506 research outputs found
Tool support for reasoning in display calculi
We present a tool for reasoning in and about propositional sequent calculi.
One aim is to support reasoning in calculi that contain a hundred rules or
more, so that even relatively small pen and paper derivations become tedious
and error prone. As an example, we implement the display calculus D.EAK of
dynamic epistemic logic. Second, we provide embeddings of the calculus in the
theorem prover Isabelle for formalising proofs about D.EAK. As a case study we
show that the solution of the muddy children puzzle is derivable for any number
of muddy children. Third, there is a set of meta-tools, that allows us to adapt
the tool for a wide variety of user defined calculi
Resource-Bound Quantification for Graph Transformation
Graph transformation has been used to model concurrent systems in software
engineering, as well as in biochemistry and life sciences. The application of a
transformation rule can be characterised algebraically as construction of a
double-pushout (DPO) diagram in the category of graphs. We show how
intuitionistic linear logic can be extended with resource-bound quantification,
allowing for an implicit handling of the DPO conditions, and how resource logic
can be used to reason about graph transformation systems
An Abstract Approach to Stratification in Linear Logic
We study the notion of stratification, as used in subsystems of linear logic
with low complexity bounds on the cut-elimination procedure (the so-called
light logics), from an abstract point of view, introducing a logical system in
which stratification is handled by a separate modality. This modality, which is
a generalization of the paragraph modality of Girard's light linear logic,
arises from a general categorical construction applicable to all models of
linear logic. We thus learn that stratification may be formulated independently
of exponential modalities; when it is forced to be connected to exponential
modalities, it yields interesting complexity properties. In particular, from
our analysis stem three alternative reformulations of Baillot and Mazza's
linear logic by levels: one geometric, one interactive, and one semantic
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Linear-Logic Based Analysis of Constraint Handling Rules with Disjunction
Constraint Handling Rules (CHR) is a declarative committed-choice programming
language with a strong relationship to linear logic. Its generalization CHR
with Disjunction (CHRv) is a multi-paradigm declarative programming language
that allows the embedding of horn programs. We analyse the assets and the
limitations of the classical declarative semantics of CHR before we motivate
and develop a linear-logic declarative semantics for CHR and CHRv. We show how
to apply the linear-logic semantics to decide program properties and to prove
operational equivalence of CHRv programs across the boundaries of language
paradigms
The polarized λ-calculus
A natural deduction system isomorphic to the focused sequent calculus for polarized intuitionistic logic is proposed. The
system comes with a language of proof-terms, named polarized λ-calculus, whose reduction rules express simultaneously a
normalization procedure and the isomorphic copy of the cut-elimination procedure pertaining to the focused sequent calculus.
Noteworthy features of this natural deduction system are: how the polarity of a connective determines the style of its
elimination rule; the existence of a proof-search strategy which is equivalent to focusing in the sequent calculus; the highlydisciplined
organization of the syntax - even atoms have introduction, elimination and normalization rules. The polarized
λ-calculus is a programming formalism close to call-by-push-value, but justified by its proof-theoretical pedigree.This research was financed by Portuguese Funds through FCT Fundac¸ao para a Ci ˜ encia ˆ
e a Tecnologia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio
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