433 research outputs found
Synthesis of Strategies Using the Hoare Logic of Angelic and Demonic Nondeterminism
We study a propositional variant of Hoare logic that can be used for
reasoning about programs that exhibit both angelic and demonic nondeterminism.
We work in an uninterpreted setting, where the meaning of the atomic actions is
specified axiomatically using hypotheses of a certain form. Our logical
formalism is entirely compositional and it subsumes the non-compositional
formalism of safety games on finite graphs. We present sound and complete
Hoare-style calculi that are useful for establishing partial-correctness
assertions, as well as for synthesizing implementations. The computational
complexity of the Hoare theory of dual nondeterminism is investigated using
operational models, and it is shown that the theory is complete for exponential
time
Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs
In this paper, we consider termination of probabilistic programs with
real-valued variables. The questions concerned are:
1. qualitative ones that ask (i) whether the program terminates with
probability 1 (almost-sure termination) and (ii) whether the expected
termination time is finite (finite termination); 2. quantitative ones that ask
(i) to approximate the expected termination time (expectation problem) and (ii)
to compute a bound B such that the probability to terminate after B steps
decreases exponentially (concentration problem).
To solve these questions, we utilize the notion of ranking supermartingales
which is a powerful approach for proving termination of probabilistic programs.
In detail, we focus on algorithmic synthesis of linear ranking-supermartingales
over affine probabilistic programs (APP's) with both angelic and demonic
non-determinism. An important subclass of APP's is LRAPP which is defined as
the class of all APP's over which a linear ranking-supermartingale exists.
Our main contributions are as follows. Firstly, we show that the membership
problem of LRAPP (i) can be decided in polynomial time for APP's with at most
demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with
angelic non-determinism; moreover, the NP-hardness result holds already for
APP's without probability and demonic non-determinism. Secondly, we show that
the concentration problem over LRAPP can be solved in the same complexity as
for the membership problem of LRAPP. Finally, we show that the expectation
problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's
without probability and non-determinism (i.e., deterministic programs). Our
experimental results demonstrate the effectiveness of our approach to answer
the qualitative and quantitative questions over APP's with at most demonic
non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201
Focusing in Asynchronous Games
Game semantics provides an interactive point of view on proofs, which enables
one to describe precisely their dynamical behavior during cut elimination, by
considering formulas as games on which proofs induce strategies. We are
specifically interested here in relating two such semantics of linear logic, of
very different flavor, which both take in account concurrent features of the
proofs: asynchronous games and concurrent games. Interestingly, we show that
associating a concurrent strategy to an asynchronous strategy can be seen as a
semantical counterpart of the focusing property of linear logic
Constructive Game Logic
Game Logic is an excellent setting to study proofs-about-programs via the
interpretation of those proofs as programs, because constructive proofs for
games correspond to effective winning strategies to follow in response to the
opponent's actions. We thus develop Constructive Game Logic which extends
Parikh's Game Logic (GL) with constructivity and with first-order programs a la
Pratt's first-order dynamic logic (DL). Our major contributions include:
1) a novel realizability semantics capturing the adversarial dynamics of
games, 2) a natural deduction calculus and operational semantics describing the
computational meaning of strategies via proof-terms, and 3) theoretical results
including soundness of the proof calculus w.r.t. realizability semantics,
progress and preservation of the operational semantics of proofs, and Existence
Properties on support of the extraction of computational artifacts from game
proofs.
Together, these results provide the most general account of a Curry-Howard
interpretation for any program logic to date, and the first at all for Game
Logic.Comment: 74 pages, extended preprint for ESO
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