-- We study time-dependent dynamics on a network of order lattices, where structure-preserving lattice maps are used to fuse lattice-valued data over vertices and edges. The principal contribution is a novel asynchronous Laplacian, generalizing the usual graph Laplacian, adapted to a network of heterogeneous lattices. The resulting gossip algorithm is shown to converge asymptotically to stable "harmonic" distributions of lattice data. This general theorem is applicable to several general problems, including lattice-valued consensus, Kripke semantics, and threat detection, all using asynchronous local update rules. The use of the graph Laplacian to diffuse information over networks is well-established in classical and contemporary work ranging from opinion dynamics [1] to distributed multi-agent consensus [2] and control [3], synchronization [4], [5], flocking [6], and much more. In the past decade, Laplacians that are adapted to handle vector-valued data, such as graph connection Laplacians [7], [8] or matrix-weighted Laplacians [9], have been revolutionary in signal processing processing [10], [11] and machine learning [12], [13]. While the ultimate form of a generalized Laplacian is as yet not present in applications, there are hints of a broader theory finding its way from algebraic topology to data science. The Laplacian from calculus class and the graph Laplacian are two extreme examples of a Hodge Laplacian .

We present a method for generating possible proofs of a query with respect to a given Answer Set Programming (ASP) rule set using an abductive process where the space of abducibles is automatically constructed just from the input rules alone. Given a (possibly empty) set of user provided facts, our method infers any additional facts that may be needed for the entailment of a query and then outputs these extra facts, without the user needing to explicitly specify the space of all abducibles. We also present a method to generate a set of directed edges corresponding to the justification graph for the query. Furthermore, through different forms of implicit term substitution, our method can take user provided facts into account and suitably modify the abductive solutions. Past work on abduction has been primarily based on goal directed methods. However these methods can result in solvers that are not truly declarative. Much less work has been done on realizing abduction in a bottom up solver like the Clingo ASP solver. We describe novel ASP programs which can be run directly in Clingo to yield the abductive solutions and directed edge sets without needing to modify the underlying solving engine.

LegalRuleML is a comprehensive XML-based representation framework for modeling and exchanging normative rules. The TPTP input and output formats, on the other hand, are general-purpose standards for the interaction with automated reasoning systems. In this paper we provide a bridge between the two communities by (i) defining a logic-pluralistic normative reasoning language based on the TPTP format, (ii) providing a translation scheme between relevant fragments of LegalRuleML and this language, and (iii) proposing a flexible architecture for automated normative reasoning based on this translation. We exemplarily instantiate and demonstrate the approach with three different normative logics.

A semantic tableau method, called an argumentation tableau, that enables the derivation of arguments, is proposed. First, the derivation of arguments for standard propositional and predicate logic is addressed. Next, an extension that enables reasoning with defeasible rules is presented. Finally, reasoning by cases using an argumentation tableau is discussed.

This paper presents an extension of Defeasible Deontic Logic to deal with the Pragmatic Oddity problem. The logic applies three general principles: (1) the Pragmatic Oddity problem must be solved within a general logical treatment of CTD reasoning; (2) non-monotonic methods must be adopted to handle CTD reasoning; (3) logical models of CTD reasoning must be computationally feasible and, if possible, efficient. The proposed extension of Defeasible Deontic Logic elaborates a preliminary version of the model proposed by Governatori and Rotolo (2019). The previous solution was based on particular characteristics of the (constructive, top-down) proof theory of the logic. However, that method introduces some degree of non-determinism. To avoid the problem, we provide a bottom-up characterisation of the logic. The new characterisation offers insights for the efficient implementation of the logic and allows us to establish the computational complexity of the problem.

We introduce Knowledge-Driven Program Synthesis (KDPS) as a variant of the program synthesis task that requires the agent to solve a sequence of program synthesis problems. In KDPS, the agent should use knowledge from the earlier problems to solve the later ones. We propose a novel method based on PushGP to solve the KDPS problem, which takes subprograms as knowledge. The proposed method extracts subprograms from the solution of previously solved problems by the Even Partitioning (EP) method and uses these subprograms to solve the upcoming programming task using Adaptive Replacement Mutation (ARM). We call this method PushGP+EP+ARM. With PushGP+EP+ARM, no human effort is required in the knowledge extraction and utilization processes. We compare the proposed method with PushGP, as well as a method using subprograms manually extracted by a human. Our PushGP+EP+ARM achieves better train error, success count, and faster convergence than PushGP. Additionally, we demonstrate the superiority of PushGP+EP+ARM when consecutively solving a sequence of six program synthesis problems.

We consider the problem of jointly minimizing forms of two Boolean functions $f, g \colon \{0,1\}^J \to \{0,1\}$ such that $f + g \leq 1$ and so as to separate disjoint sets $A \cup B \subseteq \{0,1\}^J$ such that $f(A) = \{1\}$ and $g(B) = \{1\}$. We hypothesize that this problem is easier to solve or approximate than the well-understood problem of minimizing the form of one Boolean function $h: \{0,1\}^J \to \{0,1\}$ such that $h(A) = \{1\}$ and $h(B) = \{0\}$. For a large class of forms, including binary decision trees and ordered binary decision diagrams, we refute this hypothesis. For disjunctive normal forms, we show that the problem is at least as hard as MIN-SET-COVER. For all these forms, we establish that no $o(\ln (|A| + |B| -1))$-approximation algorithm exists unless P$=$NP.

The paper aims to develop a framework for coalgebraic fuzzy geometric logic by adding modalities to the language of fuzzy geometric logic. Using the methods of coalgebra, the modal operators are introduced in the language of fuzzy geometric logic. To define the modal operators, we introduce a notion of fuzzy-open predicate lifting. Based on coalgebras for an endofunctor $T$ on the category $\textbf{Fuzzy-Top}$ of fuzzy topological spaces and fuzzy continuous maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations for the defined models are discussed in this work.

We propose to build a reinforcement learning prover of independent components: a deductive system (an environment), the proof state representation (how an agent sees the environment), and an agent training algorithm. To that purpose, we contribute an additional Vampire-based environment to $\texttt{gym-saturation}$ package of OpenAI Gym environments for saturation provers. We demonstrate a prototype of using $\texttt{gym-saturation}$ together with a popular reinforcement learning framework (Ray $\texttt{RLlib}$). Finally, we discuss our plans for completing this work in progress to a competitive automated theorem prover.

In this paper, we present a Bayesian method for statistical model checking (SMC) of probabilistic hyperproperties specified in the logic HyperPCTL* on discrete-time Markov chains (DTMCs). While SMC of HyperPCTL* using sequential probability ratio test (SPRT) has been explored before, we develop an alternative SMC algorithm based on Bayesian hypothesis testing. In comparison to PCTL*, verifying HyperPCTL* formulae is complex owing to their simultaneous interpretation on multiple paths of the DTMC. In addition, extending the bottom-up model-checking algorithm of the non-probabilistic setting is not straight forward due to the fact that SMC does not return exact answers to the satisfiability problems of subformulae, instead, it only returns correct answers with high-confidence. We propose a recursive algorithm for SMC of HyperPCTL* based on a modified Bayes' test that factors in the uncertainty in the recursive satisfiability results. We have implemented our algorithm in a Python toolbox, HyProVer, and compared our approach with the SPRT based SMC. Our experimental evaluation demonstrates that our Bayesian SMC algorithm performs better both in terms of the verification time and the number of samples required to deduce satisfiability of a given HyperPCTL* formula.