Workshop: Special Year on Logic and Algorithms - One Year Later

July 23-25, 1997
DIMACS Center, CoRE Building, Rutgers University, Piscataway, NJ

Eric Allender, Rutgers University,
Robert Kurshan, Bell Labs,
Moshe Vardi, Rice University,


We consider the problem of storing a set $S \subset \Sigma^{k}$ as a minimized deterministic finite automaton (DFA). We show that inserting a new string $\sigma \in \Sigma^{k}$ or deleting a string from the set $S$ represented as a minimized DFA can be done in time $O(k |\Sigma|)$ so that the resulting set is also represented as a minimized DFA. We then discuss some applications of this work to the problem of state space exploration.

This is joint work with Gerard Holzmann.

The talk presents a compiler development methodology based on a formal specification of a compiler. The formal specification defines a mapping from a source language of the compiler into its target language through some special predicate calculus with an interpretation domain consisting of a structured pair of source and target extended derivation trees. In the methodology suggested the compiler implementation can be split into the following phases: designing the mapping, specifying it formally, testing the mapping and coding the algorithms which implement the mapping. The methodology evolved out of a number of realistic applications including the compiler from the subset of SDL 92 into Cospan.

Message sequence charts (MSC) are popular as a low-level design of a communication system. We present a methodology for specifying and verifying MSC. Specification is given using templates, namely charts with only partial information about the participating events and their interrelated order. Verification is done by a search whose aim is to match a template against a design MSC. The search is implemented by COSPAN after the template and design MSC are translated into two COSPAN processes. If a matching sequence of events exists, the template process reaches its final state organized as a trap. This gives rise to a bad cycle and the matching sequence found is extracted from the error track reported by COSPAN.

When analyzing the complexity of decision problems on graphs, one normally assumes that the input size is polynomial in the number of vertices in the graph. Galperin and Wigderson~\cite{gw}, and later Papadimitriou and Yannakakis~\cite{py}, investigated the complexity of these problems when the input graph is represented by a polylogarithmically succinct circuit. They showed that, under such a representation, not only do certain trivial properties become intractable, but also that, in general, there is an exponential blow up in the complexity of these problems.

In this paper, we show that, even when the input graph is represented by a small OBDD, there is an exponential blow up in the complexity of most graph properties. In particular, we show that the GAP and AGAP problems become complete for PSPACE and EXP, respectively, when the graphs are succinctly represented by OBDDs.

We present a methodology for Hardware/Software co-verification. SDL is used to specify the software part of the system, Verilog or VHDL is used to specify the hardware part, and the combined system specification is compiled into S/R, the native language of the verification tool COSPAN.

The interface between HW and SW is given in SDL with a modified semantics. An interface process behaves like an SDL process from the point of view of SW and like a HW module from that of HW.

We assume that co-design properties refer either to the hardware part or to the software part. For hardware properties, we apply a HW-centric methodology in which the interface process is an abstraction of the SW part while for software properties, we apply a SW-centric methodology in which the interface is considered to be an abstraction of the HW. In the HW-centric view, one can use localization reduction. For the SW-centric view, we apply the ample-set partial order reduction technique statically to define an automaton that encodes the reduction. Compiling this automaton together with the source enables us to treat the result as a synchronous system, and thus apply a further localization reduction and apply symbolic state enumeration, all with no modification to the verification engine COSPAN. As a by-product, this method gives a way to combine partial order reduction, localization reduction and symbolic search in a single algorithm by compiling the partial order reduction into the source, relative to each instance.

Program analysis methods such as model checking are hindered by the state explosion problem. Potential solutions include various types of abstraction such as symmetry reduction methods, partial order reduction, and symbolic representations. Some relations among these methods will be discussed.

(This decribes in part some works done jointly with S. Jha, K. Namjoshi, D. Peled)

FormalCheck is a CAD tool for computer-aided verification of hardware which grew out of some important theoretical and technological advances of the last decade. I'll talk about the process through which it evolved from concepts to commerce.

The concepts were founded in logic, automata and computational complexity. The process involved jumping the hurdles of technology transfer, resistance to change and commercial justification, including the world of pay-back and back-bite. The experience plows through the center of issues such as: Is there such a thing as "Technology Transfer"? If so, how do you do it? If not, why does everyone talk about it? Is there really a role for fundamental science anymore? If so, who will support it?

This is an update of a similar talk given during the SYLA.

