Logical English meets legal English for swaps and derivatives

In this paper, we present an informal introduction to Logical English (LE) and illustrate its use to standardise the legal wording of the Automatic Early Termination (AET) clauses of International Swaps and Derivatives Association (ISDA) Agreements. LE can be viewed both as an alternative to conventional legal English for expressing legal documents, and as an alternative to conventional computer languages for automating legal documents. LE is a controlled natural language (CNL), which is designed both to be computer-executable and to be readable by English speakers without special training. The basic form of LE is syntactic sugar for logic programs, in which all sentences have the same standard form, either as rules of the form conclusion if conditions or as unconditional sentences of the form conclusion. However, LE extends normal logic programming by introducing features that are present in other computer languages and other logics. These features include typed variables signalled by common nouns, and existentially quantified variables in the conclusions of sentences signalled by indefinite articles. Although LE translates naturally into a logic programming language such as Prolog or ASP, it can also serve as a neutral standard, which can be compiled into other lower-level computer languages.

Default Negation as Explicit Negation plus Update

We argue that under the stable model semantics default negation can be read as explicit negation with update. We show that dynamic logic programming which is based on default negation, even in the heads, can be interpreted in a variant of updates with explicit negation only. As corollaries, we get an easy description of default negation in generalized and normal logic programming where initially negated literals are updated. These results are discussed with respect to the understanding of negation in logic.

Stable and Supported Semantics in Continuous Vector Spaces

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

Relating Multi-Adjoint Normal Logic Programs to Core Fuzzy Answer Set Programs from a Semantical Approach

This paper relates two interesting paradigms in fuzzy logic programming from a semantical approach: core fuzzy answer set programming and multi-adjoint normal logic programming. Specifically, it is shown how core fuzzy answer set programs can be translated into multi-adjoint normal logic programs and vice versa, preserving the semantics of the starting program. This translation allows us to combine the expressiveness of multi-adjoint normal logic programming with the compactness and simplicity of the core fuzzy answer set programming language. As a consequence, theoretical properties and results which relate the answer sets to the stable models of the respective logic programming frameworks are obtained. Among others, this study enables the application of the existence theorem of stable models developed for multi-adjoint normal logic programs to ensure the existence of answer sets in core fuzzy answer set programs.

Inconsistency Proofs for ASP: The ASP - DRUPE Format

Answer Set Programming (ASP) solvers are highly-tuned and complex procedures that implicitly solve the consistency problem, i.e., deciding whether a logic program admits an answer set. Verifying whether a claimed answer set is formally a correct answer set of the program can be decided in polynomial time for (normal) programs. However, it is far from immediate to verify whether a program that is claimed to be inconsistent, indeed does not admit any answer sets. In this paper, we address this problem and develop the new proof format ASP-DRUPE for propositional, disjunctive logic programs, including weight and choice rules. ASP-DRUPE is based on the Reverse Unit Propagation (RUP) format designed for Boolean satisfiability. We establish correctness of ASP-DRUPE and discuss how to integrate it into modern ASP solvers. Later, we provide an implementation of ASP-DRUPE into the wasp solver for normal logic programs.

On the Equivalence Between Abstract Dialectical Frameworks and Logic Programs

Abstract Dialectical Frameworks (ADFs) are argumentation frameworks where each node is associated with an acceptance condition. This allows us to model different types of dependencies as supports and attacks. Previous studies provided a translation from Normal Logic Programs (NLPs) to ADFs and proved the stable models semantics for a normal logic program has an equivalent semantics to that of the corresponding ADF. However, these studies failed in identifying a semantics for ADFs equivalent to a three-valued semantics (as partial stable models and well-founded models) for NLPs. In this work, we focus on a fragment of ADFs, called Attacking Dialectical Frameworks (ADF+s), and provide a translation from NLPs to ADF+s robust enough to guarantee the equivalence between partial stable models, well-founded models, regular models, stable models semantics for NLPs and respectively complete models, grounded models, preferred models, stable models for ADFs. In addition, we define a new semantics for ADF+s, called L-stable, and show it is equivalent to the L-stable semantics for NLPs.

The Weak Completion Semantics and Equality

The weak completion semantics is an integrated and computational cognitive theory which is based on normal logic programs,three-valued Lukasiewicz logic, weak completion, and skeptical abduction. It has been successfully applied – among others – to the suppression task, the selection task, and to human syllogistic reasoning. In order to solve ethical decision problems like – for example – trolley problems, we need to extend the weak completion semantics to deal with actions and causality. To this end we consider normal logic programs and a set E of equations as in the fluent calculus. We formally show that normal logic programs with equality admit a least E-model under the weak completion semantics and that this E-model can be computed as the least fixed point of an associated semantic operator. We show that the operator is not continuous in general, but is continuous if the logic program is a propositional, a finite-ground, or a finite datalog program and the Herbrand E-universe is finite. Finally, we show that the weak completion semantics with equality can solve a variety of ethical decision problems like the bystander case, the footbridge case, and the loop case by computing the least E-model and reasoning with respect to this E-model. The reasoning process involves counterfactuals which is necessary to model the different ethical dilemmas.

On the Equivalence between Assumption-Based Argumentation and Logic Programming (Extended Abstract)

In this work, we explain how Assumption-Based Argumentation (ABA) is subsumed by Logic Programming (LP). The translation from ABA to LP (with a few restrictions on the ABA framework) results in a normal logic program whose semantics coincide with the semantics of the underlying ABA framework. Although the precise technicalities are beyond the current extended abstract (these can be found in the associated full paper) we provide a number of examples to illustrate the general idea.