arXiv subjects

Manfred Schmidt-Schauß

Publications and source records attributed to Manfred Schmidt-Schauß.

12 recordsLinked to original sources

arXiv2022

A Probabilistic Call-by-Need Lambda-Calculus -- Extended Version

To support the understanding of declarative probabilistic programming languages, we introduce a lambda-calculus with a fair binary probabilistic choice that chooses between its arguments with equal probability. The reduction strategy of the calculus is a call-by-need strategy that performs lazy evaluation and implements sharing by recursive let-expressions. Expected convergence of expressions is the limit of the sum of all successful reduction outputs weighted by their probability. We use contextual equivalence as program semantics: two expressions are contextually equivalent if and only if the expected convergence of the expressions plugged into any program context is always the same. We develop and illustrate techniques to prove equivalences including a context lemma, two derived criteria to show equivalences and a syntactic diagram-based method. This finally enables us to show correctness of a large set of program transformations with respect to the contextual equivalence.

cs.LO↗

arXiv2022

Nominal Unification and Matching of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP.

cs.LO↗

arXiv2021

Minimal Translations from Synchronous Communication to Synchronizing Locks (Extended Version)

In order to understand the relative expressive power of larger concurrent programming languages, we analyze translations of small process calculi which model the communication and synchronization of concurrent processes. The source language SYNCSIMPLE is a minimalistic model for message passing concurrency while the target language LOCKSIMPLE is a minimalistic model for shared memory concurrency. The former is a calculus with synchronous communication of processes, while the latter has synchronizing mutable locations -- called locks -- that behave similarly to binary semaphores. The criteria for correctness of translations is that they preserve and reflect may-termination and must-termination of the processes. We show that there is no correct compositional translation from SYNCSIMPLE to LOCKSIMPLE that uses one or two locks, independent from the initialisation of the locks. We also show that there is a correct translation that uses three locks. Also variants of the locks are taken into account with different blocking behavior.

cs.LO↗

arXiv2021

Minimal Translations from Synchronous Communication to Synchronizing Locks

In order to understand the relative expressive power of larger concurrent programming languages, we analyze translations of small process calculi which model the communication and synchronization of concurrent processes. The source language SYNCSIMPLE is a minimalistic model for message passing concurrency while the target language LOCKSIMPLE is a minimalistic model for shared memory concurrency. The former is a calculus with synchronous communication of processes, while the latter has synchronizing mutable locations - called locks - that behave similarly to binary semaphores. The criteria for correctness of translations is that they preserve and reflect may-termination and must-termination of the processes. We show that there is no correct compositional translation from SYNCSIMPLE to LOCKSIMPLE that uses one or two locks, independent from the initialisation of the locks. We also show that there is a correct translation that uses three locks. Also variants of the locks are taken into account with different blocking behavior.

cs.LO↗

arXiv2020

Correctly Implementing Synchronous Message Passing in the Pi-Calculus By Concurrent Haskell's MVars

Comparison of concurrent programming languages and correctness of program transformations in concurrency are the focus of this research. As criterion we use contextual semantics adapted to concurrency, where may -- as well as should -- convergence are observed. We investigate the relation between the synchronous pi-calculus and a core language of Concurrent Haskell (CH). The contextual semantics is on the one hand forgiving with respect to the details of the operational semantics, and on the other hand implies strong requirements for the interplay between the processes after translation. Our result is that CH embraces the synchronous pi-calculus. Our main task is to find and prove correctness of encodings of pi-calculus channels by CH's concurrency primitives, which are MVars. They behave like (blocking) 1-place buffers modelling the shared-memory. The first developed translation uses an extra private MVar for every communication.We also automatically generate and check potentially correct translations that reuse the MVars where one MVar contains the message and two additional MVars for synchronization are used to model the synchronized communication of a single channel in the pi-calculus.Our automated experimental results lead to the conjecture that one additional MVar is insufficient.

