Valeriy Balabanov

Unified QBF certification and its applications

Quantified Boolean formulae (QBF) allow compact encoding of many decision problems. Their importance motivated the development of fast QBF solvers. Certifying the results of a QBF solver not only ensures correctness, but also enables certain synthesis and verification tasks. To date the certificate of a true formula can be in the form of either a syntactic cube-resolution proof or a semantic Skolem-function model whereas that of a false formula is only in the form of a syntactic clause-resolution proof. The semantic certificate for a false QBF is missing, and the syntactic and semantic certificates are somewhat unrelated. This paper identifies the missing Herbrand-function countermodel for false QBF, and strengthens the connection between syntactic and semantic certificates by showing that, given a true QBF, its Skolem-function model is derivable from its cube-resolution proof of satisfiability as well as from its clause-resolution proof of unsatisfiability under formula negation. Consequently Skolem-function derivation can be decoupled from special Skolemization-based solvers and computed from standard search-based ones. Experimental results show strong benefits of the new method.

TSV-aware analytical placement for 3D IC designs

Through-silicon vias (TSVs) are required for transmitting signals among different dies for the three-dimensional integrated circuit (3D IC) technology. The significant silicon areas occupied by TSVs bring critical challenges for 3D IC placement. Unlike most published 3D placement works that only minimize the number of TSVs during placement due to the limitations in their techniques, this paper proposes a new 3D cell placement algorithm which can additionally consider the sizes of TSVs and the physical positions for TSV insertion during placement. The algorithm consists of three stages: (1) 3D analytical global placement with density optimization and whitespace reservation for TSVs, (2) TSV insertion and TSV-aware legalization, and (3) layer-by-layer detailed placement. In particular, the global placement is based on a novel weighted-average wirelength model, giving the first model in the literature that can outperform the well-known log-sum-exp wirelength model theoretically and empirically. Further, 3D routing can easily be accomplished by traditional 2D routers since the physical positions of TSVs are determined during placement. Compared with state-of-the-art 3D cell placement works, our algorithm can achieve the best routed wirelength, TSV counts, and total silicon area, in shortest running time.

QBF Resolution Systems and Their Proof Complexities

Quantified Boolean formula (QBF) evaluation has a broad range of applications in computer science and is gaining increasing attention. Recent progress has shown that for a certain family of formulas, Q-resolution, which forms the foundation of learning in modern search-based QBF solvers, is exponentially inferior in proof size to two of its extensions: Q-resolution with resolution over universal literals (QU-resolution) and long-distance Q-resolution (LQ-resolution). The relative proof power between LQ-resolution and QU-resolution, however, remains unknown. In this paper, we show their incomparability by exponential separations on two families of QBFs, and further propose a combination of the two resolution methods to achieve an even more powerful proof system. These results may shed light on solver development with enhanced learning mechanisms. In addition, we show how QBF Skolem/Herbrand certificate extraction can benefit from polynomial LQ-resolution proofs in contrast to their exponential Q-resolution counterparts.

TSV-Aware Analytical Placement for 3-D IC Designs Based on a Novel Weighted-Average Wirelength Model

Through-silicon vias (TSVs) are required for transmitting signals among different dies for the 3-D integrated circuit (IC) technology. The significant silicon areas occupied by TSVs bring critical challenges for 3-D IC placement. Unlike most published 3-D placement works that only minimize the number of TSVs during placement due to the limitations in their techniques, this paper proposes a new 3-D cell placement algorithm that can additionally consider the sizes of TSVs and the physical positions for TSV insertion during placement. The algorithm consists of three stages: 1) 3-D analytical global placement with density optimization and whitespace reservation for TSVs; 2) TSV insertion and TSV-aware legalization; and 3) layer-by-layer detailed placement. In particular, the global placement is based on a novel weighted-average (WA) wirelength model, giving the first published model that can outperform the well-known log-sum-exp wirelength model theoretically and empirically. Also, a scheme is proposed to enhance the numerical stability of the WA wirelength model. Furthermore, 3-D routing can easily be accomplished by traditional 2-D routers since the physical positions of TSVs are determined during placement. Experimental results show the effectiveness of our algorithm. Compared with state-of-the-art 3-D cell placement works, our algorithm can achieve the best routed wirelength, TSV counts, and total silicon area, in shortest running time.

