25 research outputs found
Logical calculi for reasoning with binding
In informal mathematical usage we often reason about languages involving binding of object-variables. We find ourselves writing assertions involving meta-variables and capture-avoidance constraints on where object-variables can and cannot occur free. Formalising such assertions is problematic because the standard logical frameworks cannot express capture-avoidance constraints directly. In this thesis we make the case for extending logical frameworks with metavariables and capture-avoidance constraints. We use nominal techniques that allow for a direct formalisation of meta-level assertions, while remaining close to informal practice. Our focus is on derivability and we show that our derivation rules support the following key features of meta-level reasoning: • instantiation of meta-variables, by means of capturing substitution of terms for meta-variables; • ??-renaming of object-variables and capture-avoiding substitution of terms for object-variables in the presence of meta-variables; • generation of fresh object-variables inside a derivation. We apply our nominal techniques to the following two logical frameworks: • Equational logic. We investigate proof-theoretical properties, give a semantics in nominal sets and compare the notion of ??-renaming to existing notions of ??-equivalence with meta-variables. We also provide an axiomatisation of capture-avoiding substitution, and show that it is sound and complete with respect to the usual notion of capture-avoiding substitution. • First-order logic with equality. We provide a sequent calculus with metavariables and capture-avoidance constraints, and show that it represents schemas of derivations in first-order logic. We also show how we can axiomatise this notion of derivability in the calculus for equational logic
Placeholder calculus for first-order logic
In this paper we present an extension of first-order predicate logic with placeholders. These placeholders allow the construction of proofs for incomplete theorems. These theorems can be completed during the proof construction process. By using special definitions of substitutions and replacements, we obtain an unexpectedly simple cal- culus. Furthermore, we avoid the need of additional rules for explicit substitutions to deal with postponed substitutions in placeholders, since the definitions of substitution and replacement deal with them directly
Closed nominal rewriting and efficiently computable nominal algebra equality
We analyse the relationship between nominal algebra and nominal rewriting,
giving a new and concise presentation of equational deduction in nominal
theories. With some new results, we characterise a subclass of equational
theories for which nominal rewriting provides a complete procedure to check
nominal algebra equality. This subclass includes specifications of the
lambda-calculus and first-order logic.Comment: In Proceedings LFMTP 2010, arXiv:1009.218
Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets
We investigate a class of nominal algebraic Henkin-style models for the
simply typed lambda-calculus in which variables map to names in the denotation
and lambda-abstraction maps to a (non-functional) name-abstraction operation.
The resulting denotations are smaller and better-behaved, in ways we make
precise, than functional valuation-based models.
Using these new models, we then develop a generalisation of \lambda-term
syntax enriching them with existential meta-variables, thus yielding a theory
of incomplete functions. This incompleteness is orthogonal to the usual notion
of incompleteness given by function abstraction and application, and
corresponds to holes and incomplete objects.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
Nominal Logic with Equations Only
Many formal systems, particularly in computer science, may be captured by
equations modulated by side conditions asserting the "freshness of names";
these can be reasoned about with Nominal Equational Logic (NEL). Like most
logics of this sort NEL employs this notion of freshness as a first class
logical connective. However, this can become inconvenient when attempting to
translate results from standard equational logic to the nominal setting. This
paper presents proof rules for a logic whose only connectives are equations,
which we call Nominal Equation-only Logic (NEoL). We prove that NEoL is just as
expressive as NEL. We then give a simple description of equality in the empty
NEoL-theory, then extend that result to describe freshness in the empty
NEL-theory.Comment: In Proceedings LFMTP 2011, arXiv:1110.668
From nominal sets binding to functions and lambda-abstraction: connecting the logic of permutation models with the logic of functions
Permissive-Nominal Logic (PNL) extends first-order predicate logic with
term-formers that can bind names in their arguments. It takes a semantics in
(permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are
just term-formers satisfying axioms, and their denotation is functions on
nominal atoms-abstraction.
Then we have higher-order logic (HOL) and its models in ordinary (i.e.
Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full
or partial function spaces.
This raises the following question: how are these two models of binding
connected? What translation is possible between PNL and HOL, and between
nominal sets and functions?
We exhibit a translation of PNL into HOL, and from models of PNL to certain
models of HOL. It is natural, but also partial: we translate a restricted
subsystem of full PNL to HOL. The extra part which does not translate is the
symmetry properties of nominal sets with respect to permutations. To use a
little nominal jargon: we can translate names and binding, but not their
nominal equivariance properties. This seems reasonable since HOL---and ordinary
sets---are not equivariant.
Thus viewed through this translation, PNL and HOL and their models do
different things, but they enjoy non-trivial and rich subsystems which are
isomorphic
Semantics out of context: nominal absolute denotations for first-order logic and computation
Call a semantics for a language with variables absolute when variables map to
fixed entities in the denotation. That is, a semantics is absolute when the
denotation of a variable a is a copy of itself in the denotation. We give a
trio of lattice-based, sets-based, and algebraic absolute semantics to
first-order logic. Possibly open predicates are directly interpreted as lattice
elements / sets / algebra elements, subject to suitable interpretations of the
connectives and quantifiers. In particular, universal quantification "forall
a.phi" is interpreted using a new notion of "fresh-finite" limit and using a
novel dual to substitution.
The interest of this semantics is partly in the non-trivial and beautiful
technical details, which also offer certain advantages over existing
semantics---but also the fact that such semantics exist at all suggests a new
way of looking at variables and the foundations of logic and computation, which
may be well-suited to the demands of modern computer science