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FunctionalProgramming


#cs.LO

Origin | Interest | Match
arXiv logo
arXiv.orgTo Zip Through the Cost Analysis of Probabilistic ProgramsProbabilistic programming and the formal analysis of probabilistic algorithms are active areas of research, driven by the widespread use of randomness to improve performance. While functional correctness has seen substantial progress, automated reasoning about expected runtime remains comparatively limited. In this work, we address this challenge by introducing a refinement-typed probability monad in Liquid Haskell. Our monad enables automated reasoning about expected values and costs by encoding probabilistic behaviour directly in types. Initially defined for discrete distributions over finite support, it is extended to support infinite distributions via an axiomatic approach. By leveraging Liquid Haskell's SMT-based refinement type checking, our framework provides a high degree of automation. We evaluate our approach through four case studies: meldable heaps, coupon collector, randomised quicksort, and zip trees. The first two demonstrate automation with minimal annotation overhead. The latter two showcase how our monad integrates with interactive proofs, including the first formal verification of the expected runtime of zip trees.
#cslo
FunctionalProgramming


#cs.LO #cs.PL #math.CT #quant-ph

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arXiv.orgProto-Quipper with Reversing and ControlThe quantum programming language Quipper supports circuit operations such as reversing and controlling certain quantum circuits. Additionally, Quipper provides a function called with-computed, which can be used to program circuits of the form g; f; g-dagger. The latter is a common pattern in quantum circuit design. One benefit of using with-computed, as opposed to constructing the circuit g ; f; g-dagger directly from g, f, and g-dagger, is that it facilitates an important optimization. Namely, if the resulting circuit is later controlled, only f needs to be controlled; the circuits g and g-dagger need not even be controllable. In this paper, we formalize a semantics for reversible and controllable circuits, using a dagger symmetric monoidal category R to interpret reversible circuits, and a new notion we call a controllable category N, which encompasses the control and with-computed operations in Quipper. We extend the language Proto-Quipper with reversing, control and the with-computed operation. Since not all circuits are reversible and/or controllable, we use a type system with modalities to track reversibility and controllability. This generalizes the modality of Fu-Kishida-Ross-Selinger 2023. We give an abstract categorical semantics, and show that the type system and operational semantics are sound with respect to this semantics.
#cslo#cspl#mathct
LLMs


#cs.AI #cs.LO

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arXiv.orgWhy Cannot Large Language Models Ever Make True Correct Reasoning?Recently, with the application progress of AIGC tools based on large language models (LLMs), led by ChatGPT, many AI experts and more non-professionals are trumpeting the "understanding ability" and "reasoning ability" of the LLMs. The present author considers that the so-called "understanding ability" and "reasoning ability" of LLMs are just illusions of those people who with vague concepts. In fact, the LLMs can never have the true understanding ability and true reasoning ability. This paper intents to explain that, because the essential limitations of their working principle, the LLMs can never have the ability of true correct reasoning.
#csai#cslo
FunctionalProgramming


#cs.PL #cs.LO

Origin | Interest | Match
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arXiv.orgAbstractions of Sequences, Functions and OperatorsWe present theoretical and practical results on the order theory of lattices of functions, focusing on Galois connections that abstract (sets of) functions - a topic known as higher-order abstract interpretation. We are motivated by the challenge of inferring closed-form bounds on functions which are defined recursively, i.e. as the fixed point of an operator or, equivalently, as the solution to a functional equation. This has multiple applications in program analysis (e.g. cost analysis, loop acceleration, declarative language analysis) and in hybrid systems governed by differential equations. Our main contribution is a new family of constraint-based abstract domains for abstracting numerical functions, B-bound domains, which abstract a function f by a conjunction of bounds from a preselected set of boundary functions. They allow inferring highly non-linear numerical invariants, which classical numerical abstract domains struggle with. We uncover a convexity property in the constraint space that simplifies, and, in some cases, fully automates, transfer function design. We also introduce domain abstraction, a functor that lifts arbitrary mappings in value space to Galois connections in function space. This supports abstraction from symbolic to numerical functions (i.e. size abstraction), and enables dimensionality reduction of equations. We base our constructions of transfer functions on a simple operator language, starting with sequences, and extending to more general functions, including multivariate, piecewise, and non-discrete domains.
#cspl#cslo
FunctionalProgramming


