Exponential Blow-up Research Articles

Singly exponential translation of alternating weak Büchi automata to unambiguous Büchi automata

We introduce a method for translating an alternating weak Büchi automaton (AWA), which corresponds to a Linear Dynamic Logic (LDL) formula, to an unambiguous Büchi automaton (UBA). Our translations generalize constructions for Linear Temporal Logic (LTL), a less expressive specification language than LDL. In classical constructions, LTL formulas are first translated to alternating very weak Büchi automata (AVAs)—automata that have only singleton strongly connected components (SCCs); these AVAs are then handled by efficient disambiguation procedures. However, general AWAs can have larger SCCs, which complicates disambiguation. Currently, the only available disambiguation procedure has to go through an intermediate construction of nondeterministic Büchi automata (NBAs), which would incur an exponential blow-up of its own. We introduce a translation from general AWAs to UBAs with a singly exponential blow-up, which also immediately provides a singly exponential translation from LDL to UBAs. Interestingly, the complexity of our translation is smaller than the best known disambiguation algorithm for NBAs (broadly (0.53n)n vs. (0.76n)n), while the input of our construction can be exponentially more succinct.

Efficient Normalization of Linear Temporal Logic

In the mid 1980s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of Linear Temporal Logic (LTL) with past operators) is equivalent to a formula of the form \(\bigwedge _{i=1}^n {\mathbf {G}}{\mathbf {F}}\varphi _i \vee {\mathbf {F}}{\mathbf {G}}\psi _i\) , where φ i and ψ i contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata.

Low-Distortion Clustering with Ordinal and Limited Cardinal Information

Motivated by recent work in computational social choice, we extend the metric distortion framework to clustering problems. Given a set of n agents located in an underlying metric space, our goal is to partition them into k clusters, optimizing some social cost objective. The metric space is defined by a distance function d between the agent locations. Information about d is available only implicitly via n rankings, through which each agent ranks all other agents in terms of their distance from her. Still, even though no cardinal information (i.e., the exact distance values) is available, we would like to evaluate clustering algorithms in terms of social cost objectives that are defined using d. This is done using the notion of distortion, which measures how far from optimality a clustering can be, taking into account all underlying metrics that are consistent with the ordinal information available. Unfortunately, the most important clustering objectives (e.g., those used in the well-known k-median and k-center problems) do not admit algorithms with finite distortion. To sidestep this disappointing fact, we follow two alternative approaches: We first explore whether resource augmentation can be beneficial. We consider algorithms that use more than k clusters but compare their social cost to that of the optimal k-clusterings. We show that using exponentially (in terms of k) many clusters, we can get low (constant or logarithmic) distortion for the k-center and k-median objectives. Interestingly, such an exponential blowup is shown to be necessary. More importantly, we explore whether limited cardinal information can be used to obtain better results. Somewhat surprisingly, for k-median and k-center, we show that a number of queries that is polynomial in k and only logarithmic in n (i.e., only sublinear in the number of agents for the most relevant scenarios in practice) is enough to get constant distortion.

Parikh’s Theorem Made Symbolic

Parikh’s Theorem is a fundamental result in automata theory with numerous applications in computer science. These include software verification (e.g. infinite-state verification, string constraints, and theory of arrays), verification of cryptographic protocols (e.g. using Horn clauses modulo equational theories) and database querying (e.g. evaluating path-queries in graph databases), among others. Parikh’s Theorem states that the letter-counting abstraction of a language recognized by finite automata or context-free grammars is definable in Linear Integer Arithmetic (a.k.a. Presburger Arithmetic). In fact, there is a linear-time algorithm computing existential Presburger formulas capturing such abstractions, which enables an efficient analysis via SMT-solvers. Unfortunately, real-world applications typically require large alphabets (e.g. Unicode, containing a ‍million of characters) — which are well-known to be not amenable to explicit treatment of the alphabets — or even worse infinite alphabets. Symbolic automata have proven in the last decade to be an effective algorithmic framework for handling large finite or even infinite alphabets. A symbolic automaton employs an effective boolean algebra, which offers a symbolic representation of character sets (i.e. in terms of predicates) and often lends itself to an exponentially more succinct representation of a language. Instead of letter-counting, Parikh’s Theorem for symbolic automata amounts to counting the number of times different predicates are satisfied by an input sequence. Unfortunately, naively applying Parikh’s Theorem from classical automata theory to symbolic automata yields existential Presburger formulas of exponential size. In this paper, we provide a new construction for Parikh’s Theorem for symbolic automata and grammars, which avoids this exponential blowup: our algorithm computes an existential formula in polynomial-time over (quantifier-free) Presburger and the base theory. In fact, our algorithm extends to the model of parametric symbolic grammars, which are one of the most expressive models of languages over infinite alphabets. We have implemented our algorithm and show it can be used to solve string constraints that are difficult to solve by existing solvers.

