P-time Algorithms Research Articles

Knowing How to Plan

Various planning-based know-how logics have been studied in the recent literature. In this paper, we use such a logic to do know-how-based planning via model checking. In particular, we can handle the higher-order epistemic planning involving know-how formulas as the goal, e.g., find a plan to make sure p such that the adversary does not know how to make p false in the future. We give a PTIME algorithm for the model checking problem over finite epistemic transition systems and axiomatize the logic under the assumption of perfect recall.

On a Theorem of Lovász that (&sdot, H ) Determines the Isomorphism Type of H

Graph homomorphism has been an important research topic since its introduction [20]. Stated in the language of binary relational structures in that paper [20], Lovász proved a fundamental theorem that, for a graph H given by its 0-1 valued adjacency matrix, the graph homomorphism function G ↦ hom( G , H ) determines the isomorphism type of H . In the past 50 years, various extensions have been proved by many researchers [1, 15, 21, 24, 26]. These extend the basic 0-1 case to admit vertex and edge weights; but these extensions all have some restrictions such as all vertex weights must be positive. In this article, we prove a general form of this theorem where H can have arbitrary vertex and edge weights. A noteworthy aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective in the following sense: it provides an algorithm that for any H either outputs a P-time algorithm solving hom(&sdot, H ) or a P-time reduction from a canonical #P-hard problem to hom(&sdot, H ).

Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .

Polynomial Time Algorithms for Branching Markov Decision Processes and Probabilistic Min(Max) Polynomial Bellman Equations

We show that one can compute the least nonnegative solution (also known as the least fixed point) for a system of probabilistic min (max) polynomial equations, to any desired accuracy [Formula: see text] > 0 in time polynomial in both the encoding size of the system and in log(1/[Formula: see text]). These are Bellman optimality equations for important classes of infinite-state Markov decision processes (MDPs), including branching MDPs (BMDPs), which generalize classic multitype branching stochastic processes. We thus obtain the first polynomial time algorithm for computing, to any desired precision, optimal (maximum and minimum) extinction probabilities for BMDPs. Our algorithms are based on a novel generalization of Newton’s method, which employs linear programming in each iteration. We also provide polynomial-time (P-time) algorithms for computing an [Formula: see text]-optimal policy for both maximizing and minimizing extinction probabilities in a BMDP, whereas we note a hardness result for computing an exact optimal policy. Furthermore, improving on prior results, we provide more efficient P-time algorithms for qualitative analysis of BMDPs, that is, for determining whether the maximum or minimum extinction probability is 1, and, if so, computing a policy that achieves this. We also observe some complexity consequences of our results for branching simple stochastic games, which generalize BMDPs.

On blockwise symmetric matchgate signatures and higher domain #CSP

For any n≥3 and q≥3, we prove that the Equality function (=n) on n variables over a domain of size q cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures. This has the implication that the standard holographic algorithms based on matchgates, a methodology known to be universal for #CSP over the Boolean domain, cannot produce P-time algorithms for planar #CSP over any higher domain q≥3.

Greatest fixed points of probabilistic min/max polynomial equations, and reachability for branching Markov decision processes

We give polynomial time algorithms for quantitative (and qualitative) reachability analysis for Branching Markov Decision Processes (BMDPs). Specifically, given a BMDP, and given an initial population, where the objective of the controller is to maximize (or minimize) the probability of eventually reaching a population that contains an object of a desired (or undesired) type, we give algorithms for approximating the supremum (infimum) reachability probability, within desired precision ϵ>0, in time polynomial in the encoding size of the BMDP and in log⁡(1/ϵ). We furthermore give P-time algorithms for computing ϵ-optimal strategies for both maximization and minimization of reachability probabilities. We also give P-time algorithms for all associated qualitative analysis problems, namely: deciding whether the optimal (supremum or infimum) reachability probabilities are 0 or 1. Prior to this paper, approximation of optimal reachability probabilities for BMDPs was not even known to be decidable.Our algorithms exploit the following basic fact: we show that for any BMDP, its maximum (minimum) non-reachability probabilities are given by the greatest fixed point (GFP) solution g⁎∈[0,1]n of a corresponding monotone max (min) Probabilistic Polynomial System of equations (max/minPPS), x=P(x), which are the Bellman optimality equations for a BMDP with non-reachability objectives. We show how to compute the GFP of max/minPPSs to desired precision in P-time.We also study more general branching simple stochastic games (BSSGs) with (non-)reachability objectives. We show that: (1) the value of these games is captured by the GFP, g⁎, of a corresponding max-minPPS, x=P(x); (2) the quantitative problem of approximating the value is in TFNP; and (3) the qualitative problems associated with the value are all solvable in P-time.

