Marcos Cramer

The naproche project controlled natural language proof checking of mathematical texts

This paper discusses the semi-formal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to represent the semantics of texts written in the Naproche CNL. We discuss how the Naproche CNL can be used in formal mathematics, and present our prototypical Naproche system, a computer program for parsing texts in the Naproche CNL and checking the proofs in them for logical correctness.

Premise selection in the naproche system

Automated theorem provers (ATPs) struggle to solve problems with large sets of possibly superfluous axiom. Several algorithms have been developed to reduce the number of axioms, optimally only selecting the necessary axioms. However, most of these algorithms consider only single problems. In this paper, we describe an axiom selection method for series of related problems that is based on logical and textual proximity and tries to mimic a human way of understanding mathematical texts. We present first results that indicate that this approach is indeed useful.

Parsing and disambiguation of symbolic mathematics in the Naproche system

The Naproche system is a system for linguistically analysing and proof-checking mathematical texts written in a controlled natural language. The aim is to have an input language that is as close as possible to the language that mathematicians actually use when writing textbooks or papers. Mathematical texts consist of a combination of natural language and symbolic mathematics, with symbolic mathematics obeying its own syntactic rules. We discuss the difficulties that a program for parsing and disambiguating symbolic mathematics must face and present how these difficulties have been tackled in the Naproche system. One of these difficulties is the fact that information provided in the preceding context - including information provided in natural language - can influence the way a symbolic expression has to be disambiguated.

A Logic of Trust for Reasoning about Delegation and Revocation

In ownership-based access control frameworks with the possibility of delegating permissions and administrative rights, chains of delegated accesses will form. There are different ways to treat these delegation chains when revoking rights, which give rise to different revocation schemes. Hagström et al. [8] proposed a framework for classifying revocation schemes, in which the different revocation schemes are defined graph-theoretically; they motivate the revocation schemes in this framework by presenting various scenarios in which the agents have different reasons for revocating. This paper is based on the observation that there are some problems with Hagström et al.s definitions of the revocation schemes, which have led us to propose a refined framework with new graph-theoretic definitions of the revocation schemes. In order to formally study the merits and demerits of various definitions of revocation schemes, we propose to apply the axiomatic method originating in social choice theory to revocation schemes. For formulating an axiom, i.e. a desirable property of revocation frameworks, we propose a logic, Trust Delegation Logic TDL) , with which one can formalize the different reasons an agent may have for performing a revocation. We show that our refined graph-theoretic definitions of the revocation schemes, unlike Hagström et al.s original definitions, satisfy the desirable property that can be formulated using TDL.

Interpreting plurals in the naproche CNL

The Naproche CNL is a controlled natural language for mathematical texts. A recent addition to the Naproche CNL are plural statements. We discuss the collective-distributive ambiguity in the context of mathematical language, as well as pairwise interpretations of collective plurals. Additionally, we present a special scope ambiguity conjunctions give rise to. Finally, we describe an innovative plural interpretation algorithm implemented in Naproche for disambiguating plurals in DRT and giving them the interpretation that would normally be preferred in a mathematical context.

Postulates for Revocation Schemes

In access control frameworks with the possibility of delegating permissions and administrative rights, delegation chains can form. There are different ways to treat these delegation chains when revoking rights, which give rise to different revocation schemes. Hagstrom et al. [11] proposed a framework for classifying revocation schemes, in which the different revocation schemes are defined graph-theoretically. At the outset, we identify multiple problems with Hagstrom et al.s definitions of the revocation schemes, which can pose security risks. This paper is centered around the question how one can systematically ensure that improved definitions of the revocation schemes do not lead to similar problems. For this we propose to apply the axiomatic method originating in social choice theory to revocation schemes. Our use of the axiomatic method resembles its use in belief revision theory. This means that we define postulates that describe the desirable behaviour of revocation schemes, study which existing revocation frameworks satisfy which postulates, and show how all defined postulates can be satisfied by defining the revocation schemes in a novel way.

Resilient Delegation Revocation with Precedence for Predecessors Is NP-Complete

In ownership-based access control frameworks with the possibility of delegating permissions and administrative rights, chains of delegated accesses will form. There are different ways to treat these delegation chains when revoking rights, which give rise to different revocation schemes. One possibility studied in the literature is to revoke rights by issuing negative authorizations, meant to ensure that the revocation is resilient to a later reissuing of the rights, and to resolve conflicts between principals by giving precedence to predecessors, i.e. principals that come earlier in the delegation chain. However, the effects of negative authorizations have been defined differently by different authors. Having identified three definitions of this effect from the literature, the first contribution of this paper is to point out that two of these three definitions pose a security threat. However, avoiding this security threat comes at a price: We prove that with the safe definition of the effect of negative authorizations, deciding whether a principal does have access to a resource is an NP-complete decision problem. We discuss two limitations that can be imposed on an access-control system in order to reduce the complexity of the problem back to a polynomial complexity: Limiting the length of delegation chains to an integer m reduces the runtime complexity of determining access to O(nm), and requiring that principals form a hierarchy that graph-theoretically forms a rooted tree makes this decision problem solvable in quadratic runtime. Finally we discuss an approach that can mitigate the complexity problem in practice without fully getting rid of NP-completeness.

A Model for Regulating of Ethical Preferences in Machine Ethics

Relying upon machine intelligence with reductions in the supervision of human beings, requires us to be able to count on a certain level of ethical behavior from it. Formalizing ethical theories is one of the plausible ways to add ethical dimensions to machines. Rule-based and consequence-based ethical theories are proper candidates for Machine Ethics. It is debatable that methodologies for each ethical theory separately might result in an action that is not always justifiable by human values. This inspires us to combine the reasoning procedure of two ethical theories, deontology and utilitarianism, in a utilitarian-based deontic logic which is an extension of STIT (Seeing To It That) logic. We keep the knowledge domain regarding the methodology in a knowledge base system called IDP. IDP supports inferences to examine and evaluate the process of ethical decision making in our formalization. To validate our proposed methodology we perform a Case Study for some real scenarios in the domain of robotics and automatous agents.

ASPIC-END: Structured Argumentation with Explanations and Natural Deduction

We propose ASPIC-END, an adaptation of the structured argumentation framework ASPIC+ which can incorporate explanations and natural deduction style arguments. We discuss an instantiation of ASPIC-END that models argumentation about explanations of semantic paradoxes (e.g. the Liar paradox), and we show that ASPIC-END satisfies rationality postulates akin to those satisfied by ASPIC+.

Extended Explanatory Argumentation Frameworks

Multiple extensions of Dung’s argumentation frameworks (AFs) have been proposed in order to model features of argumentation that cannot be directly modeled in AFs. One technique that has already previously proven useful to study and combine such extensions is the meta-argumentation methodology involving the notion of a flattening. In order to faithfully model the interaction between explanation argumentation in scientific debates, Seselja and Straser have introduced Explanatory Argumentation Frameworks (EAFs). In this paper, we first prove that the flattening technique works as expected for recursive (higher-order) attacks. Then we apply this technique in order to combine EAFs with multiple other extensions that have been proposed to AFs, namely with recursive attacks, joint attacks and a support relation between arguments. This gives rise to Extended Explanatory Argumentation Frameworks (EEAFs). We illustrate the applicability of EEAFs by using them to model a piece of argumentation from a research-level philosophy book.