Resolution proofs and Skolem functions in QBF evaluation and applications

Quantified Boolean formulae (QBF) allow compact encoding of many decision problems. Their importance motivated the development of fast QBF solvers. Certifying the results of a QBF solver not only ensures correctness, but also enables certain synthesis and verification tasks particularly when the certificate is given as a set of Skolem functions. To date the certificate of a true formula can be in the form of either a (cube) resolution proof or a Skolem-function model whereas that of a false formula is in the form of a (clause) resolution proof. The resolution proof and Skolem-function model are somewhat unrelated. This paper strengthens their connection by showing that, given a true QBF, its Skolem-function model is derivable from its cube-resolution proof of satisfiability as well as from its clause-resolution proof of unsatisfiability under formula negation. Consequently Skolem-function derivation can be decoupled from Skolemization-based solvers and computed from standard search-based ones. Fundamentally different from prior methods, our derivation in essence constructs Skolem functions following the variable quantification order. It permits constructing a subset of Skolem functions of interests rather than the whole, and is particularly desirable in many applications. Experimental results show the robust scalability and strong benefits of the new method.

Henkin quantifiers and Boolean formulae: A certification perspective of DQBF

Henkin quantifiers, when applied on Boolean formulae, yielding the so-called dependency quantified Boolean formulae (DQBFs), offer succinct descriptive power specifying variable dependencies. Despite their natural applications to games with incomplete information, logic synthesis with constrained input dependencies, etc., DQBFs remain a relatively unexplored subject however. This paper investigates their basic properties, including formula negation and complement, formula expansion, prenex and non-prenex form conversions, and resolution. In particular, the proposed DQBF formulation is established from a synthesis perspective concerned with Skolem-function models and Herbrand-function countermodels. Also a generalized resolution rule is shown to be sound, but incomplete, for DQBF evaluation.

Henkin quantifiers and boolean formulae

Henkin quantifiers, when applied on Boolean formulae, yielding the so-called dependency quantified Boolean formulae (DQBF), offer succinct descriptive power specifying variable dependencies. Despite their natural applications to games with incomplete information, logic synthesis with constrained input dependencies, etc., DQBF remain a relatively unexplored subject however. This paper investigates their basic properties, including formula negation and complement, formula expansion, and prenex and non-prenex form conversions. In particular, the proposed DQBF formulation is established from a synthesis perspective concerned with Skolem-function models and Herbrand-function countermodels.

2QBF: Challenges and Solutions

2QBF is a special form \(\forall \varvec{x} \exists \varvec{y}.\phi \) of the quantified Boolean formula (QBF) restricted to only two quantification layers, where \(\phi \) is a quantifier-free formula. Despite its restricted form, it provides a framework for a wide range of applications, such as artificial intelligence, graph theory, synthesis, etc. In this work, we overview two main 2QBF challenges in terms of solving and certification. We contribute several improvements to existing solving approaches and study how the corresponding approaches affect certification. We further conduct an extensive experimental comparison on both competition and application benchmarks to demonstrate strengths of the proposed methodology.

Speeding up MUS Extraction with Preprocessing and Chunking

In this paper we present several improvements to extraction of a minimal unsatisfiable subformula (MUS) of a Boolean formula. As our first contribution, we describe model rotation on preprocessed formulas and show that preprocessing significantly improves model rotation. We find very convenient to adopt the framework of labeled CNF formulas and we present our algorithms in this more general framework. We use the assumption-based approach for computing MUSes due to its simplicity and the ability to use any SAT-solver as the back-end. However, this comes with a price: it is well-known that the assumption-based approach performs significantly worse than the resolution-based approach. This leads to our second contribution, we show how to bridge the gap between the two approaches using “chunking”. An extensive experimental evaluation shows that our method significantly outperforms state-of-the-art solutions in the context of group MUS extraction.

Flexibility and Optimization of QBF Skolem–Herbrand Certificates

Skolem and Herbrand functions are important certificates validating the truth and falsity, respectively, of quantified Boolean formulas (QBFs). They are essential in various synthesis and verification applications. Recent advancement established a linear time extraction of Skolem/Herbrand functions from QBF consensus/resolution proofs. However, the obtained functions are often excessively large and improper for practical applications. To overcome this limitation, this paper characterizes various flexibilities of QBF certificates, and exploits them for certificate simplification. Experiments show substantial reduction on QBF certificates in terms of circuit size and depth, which are of primary concerns for synthesis applications.