#cs.LO

Origin | Interest | Match
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arXiv.orgGraded Quantitative NarrowingThe recently introduced framework of Graded Quantitative Rewriting is an innovative extension of traditional rewriting systems, in which rules are annotated with degrees drawn from a quantale. This framework provides a robust foundation for equational reasoning that incorporates metric aspects, such as the proximity between terms and the complexity of rewriting-based computations. Quantitative narrowing, introduced in this paper, generalizes quantitative rewriting by replacing matching with unification in reduction steps, enabling the reduction of terms even when they contain variables, through simultaneous instantiation and rewriting. In the standard (non-quantitative) setting, narrowing has been successfully applied in various domains, including functional logic programming, theorem proving, and equational unification. Here, we focus on quantitative narrowing to solve unification problems in quantitative equational theories over Lawverean quantales. We establish its soundness and discuss conditions under which completeness can be ensured. This approach allows us to solve quantitative equations in richer theories than those addressed by previous methods.
#cslo
FunctionalProgramming


#cs.PL #cs.LO #cs.SE

Origin | Interest | Match
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arXiv.orgAn Enumerative Embedding of the Python Type System in ACL2sPython is a high-level interpreted language that has become an industry standard in a wide variety of applications. In this paper, we take a first step towards using ACL2s to reason about Python code by developing an embedding of a subset of the Python type system in ACL2s. The subset of Python types we support includes many of the most commonly used type annotations as well as user-defined types comprised of supported types. We provide ACL2s definitions of these types, as well as defdata enumerators that are customized to provide code coverage and identify errors in Python programs. Using the ACL2s embedding, we can generate instances of types that can then be used as inputs to fuzz Python programs, which allows us to identify bugs in Python code that are not detected by state-of-the-art Python type checkers. We evaluate our work against four open-source repositories, extracting their type information and generating inputs for fuzzing functions with type signatures that are in the supported subset of Python types. Note that we only use the type signatures of functions to generate inputs and treat the bodies of functions as black boxes. We measure code coverage, which ranges from about 68% to more than 80%, and identify code patterns that hinder coverage such as complex branch conditions and external file system dependencies. We conclude with a discussion of the results and recommendations for future work.
#cspl#cslo#csse
FunctionalProgramming


#cs.LO #cs.DC

Origin | Interest | Match
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arXiv.orgApplication Placement with Constraint RelaxationNovel utility computing paradigms rely upon the deployment of multi-service applications to pervasive and highly distributed cloud-edge infrastructure resources. Deciding onto which computational nodes to place services in cloud-edge networks, as per their functional and non-functional constraints, can be formulated as a combinatorial optimisation problem. Most existing solutions in this space are not able to deal with \emph{unsatisfiable} problem instances, nor preferences, i.e. requirements that DevOps may agree to relax to obtain a solution. In this article, we exploit Answer Set Programming optimisation capabilities to tackle this problem. Experimental results in simulated settings show that our approach is effective on lifelike networks and applications.
#cslo#csdc
FunctionalProgramming


#cs.LO #math.CT

Origin | Interest | Match
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arXiv.orgAn Algebraic Extension of Intuitionistic Linear Logic: The $L_!^S$-Calculus and Its Categorical ModelWe introduce the $L_!^S$-calculus, a linear lambda-calculus extended with scalar multiplication and term addition, that acts as a proof language for intuitionistic linear logic (ILL). These algebraic operations enable the direct expression of linearity at the syntactic level, a property not typically available in standard proof-term calculi. Building upon previous work, we develop the $L_!^S$-calculus as an extension of the $L^S$-calculus with the $!$ modality. We prove key meta-theoretical properties--subject reduction, confluence, strong normalisation, and an introduction property--as well as preserve the expressiveness of the original $L^S$-calculus, including the encoding of vectors and matrices, and the correspondence between proof-terms and linear functions. A denotational semantics is provided in the framework of linear categories with biproducts, ensuring a sound and adequate interpretation of the calculus. This work is part of a broader programme aiming to build a measurement-free quantum programming language grounded in linear logic.
#cslo#mathct
FunctionalProgramming