Reducing Nondeterministic Tree Automata by Adding Transitions

We introduce saturation of nondeterministic tree automata, a technique that consists of adding new transitions to an automaton while preserving its language. We implemented our algorithm on minotaut - a module of the tree automata library libvata that reduces the size of automata by merging states and removing superfluous transitions - and we show how saturation can make subsequent merge and transition-removal operations more effective. Thus we obtain a Ptime algorithm that reduces the size of tree automata even more than before. Additionally, we explore how minotaut alone can play an important role when performing hard operations like complementation, allowing to both obtain smaller complement automata and lower computation times. We then show how saturation can extend this contribution even further. We tested our algorithms on a large collection of automata from applications of libvata in shape analysis, and on different classes of randomly generated automata.

Model checking for symbolic-heap separation logic with inductive predicates

We investigate the *model checking* problem for symbolic-heap separation logic with user-defined inductive predicates, i.e., the problem of checking that a given stack-heap memory state satisfies a given formula in this language, as arises e.g. in software testing or runtime verification. First, we show that the problem is *decidable*; specifically, we present a bottom-up fixed point algorithm that decides the problem and runs in exponential time in the size of the problem instance. Second, we show that, while model checking for the full language is EXPTIME-complete, the problem becomes NP-complete or PTIME-solvable when we impose natural syntactic restrictions on the schemata defining the inductive predicates. We additionally present NP and PTIME algorithms for these restricted fragments. Finally, we report on the experimental performance of our procedures on a variety of specifications extracted from programs, exercising multiple combinations of syntactic restrictions.

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multitype branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x = P ( x ). It was shown in Etessami and Yannakakis [2009] that a decomposed variant of Newton’s method converges monotonically to the LFP solution for any MPS that has a nonnegative solution. Subsequently, Esparza et al. [2010] obtained upper bounds on the convergence rate of Newton’s method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs [Etessami et al. 2010, 2012]. However, prior to this article, for arbitrary (not necessarily strongly connected) MPSs, no upper bounds at all were known on the convergence rate of Newton’s method as a function of the encoding size |P| of the input MPS, x = P ( x ). In this article, we provide worst-case upper bounds, as a function of both the input encoding size |P|, and ε > 0, on the number of iterations required for decomposed Newton’s method (even with rounding) to converge to within additive error ε > 0 of q*, for an arbitrary MPS with LFP solution q*. Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω -regular or LTL property.

Circuit lower bounds in bounded arithmetics

We prove that TNC1, the true universal first-order theory in the language containing names for all uniform NC1 algorithms, cannot prove that for sufficiently large n, SAT is not computable by circuits of size n4kc where k≥1,c≥2 unless each function f∈SIZE(nk) can be approximated by formulas {Fn}n=1∞ of subexponential size 2O(n1/c) with subexponential advantage: Px∈{0,1}n[Fn(x)=f(x)]≥1/2+1/2O(n1/c). Unconditionally, V0 cannot prove that for sufficiently large n, SAT does not have circuits of size nlog⁡n. The proof is based on an interpretation of Krajíček's proof (Krajíček, 2011 [15]) that certain NW-generators are hard for TPV, the true universal theory in the language containing names for all p-time algorithms.

Entity resolution for probabilistic data

Entity resolution is the problem of identifying the tuples that represent the same real world entity. In this paper, we propose a complete solution to the problem of entity resolution over probabilistic data (ERPD), which arises in many applications that have to deal with probabilistic data. To deal with the ERPD problem, we distinguish between two classes of similarity functions, i.e. context-free and context-sensitive. We propose a PTIME algorithm for context-free similarity functions, and an approximation algorithm for context-sensitive similarity functions. We validated our algorithms through experiments over both synthetic and real datasets. Our extensive performance evaluation shows the effectiveness of our algorithms.

