Categorical semantics for higher order polymorphic lambda calculus

1987 ◽  
Vol 52 (4) ◽  
pp. 969-989 ◽  
Author(s):  
R. A. G. Seely
Keyword(s):  
Lambda Calculus ◽  
Higher Order ◽  
Algebraic Semantics ◽  
Categorical Structure ◽  
Categorical Semantics ◽  
Definitional Extension

AbstractA categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete. In fact, there is an equivalence between the theories and the categories. Also presented is a definitional extension of PLC including “subtypes”, for example, equality subtypes, together with a construction providing models of the extended language, and a context for Girard's extension of the Dialectica interpretation.

scholarly journals Generalized Bounded Linear Logic and its Categorical Semantics

2021 ◽  
pp. 226-246
Author(s):  
Yōji Fukihara ◽  
Shin-ya Katsumata
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Grading System ◽  
Bounded Linear ◽  
The Core ◽  
Categorical Structure ◽  
Categorical Semantics

AbstractWe introduce a generalization of Girard et al.’s called (and its affine variant ). It is designed to capture the core mechanism of dependency in , while it is also able to separate complexity aspects of . The main feature of is to adopt a multi-object pseudo-semiring as a grading system of the !-modality. We analyze the complexity of cut-elimination in , and give a translation from with constraints to with positivity axiom. We then introduce indexed linear exponential comonads (ILEC for short) as a categorical structure for interpreting the $${!}$$ ! -modality of . We give an elementary example of ILEC using folding product, and a technique to modify ILECs with symmetric monoidal comonads. We then consider a semantics of using the folding product on the category of assemblies of a BCI-algebra, and relate the semantics with the realizability category studied by Hofmann, Scott and Dal Lago.

Mechanizing proofs with logical relations – Kripke-style

2018 ◽  
Vol 28 (9) ◽  
pp. 1606-1638 ◽  
Author(s):  
ANDREW CAVE ◽  
BRIGITTE PIENTKA
Keyword(s):  
Case Studies ◽  
Operational Semantics ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Abstract Syntax ◽  
Completeness Proof ◽  
Typed Lambda Calculus ◽  
Contextual Equivalence ◽  
The Right

Proofs with logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe two case studies using the proof environmentBeluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline how to mechanize the completeness proof of algorithmic equality for simply typed lambda-terms where we reason about logically equivalent terms. The development of these proofs inBelugarelies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using higher order abstract syntax (HOAS) thereby avoiding the need to manipulate and deal with binders, renaming and substitutions, (2) we take advantage ofBeluga's support for representing derivations that depend on assumptions and first-class contexts to directly state inductive properties such as logical relations and inductive proofs, (3) we exploitBeluga's rich equational theory for simultaneous substitutions; as a consequence, users do not need to establish and subsequently use substitution properties, and proofs are not cluttered with references to them. We believe these examples demonstrate thatBelugaprovides the right level of abstractions and primitives to mechanize challenging proofs using HOAS encodings. It also may serve as a valuable benchmark for other proof environments.

An untyped higher order logic with Y combinator

2007 ◽  
Vol 72 (4) ◽  
pp. 1385-1404
Author(s):  
James H. Andrews
Keyword(s):  
Model Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Order Logic ◽  
Proof System ◽  
Cut Elimination ◽  
Higher Order Logic ◽  
Consistent Model

AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

scholarly journals Higher-order probabilistic adversarial computations: categorical semantics and program logics

2021 ◽  
Vol 5 (ICFP) ◽  
pp. 1-30
Author(s):  
Alejandro Aguirre ◽  
Gilles Barthe ◽  
Marco Gaboardi ◽  
Deepak Garg ◽  
Shin-ya Katsumata ◽  
...  
Keyword(s):  
Higher Order ◽  
General Notion ◽  
Memory Footprint ◽  
Resource Policy ◽  
Expected Values ◽  
Error Probabilities ◽  
The Cost ◽  
Relational Logic ◽  
Categorical Semantics ◽  
Develop Program

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

scholarly journals A Complete Calculus of Monotone and Antitone Higher-Order Functions

10.29007/3n54 ◽  
2018 ◽  
Author(s):  
Thomas Icard ◽  
Lawrence Moss
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Higher Order ◽  
Typed Lambda Calculus ◽  
Higher Order Functions

