Eventual linear ranking functions

Abstract

We present the first operational account of call by need that connects syntactic theory and implementation practice. Syntactic theory: the storeless operational semantics using syntax rewriting to account for demand-driven computation and for caching intermediate results. Implementational practice: the store-based operational technique using memo-thunks to implement demand-driven computation and to cache intermediate results for subsequent sharing. The implementational practice was initiated by Landin and Wadsworth and is prevalent today to implement lazy programming languages such as Haskell. The syntactic theory was initiated by Ariola, Felleisen, Maraist, Odersky and Wadler and is prevalent today to reason equationally about lazy programs, on par with Baren-dregt et al.'s term graphs. Nobody knows, however, how the theory of call by need compares to the practice of call by need: all that is known is that the theory of call by need agrees with the theory of call by name, and that the practice of call by need optimizes the practice of call by name. Our operational account takes the form of three new calculi for lazy evaluation of lambda-terms and our synthesis takes the form of three lock-step equivalences. The first calculus is a hereditarily compressed variant of Ariola et al.'s call-by-need lambda-calculus and makes "neededness" syntactically explicit. The second calculus distinguishes between strict bindings (which are induced by demand-driven computation) and non-strict bindings (which are used for caching intermediate results). The third calculus uses memo-thunks and an algebraic store. The first calculus syntactically corresponds to a storeless abstract machine, the second to an abstract machine with local stores, and the third to a lazy Kriv-ine machine, i.e., a traditional store-based abstract machine implementing lazy evaluation. The machines are intensionally compatible with extensional reasoning about lazy programs and they are lock-step equivalent. Each machine functionally corresponds to a natural semantics for call by need in the style of Launchbury, though for non-preprocessed λ-terms. Our results reveal a genuine and principled unity of computational theory and computational practice, one that readily applies to variations on the general theme of call by need.

Ranking Functions for Linear-Constraint Loops

“…This is similar to the definition of Bradley et al [2005a], however, it is more general since it does not require a fixed association of ranking-function components with the paths of the loop. Additional generalizations of linear ranking functions are eventual ranking functions [Bagnara and Mesnard 2013] and Polyranking functions [Bradley et al 2005b]. The complexity classification of the corresponding decision problems, over the integers (and in the latter case, also over rationals), is still open.…”

Ranking Templates for Linear Loops

“…Cook et al [CKRW13] compute the integral hull of transition relations in order obtain the same completeness for integers and bitvectors. Bagnara and Mesnard generalize the PR ranking template to the 2-phase ranking template, relying on nonlinear constraint solving [BM13].…”

A practical comparator of cost functions and its applications

“…The next example is borrowed from [11] and used to illustrate motivation (1) above. Solving this example requires being able to handle non-linear size relations which is rather expensive and thus several systems cannot support them (e.g., PUBS cannot solve it).…”