Efficient Biclique Counting in Large Bipartite Graphs

A (p,q)-biclique is a complete subgraph (X,Y) that |X|=p, |Y|=q. Counting (p,q)-bicliques in bipartite graphs is an important operator for many bipartite graph analysis applications. However, getting the count of (p,q)-bicliques for large p and q (e.g., p,q ≥ 10) is extremely difficult, because the number of (p,q)-bicliques increases exponentially with respect to p and q. The state-of-the-art algorithm for this problem is based on the (p,q)-biclique enumeration technique which is often costly due to the exponential blowup in the enumeration space of (p,q)-bicliques. To overcome this problem, we first propose a novel exact algorithm, called EPivoter, based on a newly-developed edge-pivoting technique. The striking feature of EPivoter is that it can count (p,q)-bicliques for all pairs of (p,q) using a combinatorial technique, instead of exhaustively enumerating all (p,q)-bicliques. Second, we propose a novel dynamic programming (DP) based h-zigzag sampling technique to provably approximate the count of the (p,q)-bicliques for all pairs of (p,q), where an h-zigzag is an ordered simple path in G with length 2h-1 (h = min{p,q}). We show that our DP-based sampling technique is very efficient. Third, to further improve the efficiency, we also propose a hybrid framework that integrates both the exact EPivoter algorithm and sampling-based algorithms. Extensive experiments on 7 real-world graphs show that our algorithms are several orders of magnitude faster than the state-of-the-art algorithm.

Verified compilation of Quantum oracles

Quantum algorithms often apply classical operations, such as arithmetic or predicate checks, over a quantum superposition of classical data; these so-called oracles are often the largest components of a quantum program. To ease the construction of efficient, correct oracle functions, this paper presents VQO, a high-assurance framework implemented with the Coq proof assistant. The core of VQO is OQASM, the oracle quantum assembly language. OQASM operations move qubits between two different bases via the quantum Fourier transform, thus admitting important optimizations, but without inducing entanglement and the exponential blowup that comes with it. OQASM’s design enabled us to prove correct VQO’s compilers—from a simple imperative language called OQIMP to OQASM, and from OQASM to SQIR, a general-purpose quantum assembly language—and allowed us to efficiently test properties of OQASM programs using the QuickChick property-based testing framework. We have used VQO to implement a variety of arithmetic and geometric operators that are building blocks for important oracles, including those used in Shor’s and Grover’s algorithms. We found that VQO’s QFT-based arithmetic oracles require fewer qubits, sometimes substantially fewer, than those constructed using “classical” gates; VQO’s versions of the latter were nevertheless on par with or better than (in terms of both qubit and gate counts) oracles produced by Quipper, a state-of-the-art but unverified quantum programming platform.

Homomorphisms of Lifted Planning Tasks: The Case for Delete-Free Relaxation Heuristics

Classical planning tasks are modelled in PDDL which is a schematic language based on first-order logic. Most of the current planners turn this lifted representation into a propositional one via a grounding process. However, grounding may cause an exponential blowup. Therefore it is important to investigate methods for searching for plans on the lifted level. To build a lifted state-based planner, it is necessary to invent lifted heuristics. We introduce maps between PDDL tasks preserving plans allowing to transform a PDDL task into a smaller one. We propose a novel method for computing lifted (admissible) delete-free relaxed heuristics via grounding of the smaller task and computing the (admissible) delete-free relaxed heuristics there. This allows us to transfer the knowledge about relaxed heuristics from the grounded level to the lifted level.

No Efficient Disjunction or Conjunction of Switch-Lists

It is shown that disjunction of two switch-lists can blow up the representation size exponentially. Since switch-lists can be negated without any increase in size, this shows that conjunction of switch-lists also leads to an exponential blow-up in general.

Partial Order Games

We introduce a non-cooperative game model in which players’ decision nodes are partially ordered by a dependence relation, which directly captures informational dependencies in the game. In saying that a decision node v is dependent on decision nodes v1,…,vk, we mean that the information available to a strategy making a choice at v is precisely the choices that were made at v1,…,vk. Although partial order games are no more expressive than extensive form games of imperfect information (we show that any partial order game can be reduced to a strategically equivalent extensive form game of imperfect information, though possibly at the cost of an exponential blowup in the size of the game), they provide a more natural and compact representation for many strategic settings of interest. After introducing the game model, we investigate the relationship to extensive form games of imperfect information, the problem of computing Nash equilibria, and conditions that enable backwards induction in this new model.

