1,236 research outputs found
On the strength of dependent products in the type theory of Martin-L\"of
One may formulate the dependent product types of Martin-L\"of type theory
either in terms of abstraction and application operators like those for the
lambda-calculus; or in terms of introduction and elimination rules like those
for the other constructors of type theory. It is known that the latter rules
are at least as strong as the former: we show that they are in fact strictly
stronger. We also show, in the presence of the identity types, that the
elimination rule for dependent products--which is a "higher-order" inference
rule in the sense of Schroeder-Heister--can be reformulated in a first-order
manner. Finally, we consider the principle of function extensionality in type
theory, which asserts that two elements of a dependent product type which are
pointwise propositionally equal, are themselves propositionally equal. We
demonstrate that the usual formulation of this principle fails to verify a
number of very natural propositional equalities; and suggest an alternative
formulation which rectifies this deficiency.Comment: 18 pages; v2: final journal versio
Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional Martin-L\"of type
theory. We use constructive ordered algebra in toposes. Identity types in the
models are given by a notion of Moore path. By considering a particular gros
topos, we show that there is such a model that is non-truncated, i.e. contains
non-trivial structure at all dimensions. In other words, in this model a type
in a nested sequence of identity types can contain more than one element, no
matter how great the degree of nesting. Although inspired by existing
non-truncated models of type theory based on simplicial and cubical sets, the
notion of model presented here is notable for avoiding any form of Kan filling
condition in the semantics of types.Comment: This is a revised and expanded version of a paper with the same name
that appeared in the proceedings of the 2nd International Conference on
Formal Structures for Computation and Deduction (FSCD 2017
Binary Relations as a Foundation of Mathematics
We describe a theory for binary relations in the Zermelo-Fraenkel style. We choose for ZFCU, a variant of ZFC Set theory in which the Axiom of Foundation is replaced by an axiom allowing for non-wellfounded sets. The theory of binary relations is shown to be equi-consistent ZFCU by constructing a model for the theory of binary relations in ZFU and vice versa. Thus, binary relations are a foundation for mathematics in the same sense as sets are
Quantifier-Free Interpolation of a Theory of Arrays
The use of interpolants in model checking is becoming an enabling technology
to allow fast and robust verification of hardware and software. The application
of encodings based on the theory of arrays, however, is limited by the
impossibility of deriving quantifier- free interpolants in general. In this
paper, we show that it is possible to obtain quantifier-free interpolants for a
Skolemized version of the extensional theory of arrays. We prove this in two
ways: (1) non-constructively, by using the model theoretic notion of
amalgamation, which is known to be equivalent to admit quantifier-free
interpolation for universal theories; and (2) constructively, by designing an
interpolating procedure, based on solving equations between array updates.
(Interestingly, rewriting techniques are used in the key steps of the solver
and its proof of correctness.) To the best of our knowledge, this is the first
successful attempt of computing quantifier- free interpolants for a variant of
the theory of arrays with extensionality
Prospects for a Naive Theory of Classes
The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Bradyâs result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak
States of Affairs as Structured Extensions in Free Logic
The search for the extensions of sentences can be guided by Fregeâs âprinciple of compositionality of extensionâ, according to which the extension of a composed expression depends only on its logical form and the extensions of its parts capable of having extensions. By means of this principle, a strict criterion for the admissibility of objects as extensions of sentences can be derived: every object is admissible as the extension of a sentence that is preserved under the substitution of co-extensional expressions. The question is: what are the extensions of elementary sentences containing empty singular terms, like âVulcan rotatesâ. It can be demonstrated that in such sentences, states of affairs as structured objects (but not truth-values) are preserved under the substitution of co-extensional expressions. Hence, such states of affairs are admissible (while truth-values are not) as extensions of elementary sentences containing empty singular terms
Syntax for free: representing syntax with binding using parametricity
We show that, in a parametric model of polymorphism, the type ââα.â((αâââα)âââα)âââ(αâââαâââα)âââα is isomorphic to closed de Bruijn terms. That is, the type of closed higher-order abstract syntax terms is isomorphic to a concrete representation. To demonstrate the proof we have constructed a model of parametric polymorphism inside the Coq proof assistant. The proof of the theorem requires parametricity over Kripke relations. We also investigate some variants of this representation
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
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