#cs.PL #cs.LO #quant-ph

Origin | Interest | Match
arXiv logo
arXiv.orgQuantum circuits are just a phaseQuantum programs today are written at a low level of abstraction - quantum circuits akin to assembly languages - and even advanced quantum programming languages essentially function as circuit description languages. This state of affairs impedes scalability, clarity, and support for higher-level reasoning. More abstract and expressive quantum programming constructs are needed. To this end, we introduce a novel yet simple quantum programming language for generating unitaries from "just a phase"; we combine a (global) phase operation that captures phase shifts with a quantum analogue of the "if let" construct that captures subspace selection via pattern matching. This minimal language lifts the focus from quantum gates to eigendecomposition, conjugation, and controlled unitaries; common building blocks in quantum algorithm design. We demonstrate several aspects of the expressive power of our language in several ways. Firstly, we establish that our representation is universal by deriving a universal quantum gate set. Secondly, we show that important quantum algorithms can be expressed naturally and concisely, including Grover's search algorithm, Hamiltonian simulation, Quantum Fourier Transform, Quantum Signal Processing, and the Quantum Eigenvalue Transformation. Furthermore, we give clean denotational semantics grounded in categorical quantum mechanics. Finally, we implement a prototype compiler that efficiently translates terms of our language to quantum circuits, and prove that it is sound with respect to these semantics. Collectively, these contributions show that this construct offers a principled and practical step toward more abstract and structured quantum programming.
#cspl#cslo#quantph
FunctionalProgramming


#cs.PL #cs.LO #quant-ph

Origin | Interest | Match
arXiv logo
arXiv.orgQuantum circuits are just a phaseQuantum programs today are written at a low level of abstraction - quantum circuits akin to assembly languages - and even advanced quantum programming languages essentially function as circuit description languages. This state of affairs impedes scalability, clarity, and support for higher-level reasoning. More abstract and expressive quantum programming constructs are needed. To this end, we introduce a novel yet simple quantum programming language for generating unitaries from "just a phase"; we combine a (global) phase operation that captures phase shifts with a quantum analogue of the "if let" construct that captures subspace selection via pattern matching. This minimal language lifts the focus from quantum gates to eigendecomposition, conjugation, and controlled unitaries; common building blocks in quantum algorithm design. We demonstrate several aspects of the expressive power of our language in several ways. Firstly, we establish that our representation is universal by deriving a universal quantum gate set. Secondly, we show that important quantum algorithms can be expressed naturally and concisely, including Grover's search algorithm, Hamiltonian simulation, Quantum Fourier Transform, Quantum Signal Processing, and the Quantum Eigenvalue Transformation. Furthermore, we give clean denotational semantics grounded in categorical quantum mechanics. Finally, we implement a prototype compiler that efficiently translates terms of our language to quantum circuits, and prove that it is sound with respect to these semantics. Collectively, these contributions show that this construct offers a principled and practical step toward more abstract and structured quantum programming.
#cspl#cslo#quantph
FunctionalProgramming


#cs.LO #math.CT

Origin | Interest | Match
arXiv logo
arXiv.orgCoinductive Streams in Monoidal CategoriesWe introduce monoidal streams. Monoidal streams are a generalization of causal stream functions, which can be defined in cartesian monoidal categories, to arbitrary symmetric monoidal categories. In the same way that streams provide semantics to dataflow programming with pure functions, monoidal streams provide semantics to dataflow programming with theories of processes represented by a symmetric monoidal category. Monoidal streams also form a feedback monoidal category. In the same way that we can use a coinductive stream calculus to reason about signal flow graphs, we can use coinductive string diagrams to reason about feedback monoidal categories. As an example, we study syntax for a stochastic dataflow language, with semantics in stochastic monoidal streams.
#cslo#mathct
LLMs


#cs.AI #cs.LG #cs.LO

Origin | Interest | Match
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arXiv.orgNeurosymbolic Feature Extraction for Identifying Forced Labor in Supply ChainsSupply chain networks are complex systems that are challenging to analyze; this problem is exacerbated when there are illicit activities involved in the supply chain, such as counterfeit parts, forced labor, or human trafficking. While machine learning (ML) can find patterns in complex systems like supply chains, traditional ML techniques require large training data sets. However, illicit supply chains are characterized by very sparse data, and the data that is available is often (purposely) corrupted or unreliable in order to hide the nature of the activities. We need to be able to automatically detect new patterns that correlate with such illegal activity over complex, even temporal data, without requiring large training data sets. We explore neurosymbolic methods for identifying instances of illicit activity in supply chains and compare the effectiveness of manual and automated feature extraction from news articles accurately describing illicit activities uncovered by authorities. We propose a question tree approach for querying a large language model (LLM) to identify and quantify the relevance of articles. This enables a systematic evaluation of the differences between human and machine classification of news articles related to forced labor in supply chains.
#csai#cslg#cslo
LLMs


