A Semantic Framework for Proof Evidence

International audienceTheorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refu-tations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs

Modality and expressibility

When embedding data are used to argue against semantic theory A and in favor of semantic theory B, it is important to ask whether A could make sense of those data. It is possible to ask that question on a case-by-case basis. But suppose we could show that A can make sense of all the embedding data which B can possibly make sense of. This would, on the one hand, undermine arguments in favor of B over A on the basis of embedding data. And, provided that the converse does not hold—that is, that A can make sense of strictly more embedding data than B can—it would also show that there is a precise sense in which B is more constrained than A, yielding a pro tanto simplicity-based consideration in favor of B. In this paper I develop tools which allow us to make comparisons of this kind, which I call comparisons of potential expressive power. I motivate the development of these tools by way of exploration of the recent debate about epistemic modals. Prominent theories which have been developed in response to embedding data turn out to be strictly less expressive than the standard relational theory, a fact which necessitates a reorientation in how to think about the choice between these theories

The Logical Burdens of Proof. Assertion and Hypothesis

The paper proposes two logical analyses of (the norms of) justification. In a first, realist-minded case, truth is logically independent from justification and leads to a pragmatic logic LP including two epistemic and pragmatic operators, namely, assertion and hypothesis. In a second, antirealist-minded case, truth is not logically independent from justification and results in two logical systems of information and justification: AR4 and AR4¢, respectively, provided with a question-answer semantics. The latter proposes many more epistemic agents, each corresponding to a wide variety of epistemic norms. After comparing the different norms of justification involved in these logical systems, two hexagons expressing Aristotelian relations of opposition will be gathered in order to clarify how (a fragment of) pragmatic formulas can be interpreted in a fuzzy-based question-answer semantics

Preferential and Preferential-discriminative Consequence relations

The present paper investigates consequence relations that are both non-monotonic and paraconsistent. More precisely, we put the focus on preferential consequence relations, i.e. those relations that can be defined by a binary preference relation on states labelled by valuations. We worked with a general notion of valuation that covers e.g. the classical valuations as well as certain kinds of many-valued valuations. In the many-valued cases, preferential consequence relations are paraconsistant (in addition to be non-monotonic), i.e. they are capable of drawing reasonable conclusions which contain contradictions. The first purpose of this paper is to provide in our general framework syntactic characterizations of several families of preferential relations. The second and main purpose is to provide, again in our general framework, characterizations of several families of preferential discriminative consequence relations. They are defined exactly as the plain version, but any conclusion such that its negation is also a conclusion is rejected (these relations bring something new essentially in the many-valued cases).Comment: team Logic and Complexity, written in 2004-200

A framework for deriving semantic web services

Web service-based development represents an emerging approach for the development of distributed information systems. Web services have been mainly applied by software practitioners as a means to modularize system functionality that can be offered across a network (e.g., intranet and/or the Internet). Although web services have been predominantly developed as a technical solution for integrating software systems, there is a more business-oriented aspect that developers and enterprises need to deal with in order to benefit from the full potential of web services in an electronic market. This ‘ignored’ aspect is the representation of the semantics underlying the services themselves as well as the ‘things’ that the services manage. Currently languages like the Web Services Description Language (WSDL) provide the syntactic means to describe web services, but lack in providing a semantic underpinning. In order to harvest all the benefits of web services technology, a framework has been developed for deriving business semantics from syntactic descriptions of web services. The benefits of such a framework are two-fold. Firstly, the framework provides a way to gradually construct domain ontologies from previously defined technical services. Secondly, the framework enables the migration of syntactically defined web services toward semantic web services. The study follows a design research approach which (1) identifies the problem area and its relevance from an industrial case study and previous research, (2) develops the framework as a design artifact and (3) evaluates the application of the framework through a relevant scenario

The pragmatic proof: hypermedia API composition and execution

Machine clients are increasingly making use of the Web to perform tasks. While Web services traditionally mimic remote procedure calling interfaces, a new generation of so-called hypermedia APIs works through hyperlinks and forms, in a way similar to how people browse the Web. This means that existing composition techniques, which determine a procedural plan upfront, are not sufficient to consume hypermedia APIs, which need to be navigated at runtime. Clients instead need a more dynamic plan that allows them to follow hyperlinks and use forms with a preset goal. Therefore, in this paper, we show how compositions of hypermedia APIs can be created by generic Semantic Web reasoners. This is achieved through the generation of a proof based on semantic descriptions of the APIs' functionality. To pragmatically verify the applicability of compositions, we introduce the notion of pre-execution and post-execution proofs. The runtime interaction between a client and a server is guided by proofs but driven by hypermedia, allowing the client to react to the application's actual state indicated by the server's response. We describe how to generate compositions from descriptions, discuss a computer-assisted process to generate descriptions, and verify reasoner performance on various composition tasks using a benchmark suite. The experimental results lead to the conclusion that proof-based consumption of hypermedia APIs is a feasible strategy at Web scale.Peer ReviewedPostprint (author's final draft

Logic and Topology for Knowledge, Knowability, and Belief - Extended Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker's core insights, and using frameworks developed by Bjorndahl and Baltag et al., we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an "evidence-in-hand" conception of knowledge and a weaker "evidence-out-there" notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.Comment: In Proceedings TARK 2017, arXiv:1707.08250. The full version of this paper, including the longer proofs, is at arXiv:1612.0205

Argument-based Belief in Topological Structures

This paper combines two studies: a topological semantics for epistemic notions and abstract argumentation theory. In our combined setting, we use a topological semantics to represent the structure of an agent's collection of evidence, and we use argumentation theory to single out the relevant sets of evidence through which a notion of beliefs grounded on arguments is defined. We discuss the formal properties of this newly defined notion, providing also a formal language with a matching modality together with a sound and complete axiom system for it. Despite the fact that our agent can combine her evidence in a 'rational' way (captured via the topological structure), argument-based beliefs are not closed under conjunction. This illustrates the difference between an agent's reasoning abilities (i.e. the way she is able to combine her available evidence) and the closure properties of her beliefs. We use this point to argue for why the failure of closure under conjunction of belief should not bear the burden of the failure of rationality.Comment: In Proceedings TARK 2017, arXiv:1707.0825