47 research outputs found
What should a generic object be?
Jacobs has proposed definitions for (weak, strong, split) generic objects for
a fibered category; building on his definition of generic object and split
generic object, Jacobs develops a menagerie of important fibrational structures
with applications to categorical logic and computer science, including higher
order fibrations, polymorphic fibrations, -fibrations, triposes, and
others. We observe that a split generic object need not in particular be a
generic object under the given definitions, and that the definitions of
polymorphic fibrations, triposes, etc. are strict enough to rule out many
fundamental examples: for instance, the fibered preorder induced by a partial
combinatory algebra in realizability is not a tripos in the sense of Jacobs. We
argue for a new alignment of terminology that emphasizes the forms of generic
object that appear most commonly in nature, i.e. in the study of internal
categories, triposes, and the denotational semantics of polymorphic types. In
addition, we propose a new class of acyclic generic objects inspired by recent
developments in the semantics of homotopy type theory, generalizing the
realignment property of universes to the setting of an arbitrary fibration
POPLMark reloaded: Mechanizing proofs by logical relations
We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks
A Logic of Blockchain Updates
Blockchains are distributed data structures that are used to achieve
consensus in systems for cryptocurrencies (like Bitcoin) or smart contracts
(like Ethereum). Although blockchains gained a lot of popularity recently,
there is no logic-based model for blockchains available. We introduce BCL, a
dynamic logic to reason about blockchain updates, and show that BCL is sound
and complete with respect to a simple blockchain model
Automata-Based Software Model Checking of Hyperproperties
We develop model checking algorithms for Temporal Stream Logic (TSL) and
Hyper Temporal Stream Logic (HyperTSL) modulo theories. TSL extends Linear
Temporal Logic (LTL) with memory cells, functions and predicates, making it a
convenient and expressive logic to reason over software and other systems with
infinite data domains. HyperTSL further extends TSL to the specification of
hyperproperties - properties that relate multiple system executions. As such,
HyperTSL can express information flow policies like noninterference in software
systems. We augment HyperTSL with theories, resulting in HyperTSL(T),and build
on methods from LTL software verification to obtain model checking algorithms
for TSL and HyperTSL(T). This results in a sound but necessarily incomplete
algorithm for specifications contained in the forall*exists* fragment of
HyperTSL(T). Our approach constitutes the first software model checking
algorithm for temporal hyperproperties with quantifier alternations that does
not rely on a finite-state abstraction
Informe bibliográfico sobre la lógica (epistémica) de la conciencia
Awareness Logic is an extension of epistemic logic that solves the problem of logical omniscience by incorporating an awareness operator that separates the explicit knowledge from the implicit one. This report collects the most prominent works regarding the beginnings of this logic, as well as its developments in the past three decades. Specifically, it reviews the approaches from Dynamic Epistemic Logic, from the ones that combine other logics with Awareness Logic and those from Game Theory.La lógica de la conciencia es una extensión de la lógica epistémica que solventa el problema de la omnisciencia lógica incorporando un operador de conciencia para separar el conocimiento explÃcito del implÃcito. Este informe recopila los principales textos tanto de los orÃgenes de esta lógica, asà como de sus desarrollos en las últimas tres décadas. En concreto analiza los enfoques desde la lógica epistémica dinámica, desde su combinación con otras lógicas y los enfoques de teorÃa de juegos
Normal Form Bisimulations By Value
Normal form bisimilarities are a natural form of program equivalence resting
on open terms, first introduced by Sangiorgi in call-by-name. The literature
contains a normal form bisimilarity for Plotkin's call-by-value
-calculus, Lassen's \emph{enf bisimilarity}, which validates all of
Moggi's monadic laws and can be extended to validate . It does not
validate, however, other relevant principles, such as the identification of
meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the
commutation of \letexps. These shortcomings are due to issues with open terms
of Plotkin's calculus. We introduce a new call-by-value normal form
bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's
and satisfying the additional principles. We develop it on top of an existing
formalism designed for dealing with open terms in call-by-value. It turns out
that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity
does not validate Moggi's laws nor . Moreover, there is no easy way to
merge them. To better understand the situation, we provide an analysis of the
rich range of possible call-by-value normal form bisimilarities, relating them
to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method
on naive similarity) -- Submitted to POPL2
Parameterized aspects of team-based formalisms and logical inference
Parameterized complexity is an interesting subfield of complexity theory that has received a lot of attention in recent years. Such an analysis characterizes the complexity of (classically) intractable problems by pinpointing the computational hardness to some structural aspects of the input. In this thesis, we study the parameterized complexity of various problems from the area of team-based formalisms as well as logical inference.
In the context of team-based formalism, we consider propositional dependence logic (PDL). The problems of interest are model checking (MC) and satisfiability (SAT). Peter Lohmann studied the classical complexity of these problems as a part of his Ph.D. thesis proving that both MC and SAT are NP-complete for PDL. This thesis addresses the parameterized complexity of these problems with respect to a wealth of different parameterizations.
Interestingly, SAT for PDL boils down to the satisfiability of propositional logic as implied by the downwards closure of PDL-formulas. We propose an interesting satisfiability variant (mSAT) asking for a satisfiable team of size m. The problem mSAT restores the ‘team semantic’ nature of satisfiability for PDL-formulas. We propose another problem (MaxSubTeam) asking for a maximal satisfiable team if a given team does not satisfy the input formula.
From the area of logical inference, we consider (logic-based) abduction and argumentation. The problem of interest in abduction (ABD) is to determine whether there is an explanation for a manifestation in a knowledge base (KB). Following Pfandler et al., we also consider two of its variants by imposing additional restrictions over the size of an explanation (ABD and ABD=). In argumentation, our focus is on the argument existence (ARG), relevance (ARG-Rel) and verification (ARG-Check) problems. The complexity of these problems have been explored already in the classical setting, and each of them is known to be complete for the second level of the polynomial hierarchy (except for ARG-Check which is DP-complete) for propositional logic. Moreover, the work by Nord and Zanuttini (resp., Creignou et al.) explores the complexity of these problems with respect to various restrictions over allowed KBs for ABD (ARG). In this thesis, we explore a two-dimensional complexity analysis for these problems. The first dimension is the restrictions over KB in Schaefer’s framework (the same direction as Nord and Zanuttini and Creignou et al.). What differentiates the work in this thesis from an existing research on these problems is that we add another dimension, the parameterization.
The results obtained in this thesis are interesting for two reasons. First (from a theoretical point of view), ideas used in our reductions can help in developing further reductions and prove (in)tractability results for related problems. Second (from a practical point of view), the obtained tractability results might help an agent designing an instance of a problem come up with the one for which the problem is tractable
Uncertain Reasoning in Justification Logic
This thesis studies the combination of two well known formal systems for knowledge representation: probabilistic logic and justification logic. Our aim is to design a formal framework that allows the analysis of epistemic situations with incomplete information. In order to achieve this we introduce two probabilistic justification logics, which are defined by adding probability operators to the minimal justification logic J. We prove soundness and completeness theorems for our logics and establish decidability procedures. Both our logics rely on an infinitary rule so that strong completeness can be achieved. One of the most interesting mathematical results for our logics is the fact that adding only one iteration of the probability operator to the justification logic J does not increase the computational complexity of the logic