17 research outputs found
Bisimulation maps in presheaf categories
The category of presheaves on a (small) category is a suitable semantic universe to study behaviour of various dynamical systems. In particular, presheaves can be used to record the executions of a system and their morphisms correspond to simulation maps for various kinds of state-based systems. In this paper, we introduce a notion of bisimulation maps between presheaves (or executions) to capture well known behavioural equivalences in an abstract way. We demonstrate the versatility of this framework by working out the characterisations for standard bisimulation, ∀-fair bisimulation, and branching bisimulation
Minimization and Canonization of GFG Transition-Based Automata
While many applications of automata in formal methods can use
nondeterministic automata, some applications, most notably synthesis, need
deterministic or good-for-games(GFG) automata. The latter are nondeterministic
automata that can resolve their nondeterministic choices in a way that only
depends on the past. The minimization problem for deterministic B\"uchi and
co-B\"uchi word automata is NP-complete. In particular, no canonical minimal
deterministic automaton exists, and a language may have different minimal
deterministic automata. We describe a polynomial minimization algorithm for GFG
co-B\"uchi word automata with transition-based acceptance. Thus, a run is
accepting if it traverses a set of designated transitions only
finitely often. Our algorithm is based on a sequence of transformations we
apply to the automaton, on top of which a minimal quotient automaton is
defined. We use our minimization algorithm to show canonicity for
transition-based GFG co-B\"uchi word automata: all minimal automata have
isomorphic safe components (namely components obtained by restricting the
transitions to these not in ) and once we saturate the automata with
-transitions, we get full isomorphism.Comment: 28 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:2009.1088
Multipebble Simulations for Alternating Automata - (Extended Abstract)
Abstract. We study generalized simulation relations for alternating Büchi automata (ABA), as well as alternating finite automata. Having multiple pebbles allows the Duplicator to “hedge her bets ” and delay decisions in the simulation game, thus yielding a coarser simulation relation. We define (k1, k2)-simulations, with k1/k2 pebbles on the left/right, respectively. This generalizes previous work on ordinary simulation (i.e., (1, 1)-simulation) for nondeterministic Büchi automata (NBA) in [3] and ABA in [4], and (1, k)-simulation for NBA in [2]. We consider direct, delayed and fair simulations. In each case, the (k1, k2)simulations induce a complete lattice of simulations where (1,1)- and (n, n)simulations are the bottom and top element (if the automaton has n states), respectively, and the order is strict. For any fixed k1, k2, the (k1, k2)-simulation implies (ω-)language inclusion and can be computed in polynomial time. Furthermore, quotienting an ABA w.r.t. (1, n)-delayed simulation preserves its language. Finally, multipebble simulations yield new insights into the Miyano-Hayashi construction [10] on ABA.
Minimization and Canonization of GFG Transition-Based Automata
While many applications of automata in formal methods can use
nondeterministic automata, some applications, most notably synthesis, need
deterministic or good-for-games (GFG) automata. The latter are nondeterministic
automata that can resolve their nondeterministic choices in a way that only
depends on the past. The minimization problem for deterministic B\"uchi and
co-B\"uchi word automata is NP-complete. In particular, no canonical minimal
deterministic automaton exists, and a language may have different minimal
deterministic automata. We describe a polynomial minimization algorithm for GFG
co-B\"uchi word automata with transition-based acceptance. Thus, a run is
accepting if it traverses a set of designated transitions only
finitely often. Our algorithm is based on a sequence of transformations we
apply to the automaton, on top of which a minimal quotient automaton is
defined. We use our minimization algorithm to show canonicity for
transition-based GFG co-B\"uchi word automata: all minimal automata have
isomorphic safe components (namely components obtained by restricting the
transitions to these not in ) and once we saturate the automata with
-transitions, we get full isomorphism
Efficient reduction of nondeterministic automata with application to language inclusion testing
We present efficient algorithms to reduce the size of nondeterministic
B\"uchi word automata (NBA) and nondeterministic finite word automata (NFA),
while retaining their languages. Additionally, we describe methods to solve
PSPACE-complete automata problems like language universality, equivalence, and
inclusion for much larger instances than was previously possible (
states instead of 10-100). This can be used to scale up applications of
automata in formal verification tools and decision procedures for logical
theories. The algorithms are based on new techniques for removing transitions
(pruning) and adding transitions (saturation), as well as extensions of classic
quotienting of the state space. These techniques use criteria based on
combinations of backward and forward trace inclusions and simulation relations.
Since trace inclusion relations are themselves PSPACE-complete, we introduce
lookahead simulations as good polynomial time computable approximations
thereof. Extensive experiments show that the average-case time complexity of
our algorithms scales slightly above quadratically. (The space complexity is
worst-case quadratic.) The size reduction of the automata depends very much on
the class of instances, but our algorithm consistently reduces the size far
more than all previous techniques. We tested our algorithms on NBA derived from
LTL-formulae, NBA derived from mutual exclusion protocols and many classes of
random NBA and NFA, and compared their performance to the well-known automata
tool GOAL.Comment: 69 pages. arXiv admin note: text overlap with arXiv:1210.662
Generalized simulation relations with applications in automata theory
Finite-state automata are a central computational model in computer science, with
numerous and diverse applications. In one such application, viz. model-checking, automata
over infinite words play a central rˆole. In this thesis, we concentrate on B¨uchi automata
(BA), which are arguably the simplest finite-state model recognizing languages
of infinite words. Two algorithmic problems are paramount in the theory of automata:
language inclusion and automata minimization. They are both PSPACE-complete, thus
under standard complexity-theoretic assumptions no deterministic algorithm with worst
case polynomial time can be expected. In this thesis, we develop techniques to tackle
these problems.
In automata minimization, one seeks the smallest automaton recognizing a given
language (“small” means with few states). Despite PSPACE-hardness of minimization,
the size of an automaton can often be reduced substantially by means of quotienting.
In quotienting, states deemed equivalent according to a given equivalence are merged
together; if this merging operation preserves the language, then the equivalence is
said to be Good for Quotienting (GFQ). In general, quotienting cannot achieve exact
minimization, but, in practice, it can still offer a very good reduction in size. The central
topic of this thesis is the design of GFQ equivalences for B¨uchi automata.
A particularly successful approach to the design of GFQ equivalences is based on
simulation relations. Simulation relations are a powerful tool to compare the local
behavior of automata. The main contribution of this thesis is to generalize simulations,
by relaxing locality in three perpendicular ways: by fixing the input word in advance
(fixed-word simulations, Ch. 3), by allowing jumps (jumping simulations, Ch. 4), and by
using multiple pebbles (multipebble simulations for alternating BA, Ch. 5). In each case,
we show that our generalized simulations induce GFQ equivalences. For fixed-word
simulation, we argue that it is the coarsest GFQ simulation implying language inclusion,
by showing that it subsumes a natural hierarchy of GFQ multipebble simulations.
From a theoretical perspective, our study significantly extends the theory of simulations
for BA; relaxing locality is a general principle, and it may find useful applications
outside automata theory. From a practical perspective, we obtain GFQ equivalences
coarser than previously possible. This yields smaller quotient automata, which is beneficial
in applications. Finally, we show how simulation relations have recently been
applied to significantly optimize exact (exponential) language inclusion algorithms
(Ch. 6), thus extending their practical applicability
Tools and Algorithms for the Construction and Analysis of Systems
This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers