27 research outputs found
Kripke Semantics for Martin-L\"of's Extensional Type Theory
It is well-known that simple type theory is complete with respect to
non-standard set-valued models. Completeness for standard models only holds
with respect to certain extended classes of models, e.g., the class of
cartesian closed categories. Similarly, dependent type theory is complete for
locally cartesian closed categories. However, it is usually difficult to
establish the coherence of interpretations of dependent type theory, i.e., to
show that the interpretations of equal expressions are indeed equal. Several
classes of models have been used to remedy this problem. We contribute to this
investigation by giving a semantics that is standard, coherent, and
sufficiently general for completeness while remaining relatively easy to
compute with. Our models interpret types of Martin-L\"of's extensional
dependent type theory as sets indexed over posets or, equivalently, as
fibrations over posets. This semantics can be seen as a generalization to
dependent type theory of the interpretation of intuitionistic first-order logic
in Kripke models. This yields a simple coherent model theory, with respect to
which simple and dependent type theory are sound and complete
A Proof of Strong Normalization for the Theory of Constructions Using a Kripke-Like Interpretation
We give a proof that all terms that type-check in the theory of contructions are strongly normalizing (under ß-reduction). The main novelty of this proof is that it uses a Kripke-like interpretation of the types and kinds, and that it does not use infinite contexts. We explore some consequences of strong normalization, consistency and decidability of typechecking. We also show that our proof yields another proof of strong normalization for LF (under ß-reduction), using the reducibility method
Kripke Semantics for Dependent Type Theory and Realizability Interpretations
Abstract Constructive reasoning has played an increasingly important role in the development of provably correct software. Both typed and type-free frameworks stemming from ideas of Heyting, Kleene, and Curry have been developed for extracting computations from constructive specifications. These include Realizability, and Theories based on the Curry-Howard isomorphism. Realizability -in its various typed and type-free formulations -brings out the algorithmic content of theories and proofs and supplies models of the "recursive universe". Formal systems based on the propositions-as-types paradigm, such as Martin-Löf's dependent type theories, incorporate term extraction into the logic itself. Another, major tradition in constructive semantics originated in the model theory developed by Gödel, Herbrand and Tarski, resulting in the interpretations developed by Kripke and Beth, and in subsequent categorical generalizations. They provide a complete semantics for constructive logic. These models are a powerful tool for building counterexamples and establishing independence and conservativity results, but they are often less constructive and less computationally oriented. It is highly desirable to combine the power of these approaches to constructive semantics, and to elucidate some connections between them. We define modified Kripke and Beth models for syntactic Realizability and Dependent Type theory, in particular for the one-universe Intensional Martin-Löf Theory ML i 0 . These models provide a new framework for reasoning about computational evidence and the process of term-extraction. They are defined over a constructive type-free metatheory based on the Feferman-Beeson theories of abstract applicative structure. Our models have a feature which is shared by all published constructive completeness theorems for intuitionistic logic, known in the literature as "fallibility": there may be worlds in which some sentences are both false and true, a phenomenon which corresponds to the presence of empty types in various type disciplines. We also identify a natural lattice of truth values associated with type theory and realizability: the degrees of inhabitation
Kripke Semantics for Intersection Formulas
We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system
Semantics of Separation-Logic Typing and Higher-order Frame Rules for<br> Algol-like Languages
We show how to give a coherent semantics to programs that are well-specified
in a version of separation logic for a language with higher types: idealized
algol extended with heaps (but with immutable stack variables). In particular,
we provide simple sound rules for deriving higher-order frame rules, allowing
for local reasoning
A Simplicial Model for : Epistemic Logic with Agents that May Die
The standard semantics of multi-agent epistemic logic S5 is based on Kripke
models whose accessibility relations are reflexive, symmetric and transitive.
This one dimensional structure contains implicit higher-dimensional information
beyond pairwise interactions, that we formalized as pure simplicial models in a
previous work (Information and Computation, 2021). Here we extend the theory to
encompass simplicial models that are not necessarily pure. The corresponding
class of Kripke models are those where the accessibility relation is symmetric
and transitive, but might not be reflexive. Such models correspond to the
epistemic logic KB4 . Impure simplicial models arise in situations where two
possible worlds may not have the same set of agents. We illustrate it with
distributed computing examples of synchronous systems where processes may
crash
Kripke Semantics for a Logical Framework
We present a semantics (using Kripke lambda models) for a logical framework (minimal implicational predicate logic with quantification over all higher types). We apply the semantics to obtain straightforward adequacy proofs for encodings of logics in the framework. 1 Introduction There has been much recent interest in the development and use of logical frameworks. A logical framework is a formal system within which many different logics can be easily represented. It is hoped that such frameworks will facilitate the rapid development of proof assistants for the wide variety of different logics used in computer science and other fields. In this paper we give a semantic analysis (using Kripke lambda models) of the use of minimal implicational predicate logic (with quantification over all higher types) as a logical framework. We choose this framework because it is relatively straightforward to give it a useful semantics. The use of such a logic as a framework is not new. Similar logics ha..