New Techniques for Pairwise Symmetry Breaking in Multi-Agent Path Finding

We consider two new types of pairwise path symmetries which appear in the context of Multi-Agent Path Finding (MAPF). The first of them, corridor symmetry, arises when two agents attempt to pass through the same narrow passage but in opposite directions. The second, target symmetry, arises when the shortest path of one agent requires the target location of a second agent after the second agent has already arrived. These symmetries can produce an exponential blowup in the space of possible collision resolutions, leading to timeout failure even for state-of-the-art algorithms such as Conflict-Based Search. We propose to break symmetries using new reasoning techniques that: (1) detect each type of situation and, (2) resolve them by introducing specialized constraints. We implement our ideas in the context of Conflict-Based Search where, in a range of experiments, we report up to an order-of-magnitude improvement in runtime performance and, in some cases, more than a doubling in success rate.

Exponential decay and blow-up results for a viscoelastic equation with variable sources

This work deals with a class of nonlinear viscoelastic wave equations with strong damping and variable exponent sources. The main interest of this paper compared to many previous works in the literature is able to use the idea of potential well method to study both the finite time blow-up for solutions starting from the unstable sets and decay estimate for global solutions starting in the potential wells.

Landmark Heuristics for Lifted Planning - Extended Abstract

Planning problems are usually modeled using lifted representations, they specify predicates and action schemas using variables over a finite universe of objects. However, current planning systems like Fast Downward need a grounded (propositional) input model. The process of grounding might result in an exponential blowup of the model size. This limits the application of grounded planning systems in practical applications. Recent work introduced an efficient planning system for lifted heuristic search, but the work on lifted heuristics is still limited. In this extended abstract, we introduce a novel lifted heuristic based on landmarks, which we extract from the lifted problem representation. Preliminary results on a benchmark set specialized to lifted planning show that there are domains where our approach finds enough landmarks to guide the search more effective than the heuristics available.

The descriptional power of queue automata of constant length

We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

Synthesis of covert actuator and sensor attackers

In this work, we shall investigate the problem of covert attacker synthesis in the framework of supervisory control of discrete-event systems. Intuitively, the covertness property says that the attacker cannot reach a situation where its existence has been detected by the supervisor while no damage can be caused. We consider covert attackers that can exercise both actuator attacks (including enablement attacks and disablement attacks) and sensor attacks (restricted to sensor replacement attacks), where the (partial-observation) attackers may or may not eavesdrop the control commands issued by the supervisor. We shall develop an exponential time reduction from the covert attacker synthesis problem to the well studied Ramadge–Wonham supervisor synthesis problem, which generalizes our previous work on a reduction based approach for covert actuator attacker synthesis, for both the damage-reachable goal and the damage-nonblocking goal. We also provide discussions on conditions under which the exponential blowup in state sizes, due to the reduction construction, can be avoided.

Endomorphisms of Lifted Planning Problems

Classical planning tasks are usually modelled in the PDDL which is a schematic language based on first-order logic. Nevertheless, most of the current planners turn this first-order representation into a propositional one via the grounding process. It is well known that the grounding process may cause an exponential blowup. Therefore it is important to detect which grounded atoms are redundant in a sense that they are not necessary for finding a plan and therefore the grounding process does not need to generate them. This is usually done by a relaxed reachability analysis, which can be improved by employing structural symmetries. Symmetries are bijective self-maps preserving the structure of the PDDL task. In this paper, we introduce a new method which is based on self-maps preserving the structure but which need not be bijective. We call these maps PDDL endomorphisms and we show that they can be used for pruning of redundant objects even if they appear in a reachable atom. We formulate the computation of endomorphisms as a constraint satisfaction problem (CSP) that can be solved by an off-the-shelf CSP solver.

Specify and measure, cover and reveal: A unified framework for automated test generation

• Automatic test input generation is a major topic in software engineering. • We define a new coverage specification mechanism, called labels. • We propose innovative testing techniques for labels. • Label coverage provides a rich framework to handle test objectives. Automatic test input generation (ATG) is a major topic in software engineering, analysis and security. In this paper, we bridge the gap between state-of-the-art white-box ATG techniques, especially Dynamic Symbolic Execution, and the diversity of test objectives that they may be used to cover in practice, including many of those defined by common source-code coverage criteria. We define a new coverage specification mechanism, called labels, for specifying test objectives, and prove it to be both expressive and amenable to efficient automation. We present an efficient approach for detecting – revealing – infeasible (i.e. uncoverable) test objectives expressed as labels. We demonstrate that measuring the achieved coverage can be efficiently performed for labels. Finally, we propose an innovative extension of DSE resulting in an efficient support for label coverage , while the existing naive approach induces an exponential blow-up of the search space. Experiments show that our ATG technique yields very significant savings and confirm the interest of infeasible label detection, enabling to lift DSE to label coverage with only a slight overhead. Overall, we show that label coverage provides the basis of a rich framework allowing one to express and handle test objectives from various contexts in an efficient and generic manner. To illustrate this framework, we describe LTest , an all-in-one testing toolset based on labels and used in the industry, which offers automatic program annotation, ATG, coverage measurement and detection of infeasible test objectives.