cs.PL↗

arXiv2019

Optimizing Space of Parallel Processes

This paper is a contribution to exploring and analyzing space-improvements in concurrent programming languages, in particular in the functional process-calculus CHF. Space-improvements are defined as a generalization of the corresponding notion in deterministic pure functional languages. The main part of the paper is the O(n*log n) algorithm SpOptN for offline space optimization of several parallel independent processes. Applications of this algorithm are: (i) affirmation of space improving transformations for particular classes of program transformations; (ii) support of an interpreter-based method for refuting space-improvements; and (iii) as a stand-alone offline-optimizer for space (or similar resources) of parallel processes.

cs.PL↗

arXiv2018

Space Improvements and Equivalences in a Functional Core Language

We explore space improvements in LRP, a polymorphically typed call-by-need functional core language. A relaxed space measure is chosen for the maximal size usage during an evaluation. It abstracts from the details of the implementation via abstract machines, but it takes garbage collection into account and thus can be seen as a realistic approximation of space usage. The results are: a context lemma for space improving translations and for space equivalences, all but one reduction rule of the calculus are shown to be space improvements, and for the exceptional one we show bounds on the space increase. Several further program transformations are shown to be space improvements or space equivalences in particular the translation into machine expressions is a space equivalence. We also classify certain space-worsening transformations as space-leaks or as space-safe. These results are a step forward in making predictions about the change in runtime space behavior of optimizing transformations in call-by-need functional languages.

cs.PL↗

arXiv2016

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.

cs.PL↗

arXiv2015

Simulation in the Call-by-Need Lambda-Calculus with Letrec, Case, Constructors, and Seq

This paper shows equivalence of several versions of applicative similarity and contextual approximation, and hence also of applicative bisimilarity and contextual equivalence, in LR, the deterministic call-by-need lambda calculus with letrec extended by data constructors, case-expressions and Haskell's seq-operator. LR models an untyped version of the core language of Haskell. The use of bisimilarities simplifies equivalence proofs in calculi and opens a way for more convenient correctness proofs for program transformations. The proof is by a fully abstract and surjective transfer into a call-by-name calculus, which is an extension of Abramsky's lazy lambda calculus. In the latter calculus equivalence of our similarities and contextual approximation can be shown by Howe's method. Similarity is transferred back to LR on the basis of an inductively defined similarity. The translation from the call-by-need letrec calculus into the extended call-by-name lambda calculus is the composition of two translations. The first translation replaces the call-by-need strategy by a call-by-name strategy and its correctness is shown by exploiting infinite trees which emerge by unfolding the letrec expressions. The second translation encodes letrec-expressions by using multi-fixpoint combinators and its correctness is shown syntactically by comparing reductions of both calculi. A further result of this paper is an isomorphism between the mentioned calculi, which is also an identity on letrec-free expressions.

cs.LO↗

arXiv2010

Towards Correctness of Program Transformations Through Unification and Critical Pair Computation

Correctness of program transformations in extended lambda calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach to proving correctness is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, and then of so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study we apply the method to a lambda calculus with recursive let-expressions and describe an effective unification algorithm to determine all overlaps of a set of transformations with all reduction rules. The unification algorithm employs many-sorted terms, the equational theory of left-commutativity modelling multi-sets, context variables of different kinds and a mechanism for compactly representing binding chains in recursive let-expressions.

cs.PL↗

arXiv2010

Unification and Matching on Compressed Terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.

cs.LO↗

arXiv2000

A Lambda-Calculus with letrec, case, constructors and non-determinism

A non-deterministic call-by-need lambda-calculus \calc with case, constructors, letrec and a (non-deterministic) erratic choice, based on rewriting rules is investigated. A standard reduction is defined as a variant of left-most outermost reduction. The semantics is defined by contextual equivalence of expressions instead of using $αβ(η)$-equivalence. It is shown that several program transformations are correct, for example all (deterministic) rules of the calculus, and in addition the rules for garbage collection, removing indirections and unique copy. This shows that the combination of a context lemma and a meta-rewriting on reductions using complete sets of commuting (forking, resp.) diagrams is a useful and successful method for providing a semantics of a functional programming language and proving correctness of program transformations.

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