Algebraic proof systems are proof systems that operate with (commutative) multi-variable polynomials over some ring and prove facts of the form ``{\em every solution $(a_1,\ldots,a_n)$ to some system of polynomial equations $f_1(x_1,\ldots,x_n)=\ldots=f_m(x_1,\ldots,x_n)=0$ is also a solution to another equation $g(x_1,\ldots,x_n)=0$}''. Of especial interest to us in this talk is the ``combinatorial'' framework in which we include into the set of initial equations $f_1=\ldots=f_m=0$ all equations $x_i^2-x_i=0$ as default axioms (this ensures us that we are in the Boolean world, where $a_i\in\{0,1\}$), and pay a special attention to those implications $(f_1=\ldots=f_m=0)\Rightarrow g=0$ that reflect {\em natural} combinatorial principles (which is to make people inhabiting that world excited).

Algebraic proof systems originally emerged in the paper ``Lower bounds on Hilbert's Nullstellensatz\ldots'' by Beame et al\ (FOCS 1994) as a tool for proving lower bounds in the {\em propositional\/} proof complexity (more specifically, for attacking the system $F_d+MOD_p$ of constant-depth Frege proofs allowing also modular gates modulo $p$). It soon became clear, however, that the importance of algebraic proof systems goes far beyond this particular task: they provide natural and elegant models for studying (our way of thinking about) the most basic algebraic facts and constructions. At the same time, in many aspects they are closely related to such already established areas as propositional proof complexity or automatic theorem proving. Somewhat surprising is also the fact that lower bounds for algebraic proof systems are rather hard to come by, despite the fact that they amount to some very ``elementary'' linear algebra.

In this talk we firstly survey various connections between algebraic proof systems on the one hand, and such areas as propositional proof systems or automatic theorem proving on the other. In particular, we try to explain from the complexity perspective what makes algebraic proof systems very unique for the automatic proof generation.

Then we survey lower bounds known for algebraic proof systems. We pay a special attention to two recent advances in the area directly influenced by the 1996 special year at DIMACS. One result establishes strong lower bounds on the complexity of Nullstellensatz proofs for the {\em onto} version of the pigeon-hole principle PHP (Beame, Riis). Another result shows the first non-trivial lower bounds on proofs in the general polynomial calculus for the (ordinary) PHP (Razborov).

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? The question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, Gurevich conjectured the negative answer. The problem motivated a quest for stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras) operating on the given input structure plus a universe of finite sets built from it. The idea is to replace arbitrary choice with parallel execution. The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all of PTime.

The development of Descriptive Complexity suggests a very close connection between proving lower bounds in complexity theory and proving inexpressibility results in logic. The latter are of the form "property P cannot be expressed in logic L over the class of finite models." Developing tools for proving such expressivity bounds is one of the central problems in Finite-Model Theory. The Ehrenfeucht-Fraisse (EF) games remain the dominant tool; however, as we are interested in more complex problems and logics, playing a game is often a nontrivial task, and thus it becomes important to find easier tools for proving expressivity bounds.

In this talk, I survey tools based on "local" properties of logics. Intuitively, such local properties say that a given logic cannot grasp the whole structure, but rather only a small part of it. I start by reviewing various locality theorems proved for first-order logic, FO (theorems by Gaifman, Hanf, Fagin-Stockmeyer-Vardi, and corollaries) and show how they can be used to prove expressivity bounds. I also show how locality notions provide winning strategies in various EF games that capture expressivity in FO logic and beyond.

In the second part of the talk, I show how to extract four abstract notions of locality by examining locality theorems, and show a chain of implications among them. This chain of implications immediately gives new expressivity bounds for some extensions of FO (for example, with counting and unary generalized quantifiers). I also give characterization of those notions on structures of small degree.

At the end, I'll show two applications of the locality techniques. One deals with a descriptive complexity approach to the TC^0 vs. Logspace problem. The other is about expressive power of database query languages with aggregate functions (essentially SQL-like languages).

Logics with only two element variables play an important role as process logics, as temporal logics or as logics for knowledge representation. Typical applications of logics in these areas hinge on the tractability of their satisfiability problem.

By a classical result of Mortimer's, two-variable first-order logic has the finite model property and is decidable for satisfiability. Recent studies by a number of different authors have addressed the decision problems (of satisfiability in finite or in general models) for logics in the vicinity of two-variable first-order logic -- guided by the above-mentioned applications. This programme has been carried out also with a view to obtaining extensions that combine features of decidable extensions of modal logic with those of two-variable first-order logic (viz. global quantification).

Concerning decidability of extensions, however, two-variable first-order logic turns out to be nowhere near as robust as modal logic. For instance, least fixed points (as in mu-calculus) or transitive closures (as in PDL or CTL), when added to two-variable first-order logic lead to high degrees of undecidability. One important exception concerns the extension of two-variable first-order logic by counting quantifiers (as with graded modalities). Even though this logic does not have the finite model property, it is decidable (for satisfiability in finite as well as in general models), and good upper complexity bounds have been be obtained.

