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We present a novel method for working with the physicist's method of amortized resource analysis, which we call the quantum physicist's method. These principles allow for more precise analyses of resources that are not monotonically consumed, like stack. This method takes its name from its two major features, worldviews and resource tunneling, which behave analogously to quantum superposition and quantum tunneling. We use the quantum physicist's method to extend the Automatic Amortized Resource Analysis (AARA) type system, enabling the derivation of resource bounds based on tree depth. In doing so, we also introduce remainder contexts, which aid bookkeeping in linear type systems. We then evaluate this new type system's performance by bounding stack use of functions in the Set module of OCaml's standard library. Compared to state-of-the-art implementations of AARA, our new system derives tighter bounds with only moderate overhead. 

This paper presents λ-amor, a new type-theoretic framework for amortized cost analysis of higher-order functional programs and shows that existing type systems for cost analysis can be embedded in it. λ-amor introduces a new modal type for representing potentials – costs that have been accounted for, but not yet incurred, which are central to amortized analysis. Additionally, λ-amor relies on standard type-theoretic concepts like affineness, refinement types and an indexed cost monad. λ-amor is proved sound using a rather simple logical relation. We embed two existing type systems for cost analysis in λ-amor showing that, despite its simplicity, λ-amor can simulate cost analysis for different evaluation strategies (call-by-name and call-by-value), in different styles (effect-based and coeffect-based), and with or without amortization. One of the embeddings also implies that λ-amor is relatively complete for all terminating PCF programs. 

This article presents liquid resource types, a technique for automatically verifying the resource consumption of functional programs. Existing resource analysis techniques trade automation for flexibility -- automated techniques are restricted to relatively constrained families of resource bounds, while more expressive proof techniques admitting value-dependent bounds rely on handwritten proofs. Liquid resource types combine the best of these approaches, using logical refinements to automatically prove precise bounds on a program's resource consumption. The type system augments refinement types with potential annotations to conduct an amortized resource analysis. Importantly, users can annotate data structure declarations to indicate how potential is allocated within the type, allowing the system to express bounds with polynomials and exponentials, as well as more precise expressions depending on program values. We prove the soundness of the type system, provide a library of flexible and reusable data structures for conducting resource analysis, and use our prototype implementation to automatically verify resource bounds that previously required a manual proof. 

Automatic amortized resource analysis (AARA) is a type-based technique for inferring concrete (non-asymptotic) bounds on a program's resource usage. Existing work on AARA has focused on bounds that are polynomial in the sizes of the inputs. This paper presents and extension of AARA to exponential bounds that preserves the benefits of the technique, such as compositionality and efficient type inference based on linear constraint solving. A key idea is the use of the Stirling numbers of the second kind as the basis of potential functions, which play the same role as the binomial coefficients in polynomial AARA. To formalize the similarities with the existing analyses, the paper presents a general methodology for AARA that is instantiated to the polynomial version, the exponential version, and a combined system with potential functions that are formed by products of Stirling numbers and binomial coefficients. The soundness of exponential AARA is proved with respect to an operational cost semantics and the analysis of representative example programs demonstrates the effectiveness of the new analysis. 

This article presents resource-guided synthesis, a technique for synthesizing recursive programs that satisfy both a functional specification and a symbolic resource bound. The technique is type-directed and rests upon a novel type system that combines polymorphic refinement types with potential annotations of automatic amortized resource analysis. The type system enables efficient constraint-based type checking and can express precise refinement-based resource bounds. The proof of type soundness shows that synthesized programs are correct by construction. By tightly integrating program exploration and type checking, the synthesizer can leverage the user-provided resource bound to guide the search, eagerly rejecting incomplete programs that consume too many resources. An implementation in the resource-guided synthesizer ReSyn is used to evaluate the technique on a range of recursive data structure manipulations. The experiments show that ReSyn synthesizes programs that are asymptotically more efficient than those generated by a resource-agnostic synthesizer. Moreover, synthesis with ReSyn is faster than a naive combination of synthesis and resource analysis. ReSyn is also able to generate implementations that have a constant resource consumption for fixed input sizes, which can be used to mitigate side-channel attacks. 

This article introduces a novel system for deriving upper bounds on the heap-space requirements of functional programs with garbage collection. The space cost model is based on a perfect garbage collector that immediately deallocates memory cells when they become unreachable. Heap-space bounds are derived using type-based automatic amortized resource analysis (AARA), a template-based technique that efficiently reduces bound inference to linear programming. The first technical contribution of the work is a new operational cost semantics that models a perfect garbage collector. The second technical contribution is an extension of AARA to take into account automatic deallocation. A key observation is that deallocation of a perfect collector can be modeled with destructive pattern matching if data structures are used in a linear way. However, the analysis uses destructive pattern matching to accurately model deallocation even if data is shared. The soundness of the extended AARA with respect to the new cost semantics is proven in two parts via an intermediate linear cost semantics. The analysis and the cost semantics have been implemented as an extension to Resource Aware ML (RaML). An experimental evaluation shows that the system is able to derive tight symbolic heap-space bounds for common algorithms. Often the bounds are asymptotic improvements over bounds that RaML derives without taking into account garbage collection.