Minimization of Deterministic Fuzzy Tree Automata

Until now, some methods for minimizing deterministic fuzzy finite tree automata (DFFTA) and weighted tree automata have been established by researchers. Those methods are language preserving, but the behavior of original automata and minimized one may be different. This paper, considers both language preserving and behavior preserving in minimization process. We drive Myhill-Nerode kind theorems corresponding to each proposed method and introduce PTIME algorithms for behaviorally and linguistically minimization. The proposed minimization algorithms are based on two main steps. The first step includes finding dependency between equivalency of states, according to the set of transition rules of DFFTA, and making merging dependency graph (MDG). The second step is refinement of MDG and making minimization equivalency set (MES). Additionally, behavior preserving minimization of DFFTA requires a pre-processing for modifying fuzzy membership grade of rules and final states, which is called normalization.

On the hardness of learning queries from tree structured data

The problem of learning queries from tree structured data is studied by this paper. A tree structured data is modeled as a node-labeled tree $$T$$ T , and applying a query $$q$$ q on $$T$$ T will return a set $$q(T)$$ q ( T ) which is a subset of nodes in $$T$$ T . For a tree-node pair $$(T,t)$$ ( T , t ) where $$t$$ t is a node in $$T$$ T , $$q$$ q is called to accept the pair if $$t\in {q(T)}$$ t ? q ( T ) , and reject the pair if $$t\notin {q(T)}$$ t ? q ( T ) . For some query class $$\mathcal{L }$$ L , given tree-node pair sets $$E_p$$ E p and $$E_n$$ E n , the tree query learning problem is to find a query $$q\in \mathcal{L }$$ q ? L such that (1) $$q$$ q rejects all pairs in $$E_n$$ E n , and (2) the size of pairs in $$E_p$$ E p accepted by $$q$$ q is maximized. On four different query classes $$\mathcal Q ^{\tiny /}$$ Q / , $$\mathcal Q ^{\tiny /,*}$$ Q / , ? , $$\mathcal Q ^{\tiny /,//}$$ Q / , / / and $$\mathcal Q ^{\tiny /,[]}$$ Q / , [ ] , this paper studies the hardness of the corresponding tree query learning problems. For $$\mathcal Q ^{\tiny /}$$ Q / , a PTime algorithm is given. For $$\mathcal Q ^{\tiny /,*}$$ Q / , ? and $$\mathcal Q ^{\tiny /,//}$$ Q / , / / , the NP-complete results are shown. For $$\mathcal Q ^{\tiny /,[]}$$ Q / , [ ] , the problem is shown to be NP-hard by considering two constrained fragments of $$\mathcal Q ^{\tiny /,[]}$$ Q / , [ ] . Also, for $$\mathcal Q ^{\tiny /,*}$$ Q / , ? , $$\mathcal Q ^{\tiny /,[]}$$ Q / , [ ] and $$\mathcal Q ^{\tiny /,//}$$ Q / , / / , it is shown that there are no $$n^{1-\epsilon }$$ n 1 ? ∈ -approximation algorithms for any $$\epsilon >0$$ ∈ > 0 .

Querying Probabilistic Business Processes for Sub-Flows

A Business Process (BP for short) consists of a set of activities which, combined in a flow, achieve some business goal. A given BP may have a large, possibly infinite, number of possible execution flows (EX-flows for short), each having some probability to occur at run time. This paper studies query evaluation over such probabilistic BPs. We focus on two important classes of queries, namely boolean queries that compute the probability that a random EX-flow of a BP satisfies a given property, and projection queries focusing on portions of EX-flows that are of interest to the user. For the latter queries the answer consists of the top-k instances of these portions that are most likely to occur at run-time. We study the complexity of query evaluation for both kinds of queries, showing in particular that projection queries may be harder to evaluate than boolean queries. We present a picture of which combinations of BP classes and query features lead to PTIME algorithms and which to NP-hard or infeasible problems.

