We study the problem of automatically analyzing the worst-case resource usage of procedures with several arguments. Existing automatic analyses based on amortization, or sized types bound the resource usage or result size of such a procedure by a sum of unary functions of the sizes of the arguments. In this paper we generalize this to arbitrary multivariate polynomial functions thus allowing bounds of the form mn which had to be grossly overestimated by m 2 +n 2 before. Our framework even encompasses bounds like ∗ i,j≤n m_i m j where the m i are the sizes of the entries of a list of length n . This allows us for the first time to derive useful resource bounds for operations on matrices that are represented as lists of lists and to considerably improve bounds on other super-linear operations on lists such as longest common subsequence and removal of duplicates from lists of lists. Furthermore, resource bounds are now closed under composition which improves accuracy of the analysis of composed programs when some or all of the components exhibit super-linear resource or size behavior. The analysis is based on a novel multivariate amortized resource analysis. We present it in form of a type system for a simple first-order functional language with lists and trees, prove soundness, and describe automatic type inference based on linear programming. We have experimentally validated the automatic analysis on a wide range of examples from functional programming with lists and trees. The obtained bounds were compared with actual resource consumption. All bounds were asymptotically tight, and the constants were close or even identical to the optimal ones.

We study the problem of automatically analyzing the worst-case resource usage of procedures with several arguments. Existing automatic analyses based on amortization, or sized types bound the resource usage or result size of such a procedure by a sum of unary functions of the sizes of the arguments.In this paper we generalize this to arbitrary multivariate polynomial functions thus allowing bounds of the form mn which had to be grossly overestimated by m 2 + n 2 before. Our framework even encompasses bounds like i,j≤n mimj where the mi are the sizes of the entries of a list of length n.This allows us for the first time to derive useful resource bounds for operations on matrices that are represented as lists of lists and to considerably improve bounds on other super-linear operations on lists such as longest common subsequence and removal of duplicates from lists of lists. Furthermore, resource bounds are now closed under composition which improves accuracy of the analysis of composed programs when some or all of the components exhibit super-linear resource or size behavior.The analysis is based on a novel multivariate amortized resource analysis. We present it in form of a type system for a simple firstorder functional language with lists and trees, prove soundness, and describe automatic type inference based on linear programming.We have experimentally validated the automatic analysis on a wide range of examples from functional programming with lists and trees. The obtained bounds were compared with actual resource consumption. All bounds were asymptotically tight, and the constants were close or even identical to the optimal ones.

We study the problem of automatically analyzing the worst-case resource usage of procedures with several arguments. Existing automatic analyses based on amortization or sized types bound the resource usage or result size of such a procedure by a sum of unary functions of the sizes of the arguments.In this article we generalize this to arbitrary multivariate polynomial functions thus allowing bounds of the form mn which had to be grossly overestimated by m 2 + n 2 before. Our framework even encompasses bounds like i, j≤n m i m j where the m i are the sizes of the entries of a list of length n.This allows us for the first time to derive useful resource bounds for operations on matrices that are represented as lists of lists and to considerably improve bounds on other superlinear operations on lists such as longest common subsequence and removal of duplicates from lists of lists. Furthermore, resource bounds are now closed under composition which improves accuracy of the analysis of composed programs when some or all of the components exhibit superlinear resource or size behavior.The analysis is based on a novel multivariate amortized resource analysis. We present it in form of a type system for a simple first-order functional language with lists and trees, prove soundness, and describe automatic type inference based on linear programming.We have experimentally validated the automatic analysis on a wide range of examples from functional programming with lists and trees. The obtained bounds were compared with actual resource consumption. All bounds were asymptotically tight, and the constants were close or even identical to the optimal ones.

The automatic determination of the quantitative resource consumption of programs is a classic research topic which has many applications in software development. Recently, we developed a novel multivariate amortized resource analysis that automatically computes polynomial resource bounds for first-order functional programs. In this tool paper, we describe Resource Aware ML (RAML), a functional programming language that implements our analysis. Other than in earlier articles, we focus on the practical aspects of the implementation. We describe the syntax of RAML, the code transformation prior to the analysis, the web interface, the output of the analysis, and the results of our experiments with the analysis of example programs.

Abstract. Recent work by Knapik, Niwiński and Urzyczyn (in FOS-SACS 2002) has revived interest in the connexions between higher-order grammars and higher-order pushdown automata. Both devices can be viewed as definitions for term trees as well as string languages. In the latter setting we recall the extensive study by Damm (1982), and Damm and Goerdt (1986). There it was shown that a language is accepted by a level-n pushdown automaton if and only if the language is generated by a safe level-n grammar. We show that at level 2 the safety assumption may be removed. It follows that there are no inherently unsafe string languages at level 2.

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Abstract. Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees.A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.

A purely syntactic and untyped variant of Normalisation by Evaluation for the λ-calculus is presented in the framework of a two-level λ-calculus with rewrite rules to model the inverse of the evaluation functional. Among its operational properties there is a standardisation theorem that formally establishes the adequacy of implementation in functional programming languages. An example implementation in Haskell is provided. The relation to the usual type-directed Normalisation by Evaluation is highlighted, using a short analysis of η-expansion that leads to a perspicuous strong normalisation and confluence proof for βη↑-reduction as a byproduct.