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Register a new userFull abstraction for probabilistic PCF
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Added by
Thomas Ehrhard
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Authors
Thomas Ehrhard
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We present a probabilistic version of PCF, a well-known simply typed universal functional language. The type hierarchy is based on a single ground type of natural numbers. Even if the language is globally call-by-name, we allow a call-by-value evaluation for ground type arguments in order to provide the language with a suitable algorithmic expressiveness. We describe a denotational semantics based on probabilistic coherence spaces, a model of classical Linear Logic developed in previous works. We prove an adequacy and an equational full abstraction theorem showing that equality in the model coincides with a natural notion of observational equivalence.
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In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent presheaf semantics and as a concurrent game semantics. It is here proved that a behavioural equivalence induced by this semantics on CCS processes is fully abstract for fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the rule of the game. From any playground, two languages, equipped with labelled transition systems, are derived, as well as a strong, functional bisimulation between them.
In previous work with Pous, we defined a semantics for CCS which may both be viewed as an innocent form of presheaf semantics and as a concurrent form of game semantics. We define in this setting an analogue of fair testing equivalence, which we prove fully abstract w.r.t. standard fair testing equivalence. The proof relies on a new algebraic notion called playground, which represents the `rule of the game. From any playground, we derive two languages equipped with labelled transition systems, as well as a strong, functional bisimulation between them.
Partially Observable Markov Decision Process (POMDP) is widely used to model probabilistic behavior for complex systems. Compared with MDPs, POMDP models a system more accurate but solving a POMDP generally takes exponential time in the size of its state space. This makes the formal verification and synthesis problems much more challenging for POMDPs, especially when multiple system components are involved. As a promising technique to reduce the verification complexity, the abstraction method tries to find an abstract system with a smaller state space but preserves enough properties for the verification purpose. While abstraction based verification has been explored extensively for MDPs, in this paper, we present the first result of POMDP abstraction and its refinement techniques. The main idea follows the counterexample-guided abstraction refinement (CEGAR) framework. Starting with a coarse guess for the POMDP abstraction, we iteratively use counterexamples from formal verification to refine the abstraction until the abstract system can be used to infer the verification result for the original POMDP. Our main contributions have two folds: 1) we propose a novel abstract system model for POMDP and a new simulation relation to capture the partial observability then prove the preservation on a fragment of Probabilistic Computation Tree Logic (PCTL); 2) to find a proper abstract system that can prove or disprove the satisfaction relation on the concrete POMDP, we develop a novel refinement algorithm. Our work leads to a sound and complete CEGAR framework for POMDP.
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Relational verification is a technique that aims at proving properties that relate two different program fragments, or two different program runs. It has been shown that constrained Horn clauses (CHCs) can effectively be used for relational verification by applying a CHC transformation, called predicate pairing, which allows the CHC solver to infer relations among arguments of different predicates. In this paper we study how the effects of the predicate pairing transformation can be enhanced by using various abstract domains based on linear arithmetic (i.e., the domain of convex polyhedra and some of its subdomains) during the transformation. After presenting an algorithm for predicate pairing with abstraction, we report on the experiments we have performed on over a hundred relational verification problems by using various abstract domains. The experiments have been performed by using the VeriMAP transformation and verification system, together with the Parma Polyhedra Library (PPL) and the Z3 solver for CHCs.
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