The original formulation of the Curry---Howard correspondence relates propositional logic to the simply-typed $$\lambda $$λ-calculus at three levels: the syntax of propositions corresponds to the syntax of types; the proofs of propositions correspond to programs of the corresponding types; and the normalization of proofs corresponds to the evaluation of programs. This rich correspondence has inspired our community for half a century and has been generalized to deal with more advanced logics and programming models. We propose a variant of this correspondence which is inspired by conservation of information and recent homotopy theoretic approaches to type theory.

Our proposed correspondence naturally relates semirings to reversible programming languages: the syntax of semiring elements corresponds to the syntax of types; the proofs of semiring identities correspond to reversible programs of the corresponding types; and equivalences between algebraic proofs correspond to meaning-preserving program transformations and optimizations. These latter equivalences are not ad hoc: the same way semirings arise naturally out of the structure of types, a categorical look at the structure of proof terms gives rise to at least a weak rig groupoid structure, and the coherence laws are exactly the program transformations we seek. Thus it is algebra, rather than logic, which finally leads us to our correspondence.

Computing with Semirings and Weak Rig Groupoids

We present a new sound and complete axiomatization of regular expression containment. It consists of the conventional axiomatization of concatenation, alternation, empty set and (the singleton set containing) the empty string as an idempotent semiring, the fixed- point rule E* = 1 + E × E* for Kleene-star, and a general coinduction rule as the only additional rule.Our axiomatization gives rise to a natural computational interpretation of regular expressions as simple types that represent parse trees, and of containment proofs as coercions. This gives the axiom- atization a Curry-Howard-style constructive interpretation: Containment proofs do not only certify a language-theoretic contain- ment, but, under our computational interpretation, constructively transform a membership proof of a string in one regular expression into a membership proof of the same string in another regular expression.We show how to encode regular expression equivalence proofs in Salomaa's, Kozen's and Grabmayer's axiomatizations into our containment system, which equips their axiomatizations with a computational interpretation and implies completeness of our axiomatization. To ensure its soundness, we require that the computational interpretation of the coinduction rule be a hereditarily total function. Hereditary totality can be considered the mother of syn- tactic side conditions: it "explains" their soundness, yet cannot be used as a conventional side condition in its own right since it turns out to be undecidable.We discuss application of regular expressions as types to bit coding of strings and hint at other applications to the wide-spread use of regular expressions for substring matching, where classical automata-theoretic techniques are a priori inapplicable.Neither regular expressions as types nor subtyping interpreted coercively are novel per se. Somewhat surprisingly, this seems to be the first investigation of a general proof-theoretic framework for the latter in the context of the former, however.

Regular expression containment: coinductive axiomatization and computational interpretation

In reversible computing, the management of space is subject to two broad classes of constraints. First, as with general-purpose computation, every allocation must be paired with a matching de-allocation. Second, space can only be safely de-allocated if its contents are restored to their initial value from allocation time. Generally speaking, the state of the art provides limited partial solutions that address the first constraint by imposing a stack discipline and by leaving the second constraint to programmers' assertions. We propose a novel approach based on the idea of fractional types. As a simple intuitive example, allocation of a new boolean value initialized to $\texttt{false}$ also creates a value $1/{\texttt{false}}$ that can be thought of as a garbage collection (GC) process specialized to reclaim, and only reclaim, storage containing the value $\texttt{false}$. This GC process is a first-class entity that can be manipulated, decomposed into smaller processes and combined with other GC processes. We formalize this idea in the context of a reversible language founded on type isomorphisms, prove its fundamental correctness properties, and illustrate its expressiveness using a wide variety of examples. The development is backed by a fully-formalized Agda implementation.

Fractional Types: Expressive and Safe Space Management for Ancilla Bits

The amount of compressed data in network traffic is gradually increasing. Such traffic effectively reduces the transmission time and improves the data transmission efficiency in the network, but it poses a challenge to the existing network traffic identification mechanism. At present, the identification method used for the compression traffic is often used to recover the complete compressed data for content identification. However, this method takes a lot of time to restore the complete compressed data and subsequent content matching.This paper analyzes the specific compressed data format (gz, zip) and proposes an algorithm for decompressing compressed file fragments: decompresses the compressed file fragments and restores some of the original data. Because only the fragments of the compressed file need to be restored, the algorithm effectively improves the efficiency of network data restoration. The structure of regular expression and the string matching algorithm based on suffix search are studied. According to the distribution of target data in the actual traffic monitoring system, a regular expression matching algorithm based on preprocessing is proposed: extract the exact string of the regular expression, use the exact string as the pattern string to filter the suspicious data in the data using the suffix-based string matching algorithm, and then use the regular expression for matching verification. As a result of filtering many unrelated data during the process of searching for suspicious data, the efficiency of the algorithm is improved. In the end, the paper designs a packet capture system, which compresses compressed traffic transmitted through partial recombination of FTP and HTTP protocols, combines compressed file fragment decompression algorithm with preprocessed regular expression matching method, and realizes early detection and identification of compressed traffic.In summary, this paper proposes and implements an early analysis and detection system for compressed flow, which plays a certain role in the identification and detection of compressed data traffic.

Early Analysis and Detection of Compressed Traffic

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

Lambda Calculus with Types

This paper gives the first decidability results on type isomorphism for recursive types, establishing the explicit decidability of type isomorphism for the type theory of sums and products over an ...

Isomorphisms of generic recursive polynomial types

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Isomorphisms of generic recursive polynomial types