to rank-1 polymorphism (TF1, TH1). In addition, the prover natively supports almost every normal higher-order modal logic. Leo-III cooperates with first-order reasoning tools using translations to many-sorted first-order logic and produces verifiable proof certificates. The prover is evaluated on heterogeneous benchmark sets.

### I/O Logic in HOL --- First Steps

A semantical embedding of input/output logic in classical higher-order logic is presented. This embedding enables the mechanisation and automation of reasoning tasks in input/output logic with off-the-shelf higher-order theorem provers and proof assistants. The key idea for the solution presented here results from the analysis of an inaccurate previous embedding attempt, which we will discuss as well.

### Dichotomies in Ontology-Mediated Querying with the Guarded Fragment

We study the complexity of ontology-mediated querying when ontologies are formulated in the guarded fragment of first-order logic (GF). Our general aim is to classify the data complexity on the level of ontologies where query evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several large and relevant fragments of GF that enjoy a dichotomy between PTime and coNP, some of them additionally admitting a form of counting. In fact, almost all ontologies in the BioPortal repository fall into these fragments or can easily be rewritten to do so. We then establish a variation of Ladner's Theorem on the existence of NP-intermediate problems and use this result to show that for other fragments, there is provably no such dichotomy. Again for other fragments (such as full GF), establishing a dichotomy implies the Feder-Vardi conjecture on the complexity of constraint satisfaction problems. We also link these results to Datalog-rewritability and study the decidability of whether a given ontology enjoys PTime query evaluation, presenting both positive and negative results.

### A Language for Function Signature Representations

Recent work by (Richardson and Kuhn, 2017a,b; Richardson et al., 2018) looks at semantic parser induction and question answering in the domain of source code libraries and APIs. In this brief note, we formalize the representations being learned in these studies and introduce a simple domain specific language and a systematic translation from this language to first-order logic. By recasting the target representations in terms of classical logic, we aim to broaden the applicability of existing code datasets for investigating more complex natural language understanding and reasoning problems in the software domain.

### A General Account of Argumentation with Preferences

This paper builds on the recent ASPIC+ formalism, to develop a general framework for argumentation with preferences. We motivate a revised definition of conflict free sets of arguments, adapt ASPIC+ to accommodate a broader range of instantiating logics, and show that under some assumptions, the resulting framework satisfies key properties and rationality postulates. We then show that the generalised framework accommodates Tarskian logic instantiations extended with preferences, and then study instantiations of the framework by classical logic approaches to argumentation. We conclude by arguing that ASPIC+'s modelling of defeasible inference rules further testifies to the generality of the framework, and then examine and counter recent critiques of Dung's framework and its extensions to accommodate preferences.

### 20 free books to get started with Artificial Intelligence

This is one of the few texts that combines three essential theses in the study of logic programming: the logic that gives logic programs their unique character: the practice of programming effectively using the logic; and the efficient implementation of logic programming on computers.

### Unifying DAGs and UGs

We introduce a new class of graphical models that generalizes Lauritzen-Wermuth-Frydenberg chain graphs by relaxing the semi-directed acyclity constraint so that only directed cycles are forbidden. Moreover, up to two edges are allowed between any pair of nodes. Specifically, we present local, pairwise and global Markov properties for the new graphical models and prove their equivalence. We also present an equivalent factorization property. Finally, we present a causal interpretation of the new models.

### Incremental and Iterative Learning of Answer Set Programs from Mutually Distinct Examples

Over these years the Artificial Intelligence (AI) community has produced several datasets which have given the machine learning algorithms the opportunity to learn various skills across various domains. However, a subclass of these machine learning algorithms that aimed at learning logic programs, namely the Inductive Logic Programming algorithms, have often failed at the task due to the vastness of these datasets. This has impacted the usability of knowledge representation and reasoning techniques in the development of AI systems. In this research, we try to address this scalability issue for the algorithms that learn Answer Set Programs. We present a sound and complete algorithm which takes the input in a slightly different manner and perform an efficient and more user controlled search for a solution. We show via experiments that our algorithm can learn from two popular datasets from machine learning community, namely bAbl (a question answering dataset) and MNIST (a dataset for handwritten digit recognition), which to the best of our knowledge was not previously possible. The system is publicly available at https://goo.gl/KdWAcV.