Abstract A standard informal method for analyzing the asymptotic complexity of a program is to extract a recurrence that describes its cost in terms of the size of its input and then to compute a closed-form upper bound on that recurrence. We give a formal account of that method for functional programs in a higher order language with $\mathtt{let}$ -polymorphism. The method consists of two phases. In the first phase, a monadic translation is performed to extract a cost-annotated version of the original program. In the second phase, the extracted program is interpreted in a model. The key feature of this second phase is that different models describe different notions of size. This plays out in several ways. For example, when analyzing functions that take arguments of inductive types, different notions of size may be appropriate depending on the analysis. When analyzing polymorphic functions, our approach shows that one can formally describe the notion of size of an argument in terms of the data that is common to the notions of size for each type instance of the domain type. We give several examples of different models that formally justify various informal cost analyses to show the applicability of our approach.

A typical way of analyzing the time complexity of functional programs is to extract a recurrence expressing the running time of the program in terms of the size of its input, and then to solve the recurrence to obtain a big-O bound. For recurrence extraction to be compositional, it is also necessary to extract recurrences for the size of outputs of helper functions. Previous work has developed techniques for using logical relations to state a formal correctness theorem for a general recurrence extraction translation: a program is bounded by a recurrence when the operational cost is bounded by the extracted cost, and the output value is bounded, according to a value bounding relation defined by induction on types, by the extracted size. This previous work supports higher-order functions by viewing recurrences as programs in a lambda-calculus, or as mathematical entities in a denotational semantics thereof. In this paper, we extend these techniques to support amortized analysis, where costs are rearranged from one portion of a program to another to achieve more precise bounds. We give an intermediate language in which programs can be annotated according to the banker's method of amortized analysis; this language has an affine type system to ensure credits are not spent more than once. We give a recurrence extraction translation of this language into a recurrence language, a simply-typed lambda-calculus with a cost type, and state and prove a bounding logical relation expressing the correctness of this translation. The recurrence language has a denotational semantics in preorders, and we use this semantics to solve recurrences, e.g analyzing binary counters and splay trees.

The main way of analysing the complexity of a program is that of extracting and solving a recurrence that expresses its running time in terms of the size of its input. We develop a method that automatically extracts such recurrences from the syntax of higher-order recursive functional programs. The resulting recurrences, which are programs in a call-by-name language with recursion, explicitly compute the running time in terms of the size of the input. In order to achieve this in a uniform way that covers both call-by-name and call-by-value evaluation strategies, we use Call-by-Push-Value (CBPV) as an intermediate language. Finally, we use domain theory to develop a denotational cost semantics for the resulting recurrences.

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.

Tor is one of the more popular systems for anonymizing near-real-time communications on the Internet. Borisov et al. [2007] proposed a denial-of-service-based attack on Tor (and related systems) that significantly increases the probability of compromising the anonymity provided. In this article, we analyze the effectiveness of the attack using both an analytic model and simulation. We also describe two algorithms for detecting such attacks, one deterministic and proved correct, the other probabilistic and verified in simulation.

This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR, has its first-order programs characterize the polynomial-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook's and Leivant's implicit characterizations of polynomial-time. The time-restricting part is an affine version of Barber and Plotkin's DILL. Two semantics are constructed for ATR. The first is a pruning of the naive denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR's time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine.

Abstract. This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR, has its first-order programs characterize the polynomial-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomial-time. The time-restricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the naïve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine. 1.