968 research outputs found
Formal verification of higher-order probabilistic programs
Probabilistic programming provides a convenient lingua franca for writing
succinct and rigorous descriptions of probabilistic models and inference tasks.
Several probabilistic programming languages, including Anglican, Church or
Hakaru, derive their expressiveness from a powerful combination of continuous
distributions, conditioning, and higher-order functions. Although very
important for practical applications, these combined features raise fundamental
challenges for program semantics and verification. Several recent works offer
promising answers to these challenges, but their primary focus is on semantical
issues.
In this paper, we take a step further and we develop a set of program logics,
named PPV, for proving properties of programs written in an expressive
probabilistic higher-order language with continuous distributions and operators
for conditioning distributions by real-valued functions. Pleasingly, our
program logics retain the comfortable reasoning style of informal proofs thanks
to carefully selected axiomatizations of key results from probability theory.
The versatility of our logics is illustrated through the formal verification of
several intricate examples from statistics, probabilistic inference, and
machine learning. We further show the expressiveness of our logics by giving
sound embeddings of existing logics. In particular, we do this in a parametric
way by showing how the semantics idea of (unary and relational) TT-lifting can
be internalized in our logics. The soundness of PPV follows by interpreting
programs and assertions in quasi-Borel spaces (QBS), a recently proposed
variant of Borel spaces with a good structure for interpreting higher order
probabilistic programs
On Spatial Conjunction as Second-Order Logic
Spatial conjunction is a powerful construct for reasoning about dynamically
allocated data structures, as well as concurrent, distributed and mobile
computation. While researchers have identified many uses of spatial
conjunction, its precise expressive power compared to traditional logical
constructs was not previously known. In this paper we establish the expressive
power of spatial conjunction. We construct an embedding from first-order logic
with spatial conjunction into second-order logic, and more surprisingly, an
embedding from full second order logic into first-order logic with spatial
conjunction. These embeddings show that the satisfiability of formulas in
first-order logic with spatial conjunction is equivalent to the satisfiability
of formulas in second-order logic. These results explain the great expressive
power of spatial conjunction and can be used to show that adding unrestricted
spatial conjunction to a decidable logic leads to an undecidable logic. As one
example, we show that adding unrestricted spatial conjunction to two-variable
logic leads to undecidability. On the side of decidability, the embedding into
second-order logic immediately implies the decidability of first-order logic
with a form of spatial conjunction over trees. The embedding into spatial
conjunction also has useful consequences: because a restricted form of spatial
conjunction in two-variable logic preserves decidability, we obtain that a
correspondingly restricted form of second-order quantification in two-variable
logic is decidable. The resulting language generalizes the first-order theory
of boolean algebra over sets and is useful in reasoning about the contents of
data structures in object-oriented languages.Comment: 16 page
A Logic of Reachable Patterns in Linked Data-Structures
We define a new decidable logic for expressing and checking invariants of
programs that manipulate dynamically-allocated objects via pointers and
destructive pointer updates. The main feature of this logic is the ability to
limit the neighborhood of a node that is reachable via a regular expression
from a designated node. The logic is closed under boolean operations
(entailment, negation) and has a finite model property. The key technical
result is the proof of decidability. We show how to express precondition,
postconditions, and loop invariants for some interesting programs. It is also
possible to express properties such as disjointness of data-structures, and
low-level heap mutations. Moreover, our logic can express properties of
arbitrary data-structures and of an arbitrary number of pointer fields. The
latter provides a way to naturally specify postconditions that relate the
fields on entry to a procedure to the fields on exit. Therefore, it is possible
to use the logic to automatically prove partial correctness of programs
performing low-level heap mutations
On Role Logic
We present role logic, a notation for describing properties of relational
structures in shape analysis, databases, and knowledge bases. We construct role
logic using the ideas of de Bruijn's notation for lambda calculus, an encoding
of first-order logic in lambda calculus, and a simple rule for implicit
arguments of unary and binary predicates. The unrestricted version of role
logic has the expressive power of first-order logic with transitive closure.
Using a syntactic restriction on role logic formulas, we identify a natural
fragment RL^2 of role logic. We show that the RL^2 fragment has the same
expressive power as two-variable logic with counting C^2 and is therefore
decidable. We present a translation of an imperative language into the
decidable fragment RL^2, which allows compositional verification of programs
that manipulate relational structures. In addition, we show how RL^2 encodes
boolean shape analysis constraints and an expressive description logic.Comment: 20 pages. Our later SAS 2004 result builds on this wor
Heap Abstractions for Static Analysis
Heap data is potentially unbounded and seemingly arbitrary. As a consequence,
unlike stack and static memory, heap memory cannot be abstracted directly in
terms of a fixed set of source variable names appearing in the program being
analysed. This makes it an interesting topic of study and there is an abundance
of literature employing heap abstractions. Although most studies have addressed
similar concerns, their formulations and formalisms often seem dissimilar and
some times even unrelated. Thus, the insights gained in one description of heap
abstraction may not directly carry over to some other description. This survey
is a result of our quest for a unifying theme in the existing descriptions of
heap abstractions. In particular, our interest lies in the abstractions and not
in the algorithms that construct them.
In our search of a unified theme, we view a heap abstraction as consisting of
two features: a heap model to represent the heap memory and a summarization
technique for bounding the heap representation. We classify the models as
storeless, store based, and hybrid. We describe various summarization
techniques based on k-limiting, allocation sites, patterns, variables, other
generic instrumentation predicates, and higher-order logics. This approach
allows us to compare the insights of a large number of seemingly dissimilar
heap abstractions and also paves way for creating new abstractions by
mix-and-match of models and summarization techniques.Comment: 49 pages, 20 figure
Healthiness from Duality
Healthiness is a good old question in program logics that dates back to
Dijkstra. It asks for an intrinsic characterization of those predicate
transformers which arise as the (backward) interpretation of a certain class of
programs. There are several results known for healthiness conditions: for
deterministic programs, nondeterministic ones, probabilistic ones, etc.
Building upon our previous works on so-called state-and-effect triangles, we
contribute a unified categorical framework for investigating healthiness
conditions. We find the framework to be centered around a dual adjunction
induced by a dualizing object, together with our notion of relative
Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems
interesting in its own right in the context of monads, Lawvere theories and
enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to
LICS 201
Deciding regular grammar logics with converse through first-order logic
We provide a simple translation of the satisfiability problem for regular
grammar logics with converse into GF2, which is the intersection of the guarded
fragment and the 2-variable fragment of first-order logic. This translation is
theoretically interesting because it translates modal logics with certain frame
conditions into first-order logic, without explicitly expressing the frame
conditions.
A consequence of the translation is that the general satisfiability problem
for regular grammar logics with converse is in EXPTIME. This extends a previous
result of the first author for grammar logics without converse. Using the same
method, we show how some other modal logics can be naturally translated into
GF2, including nominal tense logics and intuitionistic logic.
In our view, the results in this paper show that the natural first-order
fragment corresponding to regular grammar logics is simply GF2 without extra
machinery such as fixed point-operators.Comment: 34 page
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