Pseudo-Deterministic Proofs

We introduce pseudo-deterministic interactive proofs (psdIP): interactive proof systems for search problems where the verifier is guaranteed with high probability to output the same output on different executions. As in the case with classical interactive proofs, the verifier is a probabilistic polynomial time algorithm interacting with an untrusted powerful prover. We view pseudo-deterministic interactive proofs as an extension of the study of pseudo-deterministic randomized polynomial time algorithms: the goal of the latter is to find canonical solutions to search problems whereas the goal of the former is to prove that a solution to a search problem is canonical to a probabilistic polynomial time verifier. Alternatively, one may think of the powerful prover as aiding the probabilistic polynomial time verifier to find canonical solutions to search problems, with high probability over the randomness of the verifier. The challenge is that pseudo-determinism should hold not only with respect to the randomness, but also with respect to the prover: a malicious prover should not be able to cause the verifier to output a solution other than the unique canonical one. The IP=PSPACE characterization implies that psdIP = IP. The challenge is to find constant round pseudo-deterministic interactive proofs for hard search problems. We show a constant round pseudo-deterministic interactive proof for the graph isomorphism problem: on any input pair of isomorphic graphs (G_0,G_1), there exist a unique isomorphism phi from G_0 to G_1 (although many isomorphism many exist) which will be output by the verifier with high probability, regardless of any dishonest prover strategy. In contrast, we show that it is unlikely that psdIP proofs with constant rounds exist for NP-complete problems by showing that if any NP-complete problem has a constant round psdIP protocol, then the polynomial hierarchy collapses

A survey of induction algorithms for machine learning

Central to all systems for machine learning from examples is an induction algorithm. The purpose of the algorithm is to generalize from a finite set of training examples a description consistent with the examples seen, and, hopefully, with the potentially infinite set of examples not seen. This paper surveys four machine learning induction algorithms. The knowledge representation schemes and a PDL description of algorithm control are emphasized. System characteristics that are peculiar to a domain of application are de-emphasized. Finally, a comparative summary of the learning algorithms is presented

Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract

On Zone-Based Analysis of Duration Probabilistic Automata

We propose an extension of the zone-based algorithmics for analyzing timed automata to handle systems where timing uncertainty is considered as probabilistic rather than set-theoretic. We study duration probabilistic automata (DPA), expressing multiple parallel processes admitting memoryfull continuously-distributed durations. For this model we develop an extension of the zone-based forward reachability algorithm whose successor operator is a density transformer, thus providing a solution to verification and performance evaluation problems concerning acyclic DPA (or the bounded-horizon behavior of cyclic DPA).Comment: In Proceedings INFINITY 2010, arXiv:1010.611