Linear Recursion II: Tail Recursion

2017 ◽  
pp. 133-170
Author(s):  
Manuel Rubio-Sánchez
Keyword(s):  
Tail Recursion ◽  
Linear Recursion

scholarly journals Ontology-Mediated Querying with the Description Logic EL: Trichotomy and Linear Datalog Rewritability

2017 ◽  
Author(s):  
Carsten Lutz ◽  
Leif Sabellek
Keyword(s):  
Description Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Data Complexity ◽  
First Order ◽  
Fine Grained ◽  
Linear Recursion

We consider ontology-mediated queries (OMQs) based on an EL ontology and an atomic query (AQ), provide an ultimately fine-grained analysis of data complexity and study rewritability into linear Datalog-aiming to capture linear recursion in SQL. Our main results are that every such OMQ is in AC0, NL-complete or PTime-complete, and that containment in NL coincides with rewritability into linear Datalog (whereas containment in AC0 coincides with rewritability into first-order logic). We establish natural characterizations of the three cases, show that deciding linear Datalog rewritability (as well as the mentioned complexities) is ExpTime-complete, give a way to construct linear Datalog rewritings when they exist, and prove that there is no constant bound on the arity of IDB relations in linear Datalog rewritings.

scholarly journals A Syntactic Correspondence between Context-Sensitive Calculi and Abstract Machines

2005 ◽  
Vol 12 (22) ◽  
Author(s):  
Malgorzata Biernacka ◽  
Olivier Danvy
Keyword(s):  
Open Problem ◽  
Lazy Evaluation ◽  
Systematic Construction ◽  
Context Sensitive ◽  
Evaluation Functions ◽  
Abstract Machines ◽  
Explicit Substitutions ◽  
Delimited Continuations ◽  
Tail Recursion

We present a systematic construction of environment-based abstract machines from context-sensitive calculi of explicit substitutions, and we illustrate it with a series of calculi and machines: Krivine's machine with call/cc, the lambda-mu-calculus, delimited continuations, i/o, stack inspection, proper tail-recursion, and lazy evaluation. Most of the machines already exist but have been obtained independently and are only indirectly related to the corresponding calculi. All of the calculi are new and they make it possible to directly reason about the execution of the corresponding machines. In connection with the functional correspondence between evaluation functions and abstract machines initiated by Reynolds, the present syntactic correspondence makes it possible to construct reduction-free normalization functions out of reduction-based ones, which was an open problem in the area of normalization by evaluation.

scholarly journals A Syntactic Correspondence between Context-Sensitive Calculi and Abstract Machines

2005 ◽  
Vol 12 (38) ◽  
Author(s):  
Malgorzata Biernacka ◽  
Olivier Danvy
Keyword(s):  
Open Problem ◽  
Lazy Evaluation ◽  
Normal Order ◽  
Systematic Construction ◽  
Context Sensitive ◽  
Evaluation Functions ◽  
Abstract Machines ◽  
Explicit Substitutions ◽  
Delimited Continuations ◽  
Tail Recursion

We present a systematic construction of environment-based abstract machines from context-sensitive calculi of explicit substitutions, and we illustrate it with ten calculi and machines for applicative order with an abort operation, normal order with generalized reduction and call/cc, the lambda-mu-calculus, delimited continuations, stack inspection, proper tail-recursion, and lazy evaluation. Most of the machines already exist but have been obtained independently and are only indirectly related to the corresponding calculi. All of the calculi are new and they make it possible to directly reason about the execution of the corresponding machines.<br /> <br />In connection with the functional correspondence between evaluation functions and abstract machines initiated by Reynolds, the present syntactic correspondence makes it possible to construct reduction-free normalization functions out of reduction-based ones, which was an open problem in the area of normalization by evaluation.

scholarly journals Fitness Conditions for fixed Point Iteration

1992 ◽  
Vol 21 (384) ◽  
Author(s):  
Flemming Nielson ◽  
Hanne Riis Nielson
Keyword(s):  
Fixed Point ◽  
Abstract Interpretation ◽  
Flow Analysis ◽  
Worst Case ◽  
Fixed Point Iteration ◽  
Average Case ◽  
Data Flow Analysis ◽  
Tail Recursion ◽  
Carry Over ◽  
Primitive Recursive

This paper provides a link between the formulation of static program analyses using the framework of abstract interpretation (popular for functional languages) and using the more classical framework of data flow analysis (popular for imperative languages). In particular we show how the classical notions of fastness, rapidity and k-boundedness carry over to the abstract interpretation framework and how this may be used to bound the number of times a functional should be unfolded in order to yield the fixed point. This is supplemented with a number of results on how to calculate the bounds for iterative forms (as for tail recursion), for linear forms (as for one nested recursive call), and for primitive recursive forms. In some cases this improves the ''worst case'' results of H.R. Nielson and F. Nielson: Bounded Fixed Point Iteration, but more importantly it gives much better ''average case'' results.

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