Denotational cost semantics for functional languages with inductive types

Abstract

Abstract. A central method for analyzing the asymptotic complexity of a functional program is to extract and then solve a recurrence that expresses evaluation cost in terms of input size. The relevant notion of input size is often specific to a datatype, with measures including the length of a list, the maximum element in a list, and the height of a tree. In this work, we give a formal account of the extraction of cost and size recurrences from higher-order functional programs over inductive datatypes. Our approach allows a wide range of programmer-specified notions of size, and ensures that the extracted recurrences correctly predict evaluation cost. To extract a recurrence from a program, we first make costs explicit by applying a monadic translation from the source language to a complexity language, and then abstract datatype values as sizes. Size abstraction can be done semantically, working in models of the complexity language, or syntactically, by adding rules to a preorder judgement. We give several different models of the complexity language, which support different notions of size. Additionally, we prove by a logical relations argument that recurrences extracted by this process are upper bounds for evaluation cost; the proof is entirely syntactic and therefore applies to all of the models we consider.

Recurrence extraction for functional programs through call-by-push-value

“…Following that, we define a map that extracts a recurrence from each term of CBV PCF, which we also denote by ∥−∥ (see Danner et al [2015] for an explanation of most of the cases). Arithmetic operations M op N are chosen to have zero cost in addition to the cost of evaluating their inputs (since for big-O bounds we mainly count recursive calls), but for their potential we distinguish some cases, guided by two constraints needed by our proof of the bounding theorem: the potential must be an upper bound on the value of the operation, and must be monotone in the potentials of the operands wherever mathematically possible Addition/multiplication are monotone in both M and N , so we use addition/multiplication to combine the potentials of M and N , which will be an upper bound on M {+, * }N .…”

“…(1) We factor the syntactic phase into a cost-preserving embedding of the source language in an intermediate language, followed by a recurrence extraction for the intermediate language in the style of Danner et al [2015]. The bounding theorem is proved once for the intermediate language, so for each source language we need only prove that the cost-preserving embeddings are correct.…”

TiML: a functional language for practical complexity analysis with invariants

“…Ghica and Smith [2011] analyzed the complexity of a concurrent Algol-like language that is compiled to hardware circuits directly, by using indices in types to control contraction in parallel compositions. Danner et al [2015] proposed a procedure to automatically transform a program into a certain form and read off the complexity-recurrence equations from there, but they did not address recurrence solving. Benzinger [2001Benzinger [ , 2004 used the Mathematica computer-algebra system to solve some forms of recurrences.…”

Monadic refinements for relational cost analysis

“…A lot of work for cost analysis relies on size types [Avanzini and Dal Lago 2017;Crary and Weirich 2000;Danner et al 2015;Serrano et al 2013]. Size types only specify the sizes of data structures, so they must be combined with other techniques to reason about costs.…”

“…Size types only specify the sizes of data structures, so they must be combined with other techniques to reason about costs. One approach is to use a cost-passing encoding to make cost an explicit output and to reason about its size [Avanzini and Dal Lago 2017;Danner et al 2015]. Crary and Weirich [2000] take a different approach that resembles a type-and-effect system.…”