This paper adds monotonicity and antitonicity information to the typed lambda calculus, thereby providing a foundation for the Monotonicity Calculus first developed by van Benthem and others. We establish properties of the type system, propose a syntax, semantics, and proof calculus, and prove completeness for the calculus with respect to hierarchies of monotone and antitone functions over base preorders.

scholarly journals HOPLA--A Higher-Order Process Language

2002 ◽  
Vol 9 (49) ◽  
Author(s):  
Mikkel Nygaard ◽  
Glynn Winskel
Keyword(s):  
Operational Semantics ◽  
Lambda Calculus ◽  
Expressive Power ◽  
Higher Order ◽  
Domain Theory ◽  
Order Process ◽  
Encode Process ◽  
Mobile Ambients ◽  
Computation Path ◽  
Congruence Properties

A small but powerful language for higher-order nondeterministic processes is introduced. Its roots in a linear domain theory for concurrency are sketched though for the most part it lends itself to a more operational account. The language can be viewed as an extension of the lambda calculus with a ``prefixed sum'', in which types express the form of computation path of which a process is capable. Its operational semantics, bisimulation, congruence properties and expressive power are explored; in particular, it is shown how it can directly encode process languages such as CCS, CCS with process passing, and mobile ambients with public names.

scholarly journals A New One-Pass Transformation into Monadic Normal Form

2002 ◽  
Vol 9 (52) ◽  
Author(s):  
Olivier Danvy
Keyword(s):  
Normal Form ◽  
Symbolic Computation ◽  
Normal Forms ◽  
Lambda Calculus ◽  
Higher Order

We present a translation from the call-by-value lambda-calculus to monadic normal forms that includes short-cut boolean evaluation. The translation is higher-order, operates in one pass, duplicates no code, generates no chains of thunks, and is properly tail recursive. It makes a crucial use of symbolic computation at translation time.

scholarly journals Experiments with ZF Set Theory in HOL and Isabelle

1995 ◽  
Vol 2 (37) ◽  
Author(s):  
Sten Agerholm ◽  
Mike Gordon
Keyword(s):  
Set Theory ◽  
Lambda Calculus ◽  
General Purpose ◽  
Higher Order ◽  
Proof Assistants ◽  
Higher Order Logic ◽  
First Order ◽  
Advantages And Disadvantages ◽  
Theory Versus

Most general purpose proof assistants support versions of<br />typed higher order logic. Experience has shown that these logics are capable<br />of representing most of the mathematical models needed in Computer<br />Science. However, perhaps there exist applications where ZF-style<br />set theory is more natural, or even necessary. Examples may include<br />Scott's classical inverse-limit construction of a model of the untyped lambda-calculus<br /> (D_inf) and the semantics of parts of the Z specification notation.<br /><br />This paper compares the representation and use of ZF set theory within<br />both HOL and Isabelle. The main case study is the construction of D_inf.<br />The advantages and disadvantages of higher-order set theory versus first-order<br />set theory are explored experimentally. This study also provides a<br />comparison of the proof infrastructure of HOL and Isabelle.

scholarly journals A Complete, Co-Inductive Syntactic Theory of Sequential Control and State

2007 ◽  
Vol 14 (4) ◽  
Author(s):  
Kristian Støvring ◽  
Søren B. Lassen
Keyword(s):  
Normal Form ◽  
Lambda Calculus ◽  
Higher Order ◽  
Syntactic Theory ◽  
Sequential Control ◽  
Contextual Equivalence

We present a new co-inductive syntactic theory, eager normal form bisimilarity, for the untyped call-by-value lambda calculus extended with continuations and mutable references.<br /> <br />We demonstrate that the associated bisimulation proof principle is easy to use and that it is a powerful tool for proving equivalences between recursive imperative higher-order programs.<br /> <br />The theory is modular in the sense that eager normal form bisimilarity for each of the calculi extended with continuations and/or mutable references is a fully abstract extension of eager normal form bisimilarity for its sub-calculi. For each calculus, we prove that eager normal form bisimilarity is a congruence and is sound with respect to contextual equivalence. Furthermore, for the calculus with both continuations and mutable references, we show that eager normal form bisimilarity is complete: it coincides with contextual equivalence.

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