#cs.SE #cs.AI #cs.LO

Origin | Interest | Match
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arXiv.orgFuzzFeed: An Automatic Approach to Weakest Precondition Generation using LLMs and FuzzingThe weakest precondition (WP) of a program describes the largest set of initial states from which all terminating executions of the program satisfy a given postcondition. The generation of WPs is an important task with practical applications in areas ranging from verification to run-time error checking. This paper proposes the combination of Large Language Models (LLMs) and fuzz testing for generating WPs. In pursuit of this goal, we introduce Fuzzing Guidance (FG); FG acts as a means of directing LLMs towards correct WPs using program execution feedback. FG utilises fuzz testing for approximately checking the validity and weakness of candidate WPs, this information is then fed back to the LLM as a means of context refinement. We demonstrate the effectiveness of our approach on a comprehensive benchmark set of deterministic array programs in Java. Our experiments indicate that LLMs are capable of producing viable candidate WPs, and that this ability can be practically enhanced through FG.
#csse#csai#cslo
SearchEngine


#cs.LO #cs.NA #math.NA

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arXiv.orgTime-Sensitive Importance SplittingState-of-the-art methods for rare event simulation of non-Markovian models face practical or theoretical limits if observing the event of interest requires prior knowledge or information on the timed behavior of the system. In this paper, we attack both limits by extending importance splitting with a time-sensitive importance function. To this end, we perform backwards reachability search from the target states, considering information about the lower and upper bounds of the active timers in order to steer the generation of paths towards the rare event. We have developed a prototype implementation of the approach for input/output stochastic automata within the Modest Toolset. Preliminary experiments show the potential of the approach in estimating rare event probabilities for an example from reliability engineering.
#cslo#csna#mathna
FunctionalProgramming


#cs.LG #cs.LO #cs.PL

Origin | Interest | Match
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arXiv.orgFloating-Point Neural Networks Are Provably Robust Universal ApproximatorsThe classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous function $f$ with arbitrary accuracy. A recent result shows that neural networks also enjoy a more general interval universal approximation (IUA) theorem, in the sense that the abstract interpretation semantics of the network using the interval domain can approximate the direct image map of $f$ (i.e., the result of applying $f$ to a set of inputs) with arbitrary accuracy. These theorems, however, rest on the unrealistic assumption that the neural network computes over infinitely precise real numbers, whereas their software implementations in practice compute over finite-precision floating-point numbers. An open question is whether the IUA theorem still holds in the floating-point setting. This paper introduces the first IUA theorem for floating-point neural networks that proves their remarkable ability to perfectly capture the direct image map of any rounded target function $f$, showing no limits exist on their expressiveness. Our IUA theorem in the floating-point setting exhibits material differences from the real-valued setting, which reflects the fundamental distinctions between these two computational models. This theorem also implies surprising corollaries, which include (i) the existence of provably robust floating-point neural networks; and (ii) the computational completeness of the class of straight-line programs that use only floating-point additions and multiplications for the class of all floating-point programs that halt.
#cslg#cslo#cspl
FunctionalProgramming


#cs.LO

Origin | Interest | Match
arXiv.orgForward and Backward Simulations for Partially Observable ProbabilityData refinement is the standard extension of a refinement relation from programs to datatypes (i.e. a behavioural subtyping relation). Forward/backward simulations provide a tractable method for establishing data refinement, and have been thoroughly studied for nondeterministic programs. However, for standard models of mixed probability and nondeterminism, ordinary assignment statements may not commute with (variable-disjoint) program fragments. This (1) invalidates a key assumption underlying the soundness of simulations, and (2) prevents modelling probabilistic datatypes with encapsulated state. We introduce a weakest precondition semantics for Kuifje$_\sqcap$, a language for partially observable Markov decision processes, using so-called loss (function) transformers. We prove soundness of forward/backward simulations in this richer setting, modulo healthiness conditions with a remarkable duality: forward simulations cannot leak information, and backward simulations cannot exploit leaked information.
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