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Register a new userSymbolic Computation and Automated Reasoning for Program Analysis
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Laura Kovacs
Publication date
2017
fields
Informatics Engineering
and research's language is
English
Authors
Laura Kovacs
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This talk describes how a combination of symbolic computation techniques with first-order theorem proving can be used for solving some challenges of automating program analysis, in particular for generating and proving properties about the logically complex parts of software. The talk will first present how computer algebra methods, such as Groebner basis computation, quantifier elimination and algebraic recurrence solving, help us in inferring properties of program loops with non-trivial arithmetic. Typical properties inferred by our work are loop invariants and expressions bounding the number of loop iterations. The talk will then describe our work to generate first-order properties of programs with unbounded data structures, such as arrays. For doing so, we use saturation-based first-order theorem proving and extend first-order provers with support for program analysis. Since program analysis requires reasoning in the combination of first-order theories of data structures, the talk also discusses new features in firstorder theorem proving, such as inductive reasoning and built-in boolean sort. These extensions allow us to express program properties directly in first-order logic and hence use further first-order theorem provers to reason about program properties.
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A quantum circuit is a computational unit that transforms an input quantum state to an output one. A natural way to reason about its behavior is to compute explicitly the unitary matrix implemented by it. However, when the number of qubits increases, the matrix dimension grows exponentially and the computation becomes intractable. In this paper, we propose a symbolic approach to reasoning about quantum circuits. It is based on a small set of laws involving some basic manipulations on vectors and matrices. This symbolic reasoning scales better than the explicit one and is well suited to be automated in Coq, as demonstrated with some typical examples.
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Ruttens behavioural differential equations. We introduce a program logic with L{o}b induction for reasoning about the contextual equivalence of programs.
The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction, computational effects, and type abstraction. We contribute a fresh synthetic take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure. Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof-relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs -- simplifying the metatheory of strong sums over the collection of types, for although there can be no relation classifying relations, one easily accommodates a family classifying small families. Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types, iterating our modal account of the phase distinction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.
This volume contains a final and revised selection of papers presented at the Seventh International Workshop on Verification and Program Transformation (VPT 2019), which took place in Genova, Italy, on April 2nd, 2019, affiliated with Programming 2019.
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