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...n the rules might be negative. A notorious dissatisfying feature of classical big-step semantics is that it does not provide evaluation judgments for non-terminating evaluations. In a companion paper =-=[17]-=- we describe a big-step operational semantics for partial evaluations that agrees with the usual big-step semantics on terminating computations. It inductively defines statements of the form V, H ⊢ e ...

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...s to a type derivation in which the variables in the type annotations are instantiated according to the solution. In practice, the type inference is slightly more complex. Most importantly, we have to deal with resource-polymorphic recursion in many examples. This Type-Based Amortized Resource Analysis with Integers and Arrays 15 means that we need a type annotation in the recursive call that differs from the annotation in the argument and result types of the function. To infer such types we successively infer type annotations of higher and higher degree. Details can be found in previous work [17]. Moreover, we have to use algorithmic versions of the type rules in the inference in which the non-syntax-directed rules are integrated into the syntax-directed ones [13]. Finally, we use several optimizations to reduce the number of generated constraints. An concrete example of a type derivation can be found in previous work [13]. 6 Experimental Evaluation We have implemented our analysis system in Resource Aware ML (RAML) [12, 14] and tested the new analysis on multiple classic examples algorithms. In this section we describe the results of our experiments with the evaluation-step metric th...

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...thermore the introduced small-step semantics (and our main result) directly establish that any well-typed TRS is terminating, thus circumventing the notion of partial big-step semantics introduced in =-=[12]-=-. Our main result can be condensed into the following observations. The physicist’s method of amortised analysis conceptually amounts to the interpretation method if we allow for the following changes...

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... LP solver such as CLP. A solution of the linear program corresponds to a type derivation in which the variables in the type annotations are instantiated according to the solution. In practice, the type inference is slightly more complex. Most importantly, we have to deal with resource-polymorphic recursion in many examples. This means that we need a type annotation in the recursive call that differs from the annotation in the argument and result types of the function. To infer such types we successively infer type annotations of higher and higher degree. Details can be found in previous work [19]. Moreover, we have to use algorithmic versions of the type rules in the inference in which the non-syntax-directed rules are integrated into the syntax-directed ones [22]. Finally, we use several optimizations to reduce the number of generated constraints. An concrete example of a type derivation can be found in previous work [22]. 6. Nested Data Parallelism The techniques that we describe in this work for a minimal function language scale to more advanced parallel languages such as Blelloch’s NESL [6]. To describe the novel type analysis in this paper, we use a binary binding construct to in...