7,868 research outputs found
The Equational Approach to CF2 Semantics
We introduce a family of new equational semantics for argumentation networks
which can handle odd and even loops in a uniform manner. We offer one version
of equational semantics which is equivalent to CF2 semantics, and a better
version which gives the same results as traditional Dung semantics for even
loops but can still handle odd loops.Comment: 36 pages, version dated 15 February 201
The Structure of First-Order Causality
Game semantics describe the interactive behavior of proofs by interpreting
formulas as games on which proofs induce strategies. Such a semantics is
introduced here for capturing dependencies induced by quantifications in
first-order propositional logic. One of the main difficulties that has to be
faced during the elaboration of this kind of semantics is to characterize
definable strategies, that is strategies which actually behave like a proof.
This is usually done by restricting the model to strategies satisfying subtle
combinatorial conditions, whose preservation under composition is often
difficult to show. Here, we present an original methodology to achieve this
task, which requires to combine advanced tools from game semantics, rewriting
theory and categorical algebra. We introduce a diagrammatic presentation of the
monoidal category of definable strategies of our model, by the means of
generators and relations: those strategies can be generated from a finite set
of atomic strategies and the equality between strategies admits a finite
axiomatization, this equational structure corresponding to a polarized
variation of the notion of bialgebra. This work thus bridges algebra and
denotational semantics in order to reveal the structure of dependencies induced
by first-order quantifiers, and lays the foundations for a mechanized analysis
of causality in programming languages
Second-Order Algebraic Theories
Fiore and Hur recently introduced a conservative extension of universal
algebra and equational logic from first to second order. Second-order universal
algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and
parameterised metavariables. This work completes the foundations of the subject
from the viewpoint of categorical algebra. Specifically, the paper introduces
the notion of second-order algebraic theory and develops its basic theory. Two
categorical equivalences are established: at the syntactic level, that of
second-order equational presentations and second-order algebraic theories; at
the semantic level, that of second-order algebras and second-order functorial
models. Our development includes a mathematical definition of syntactic
translation between second-order equational presentations. This gives the first
formalisation of notions such as encodings and transforms in the context of
languages with variable binding
A Spatial-Epistemic Logic for Reasoning about Security Protocols
Reasoning about security properties involves reasoning about where the
information of a system is located, and how it evolves over time. While most
security analysis techniques need to cope with some notions of information
locality and knowledge propagation, usually they do not provide a general
language for expressing arbitrary properties involving local knowledge and
knowledge transfer. Building on this observation, we introduce a framework for
security protocol analysis based on dynamic spatial logic specifications. Our
computational model is a variant of existing pi-calculi, while specifications
are expressed in a dynamic spatial logic extended with an epistemic operator.
We present the syntax and semantics of the model and logic, and discuss the
expressiveness of the approach, showing it complete for passive attackers. We
also prove that generic Dolev-Yao attackers may be mechanically determined for
any deterministic finite protocol, and discuss how this result may be used to
reason about security properties of open systems. We also present a
model-checking algorithm for our logic, which has been implemented as an
extension to the SLMC system.Comment: In Proceedings SecCo 2010, arXiv:1102.516
Initial Algebra Semantics for Cyclic Sharing Tree Structures
Terms are a concise representation of tree structures. Since they can be
naturally defined by an inductive type, they offer data structures in
functional programming and mechanised reasoning with useful principles such as
structural induction and structural recursion. However, for graphs or
"tree-like" structures - trees involving cycles and sharing - it remains
unclear what kind of inductive structures exists and how we can faithfully
assign a term representation of them. In this paper we propose a simple term
syntax for cyclic sharing structures that admits structural induction and
recursion principles. We show that the obtained syntax is directly usable in
the functional language Haskell and the proof assistant Agda, as well as
ordinary data structures such as lists and trees. To achieve this goal, we use
a categorical approach to initial algebra semantics in a presheaf category.
That approach follows the line of Fiore, Plotkin and Turi's models of abstract
syntax with variable binding
A general conservative extension theorem in process algebras with inequalities
We prove a general conservative extension theorem for transition system based process theories with easy-to-check and reasonable conditions. The core of this result is another general theorem which gives sufficient conditions for a system of operational rules and an extension of it in order to ensure conservativity, that is, provable transitions from an original term in the extension are the same as in the original system. As a simple corollary of the conservative extension theorem we prove a completeness theorem. We also prove a general theorem giving sufficient conditions to reduce the question of ground confluence modulo some equations for a large term rewriting system associated with an equational process theory to a small term rewriting system under the condition that the large system is a conservative extension of the small one. We provide many applications to show that our results are useful. The applications include (but are not limited to) various real and discrete time settings in ACP, ATP, and CCS and the notions projection, renaming, stage operator, priority, recursion, the silent step, autonomous actions, the empty process, divergence, etc
States and exceptions considered as dual effects
In this paper we consider the two major computational effects of states and
exceptions, from the point of view of diagrammatic logics. We get a surprising
result: there exists a symmetry between these two effects, based on the
well-known categorical duality between products and coproducts. More precisely,
the lookup and update operations for states are respectively dual to the throw
and catch operations for exceptions. This symmetry is deeply hidden in the
programming languages; in order to unveil it, we start from the monoidal
equational logic and we add progressively the logical features which are
necessary for dealing with either effect. This approach gives rise to a new
point of view on states and exceptions, which bypasses the problems due to the
non-algebraicity of handling exceptions
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
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