40 research outputs found
Dependency Stochastic Boolean Satisfiability: A Logical Formalism for NEXPTIME Decision Problems with Uncertainty
Stochastic Boolean Satisfiability (SSAT) is a logical formalism to model
decision problems with uncertainty, such as Partially Observable Markov
Decision Process (POMDP) for verification of probabilistic systems. SSAT,
however, is limited by its descriptive power within the PSPACE complexity
class. More complex problems, such as the NEXPTIME-complete Decentralized POMDP
(Dec-POMDP), cannot be succinctly encoded with SSAT. To provide a logical
formalism of such problems, we extend the Dependency Quantified Boolean Formula
(DQBF), a representative problem in the NEXPTIME-complete class, to its
stochastic variant, named Dependency SSAT (DSSAT), and show that DSSAT is also
NEXPTIME-complete. We demonstrate the potential applications of DSSAT to
circuit synthesis of probabilistic and approximate design. Furthermore, to
study the descriptive power of DSSAT, we establish a polynomial-time reduction
from Dec-POMDP to DSSAT. With the theoretical foundations paved in this work,
we hope to encourage the development of DSSAT solvers for potential broad
applications.Comment: 10 pages, 5 figures. A condensed version of this work is published in
the AAAI Conference on Artificial Intelligence (AAAI) 202
Justicia: A Stochastic SAT Approach to Formally Verify Fairness
As a technology ML is oblivious to societal good or bad, and thus, the field
of fair machine learning has stepped up to propose multiple mathematical
definitions, algorithms, and systems to ensure different notions of fairness in
ML applications. Given the multitude of propositions, it has become imperative
to formally verify the fairness metrics satisfied by different algorithms on
different datasets. In this paper, we propose a \textit{stochastic
satisfiability} (SSAT) framework, Justicia, that formally verifies different
fairness measures of supervised learning algorithms with respect to the
underlying data distribution. We instantiate Justicia on multiple
classification and bias mitigation algorithms, and datasets to verify different
fairness metrics, such as disparate impact, statistical parity, and equalized
odds. Justicia is scalable, accurate, and operates on non-Boolean and compound
sensitive attributes unlike existing distribution-based verifiers, such as
FairSquare and VeriFair. Being distribution-based by design, Justicia is more
robust than the verifiers, such as AIF360, that operate on specific test
samples. We also theoretically bound the finite-sample error of the verified
fairness measure.Comment: 24 pages, 7 figures, 5 theorem
Generalized Craig Interpolation for Stochastic Boolean Satisfiability Problems with Applications to Probabilistic State Reachability and Region Stability
The stochastic Boolean satisfiability (SSAT) problem has been introduced by
Papadimitriou in 1985 when adding a probabilistic model of uncertainty to
propositional satisfiability through randomized quantification. SSAT has many
applications, among them probabilistic bounded model checking (PBMC) of
symbolically represented Markov decision processes. This article identifies a
notion of Craig interpolant for the SSAT framework and develops an algorithm
for computing such interpolants based on a resolution calculus for SSAT. As a
potential application area of this novel concept of Craig interpolation, we
address the symbolic analysis of probabilistic systems. We first investigate
the use of interpolation in probabilistic state reachability analysis, turning
the falsification procedure employing PBMC into a verification technique for
probabilistic safety properties. We furthermore propose an interpolation-based
approach to probabilistic region stability, being able to verify that the
probability of stabilizing within some region is sufficiently large
Stochastic Constraint Programming
To model combinatorial decision problems involving uncertainty and
probability, we introduce stochastic constraint programming. Stochastic
constraint programs contain both decision variables (which we can set) and
stochastic variables (which follow a probability distribution). They combine
together the best features of traditional constraint satisfaction, stochastic
integer programming, and stochastic satisfiability. We give a semantics for
stochastic constraint programs, and propose a number of complete algorithms and
approximation procedures. Finally, we discuss a number of extensions of
stochastic constraint programming to relax various assumptions like the
independence between stochastic variables, and compare with other approaches
for decision making under uncertainty.Comment: Proceedings of the 15th Eureopean Conference on Artificial
Intelligenc
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COMPLEXITY&APPROXIMABILITY OF QUANTIFIED&STOCHASTIC CONSTRAINT SATISFACTION PROBLEMS
Let D be an arbitrary (not necessarily finite) nonempty set, let C be a finite set of constant symbols denoting arbitrary elements of D, and let S and T be an arbitrary finite set of finite-arity relations on D. We denote the problem of determining the satisfiability of finite conjunctions of relations in S applied to variables (to variables and symbols in C) by SAT(S) (by SATc(S).) Here, we study simultaneously the complexity of decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. We present simple yet general techniques to characterize simultaneously, the complexity or efficient approximability of a number of versions/variants of the problems SAT(S), Q-SAT(S), S-SAT(S),MAX-Q-SAT(S) etc., for many different such D,C ,S, T. These versions/variants include decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. Our unified approach is based on the following two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic represent ability. Some of the results extend the earlier results in [Pa85,LMP99,CF+93,CF+94O]u r techniques and results reported here also provide significant steps towards obtaining dichotomy theorems, for a number of the problems above, including the problems MAX-&-SAT( S), and MAX-S-SAT(S). The discovery of such dichotomy theorems, for unquantified formulas, has received significant recent attention in the literature [CF+93,CF+94,Cr95,KSW97