Strengthening Probabilistic Graphical Models: The Purge-and-Merge Algorithm

Probabilistic graphical models (PGMs) are powerful tools for solving systems of complex relationships over a variety of probability distributions. However, while tree-structured PGMs always result in efficient and exact solutions, inference on graph (or loopy) structured PGMs is not guaranteed to discover the optimal solutions. It is in principle possible to convert loopy PGMs to an equivalent tree structure, but this is usually impractical for interesting problems due to exponential blow-up. To address this, we developed the purge-and-merge algorithm. This algorithm iteratively nudges a malleable graph structure towards a tree structure by selectively merging factors. The merging process is designed to avoid exponential blow-up by way of sparse structures from which redundancy is purged as the algorithm progresses. We set up tasks to test the algorithm on constraint-satisfaction puzzles such as Sudoku, Fill-a-pix, and Kakuro, and it outperformed other PGM-based approaches reported in the literature. While the tasks we set focussed on the binary logic of CSP, we believe the purge-and-merge algorithm could be extended to general PGM inference.

A Constraint-based Language for Multiparty Interactions

Multiparty interactions are common place in today's distributed systems. An agent usually communicates, in a single session, with other agents to accomplish a given task. Take for instance an online transaction including the vendor, the client, the credit card system and the bank. When specifying this kind of system, we probably observe a single transaction including several (binary) communications leading to changes in the state of all the involved agents. Multiway synchronization process calculi, that move from a binary to a multiparty synchronization discipline, have been proposed to formally study the behavior of those systems. However, adopting models such as Bodei, Brodo, and Bruni's Core Network Algebra (CNA), where the number of participants in an interaction is not fixed a priori, leads to an exponential blow-up in the number of states/behaviors that can be observed from the system. In this paper we explore mechanisms to tackle this problem. We extend CNA with constraints that declaratively allow the modeler to restrict the interaction that should actually happen. Our extended process algebra, called CCNA, finds application in balancing the interactions in a concurrent system, leading to a simple, deadlock-free and fair solution for the Dinning Philosopher problem. Our definition of constraints is general enough and it offers the possibility of accumulating costs in a multiparty negotiation. Hence, only computations respecting the thresholds imposed by the modeler are observed. We use this machinery to neatly model a Service Level Agreement protocol. We develop the theory of CCNA including its operational semantics and a behavioral equivalence that we prove to be a congruence. We also propose a prototypical implementation that allows us to verify, automatically, some of the systems explored in the paper.

Sealing pointer-based optimizations behind pure functions

Functional programming languages are particularly well-suited for building automated reasoning systems, since (among other reasons) a logical term is well modeled by an inductive type, traversing a term can be implemented generically as a higher-order combinator, and backtracking search is dramatically simplified by persistent datastructures. However, existing pure functional programming languages all suffer a major limitation in these domains: traversing a term requires time proportional to the tree size of the term as opposed to its graph size. This limitation would be particularly devastating when building automation for interactive theorem provers such as Lean and Coq, for which the exponential blowup of term-tree sizes has proved to be both common and difficult to prevent. All that is needed to recover the optimal scaling is the ability to perform simple operations on the memory addresses of terms, and yet allowing these operations to be used freely would clearly violate the basic premise of referential transparency. We show how to use dependent types to seal the necessary pointer-address manipulations behind pure functional interfaces while requiring only a negligible amount of additional trust. We have implemented our approach for the upcoming version (v4) of Lean, and our approach could be adopted by other languages based on dependent type theory as well.

Lifted Successor Generation Using Query Optimization Techniques

The standard PDDL language for classical planning uses several first-order features, such as schematic actions. Yet, most classical planners ground this first-order representation into a propositional one as a preprocessing step. While this simplifies the design of other parts of the planner, in several benchmarks the grounding process causes an exponential blowup that puts otherwise solvable tasks out of reach of the planners. In this work, we take a step towards planning with lifted representations. We tackle the successor generation task, a key operation in forward-search planning, directly on the lifted representation using well-known techniques from database theory. We show how computing the variable substitutions that make an action schema applicable in a given state is essentially a query evaluation problem. Interestingly, a large number of the action schemas in the standard benchmarks result in acyclic conjunctive queries, for which query evaluation is tractable. Our empirical results show that our approach is competitive with the standard (grounded) successor generation techniques in a few domains and outperforms them on benchmarks where grounding is challenging or infeasible.