Vertex-Bipartition Method for Colouring Minor-Closed Classes of Graphs

Thomas conjectured that there is an absolute constant c such that for every proper minor-closed class of graphs, there is a polynomial-time algorithm that can colour every G ∈ with at most χ(G) + c colours. We introduce a parameter ρ(), called the degenerate value of , which is defined to be the smallest r such that every G ∈ can be vertex-bipartitioned into a part of bounded tree-width (the bound depending only on ), and a part that is r-degenerate. Although the existence of one global bound for the degenerate values of all proper minor-closed classes would imply Thomas's conjecture, we prove that the values ρ() can be made arbitrarily large. The problem lies in the clique sum operation. As our main result, we show that excluding a planar graph with a fixed number of apex vertices gives rise to a minor-closed class with small degenerate value. As corollaries, we obtain that (i) the degenerate value of every class of graphs of bounded local tree-width is at most 6, and (ii) the degenerate value of the class of Kn-minor-free graphs is at most n + 1. These results give rise to P-time approximation algorithms for colouring any graph in these classes within an error of at most 7 and n + 2 of its chromatic number, respectively.

Hard problems in max-algebra, control theory, hypergraphs and other areas

We introduce the max-atom problem (MAP): solving (in Z) systems of inequations of the form max(x, y)+k z, where x, y, z are variables and k ∈ Z. Our initial motivation for MAP was reasoning on delays in circuits using SAT Modulo Theories [10], viewing MAP as a natural extension of Difference Logic, i.e., inequations of the form x+ k y. Here we show that MAP is PTIME-equivalent to several rather different well-known problems for which no PTIME algorithm has been found so far, in spite of decades of independent efforts. One is on solving two-sided linear max-plus systems (Section 3 of this paper) that arise in Control Theory when modeling Discrete Event Systems, and another one on shortest paths in directed weighted hypergraphs (Section 4).

Type inference and type checking for queries on execution traces

This paper studies, for the first time, the management of type information for an important class of semi-structured data: nested DAGs (Directed Acyclic Graphs) that describe execution traces of business processes (BPs for short). Specifically, we consider here type inference and type checking for queries over BP execution traces. The queries that we consider select portions of the traces that are of interest to the user; the types describe the possible shape of the execution traces in the input/output of the query. We formally define and characterize here three common classes of BP execution traces and their respective notions of type inference and type checking. We study the complexity of the two problems for query languages of varying expressive power and present efficient type inference/checking algorithms whenever possible. Our analysis offers a nearly complete picture of which combinations of trace classes and query features lead to PTIME algorithms and which to NP-complete or undecidable problems.

Model checking restricted sets of timed paths

In this paper, we study the complexity of model-checking formulas of four important real-time logics ( TPTL , MTL , MITL , and TCTL ) over restricted sets of timed paths. The classes of restricted sets of timed paths that we consider are (i) a single finite (or ultimately periodic) timed path, (ii) an infinite set of finite (or infinite) timed paths defined by a finite (or ultimately periodic) path in a region graph, (iii) an infinite set of finite (or infinite) timed paths defined by a finite (or ultimately periodic) path in a zone graph. Several results are quite negative: TPTL and MTL remain undecidable along region- and zone-paths. On the other hand, we obtained PTIME algorithms for model-checking TCTL along a region path, and for MTL along a single timed path.

Dual weak pigeonhole principle, Boolean complexity, and derandomization

We study the extension (introduced as BT by Krajı́ček in Fund. Math. 170 (2001) 123) of the theory S 2 1 by instances of the dual (onto) weak pigeonhole principle for p-time functions, dWPHP( PV) x 2 x . We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S 2 1+ dWPHP( PV). We construct a propositional proof system WF (based on a reformulation of Extended Frege in terms of Boolean circuits), which captures the ∀ Π 1 b -consequences of S 2 1+ dWPHP( PV). We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. (Comput. Complexity 6 (1996/1997) 256). We prove that dWPHP( PV) is (over S 2 1) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction (derandomization of probabilistic p-time algorithms) in a conservative extension of S 2 1+ dWPHP( PV).