Finite model theory has been primarily concerned with the properties of logics (first order and otherwise) on finite models. This is analogous to the focus of first order model theory until the early 60's. Since then, first order model theory has progressed by studying the type structure of complete first order theories. We will pose a number of questions concerning the development of a classification theory (analogous to stability theory) for complete theories in $k$-variable logic with arbitrarily large finite models.

A first order theory is unstable if and only if it has the independence property or the strict order property. Theorem. (Baldwin and Dawar) A complete theory in $k$-variable logic with arbitrarily large finite models does not have the strict order property. Conjecture: A complete theory in $k$-variable logic with arbitrarily large finite models is either stable or has the independence property. The conjecture does not follow easily from the theorem because of a number of complexities in rephrasing first order notions for $k$-variable logic.

We discuss the role of the finite-variable fragments of first-order logic in descriptive complexity theory. Taking the number of variables as a complexity measure sounds strange first, but has turned out to be quite useful. This can best be seen from the following two fundamental results (due to Vardi and Immerman/Lander):

Actually, both problems are PTIME complete (for k>1). This suggests a close connection between the number of variables needed to express a problem and the exponent of a polynomial bounding the time needed to compute the problem.

Starting from the results above, the finite variable logics have been developed into the framework for important questions in finite model theory, which are in particular closely related to the problem of finding a logic for PTIME.

In this talk we report on recent advances in this area.

We show that the following three statements are equivalent: QPV is conservative over QALV, QALV proves its open induction formulas, and QALV proves P = NC1. Here QPV and QALV are first order theories whose function symbols range over polynomial-time and NC1 functions, respectively.

Three well-studied bounded arithmetic theories are $T^i_2$, $S^i_2$, $R^i_2$. It is known that $S^{i+1}_2$ is $\Sigma^b_{i+1}$-conservative over $T^i_2$ and that the $\Sigma^b_{i+1}$-definable functions of $S^{i+1}_2$ are the class $FP^{\Sigma^p_i}$. The $\Sigma^b_{i+1}$-multifunctions of $S^i_2$ are the class $FP^{\Sigma^p_i}(wit,log)$. Definable multifunctions for the classes $R^i_2$ are less well understood. The main known result is that the $\Sigma^b_1$-functions of $R^1_2$ are the class $FNC$. In this talk I will discuss results which arose while trying to understand the $\Sigma^b_{i+1}$ and $\Sigma^b_i$-definable multifunctions of $R^i_2$. One tactic which allowed me to derive my results was to consider only prenex $\Sigma^b_i$-definability in theories whose induction schemas were restricted to prenex formulas. The prenex versions of $S^i_2$ and $T^i_2$ turn out to be the same as the usual versions. However, this does not appear to be the case for $R^i_2$. Nevertheless, $R^i_2$ is $\Sigma^b_i$-conservative over its prenex version. Results from my thesis about prenex $R^{i+1}_2$ enable one to show the $\Sigma^b_{i+1}$-multifunctions of $R^{i+1}_2$ are $FP^{\Sigma^p_i}(wit,log^{O(1)})$ and the $\Sigma^b_{i+1}$-functions are $FNC^{\Sigma^p_i}$. As well as the above in this talk I will discuss how prenex theories allow one to generalize the above definability results to much weaker theories than $T^i_2$, $S^i_2$, or $R^i_2$. I will also describe a nice generalization of the conservation result between $S^{i+1}_2$ and $T^i_2$ which holds for prenex theories. Finally, I will discuss how a theorem of Kadin allows one to show various weak equalities among bounded arithmetic theories imply the collapse of the polynomial hierarchy.

A polynomial time computable function $h:\Sigma^*\to\Sigma^*$ whose range is the set of tautologies in Propositional Logic (TAUT), is called a proof system. Cook and Reckhow defined this concept and in order to compare the relative strenth of different proof systems, they considered the notion of p-simulation. Intuitively a proof system $h$ p-simulates a second one $h'$ if there is a polynomial time computable function $\gamma$ translating proofs in $h'$ into proofs in $h$. A proof system is called optimal if it p-simulates every other proof system. The question of whether p-optimal proof systems exist is an important one in the field. Kraj\'{\i}\v{c}ek and Pudl\'ak have given a sufficient condition for the existence of such optimal systems, showing that if the deterministic and nondeterministic exponential time classes coincide, then p-optimal proof systems exist. They also give a condition implying the existence of optimal proof systems (a related concept to the one of p-optimal systems) exist. In this paper we improve this result giving a weaker sufficient condition for this fact. We show that if a particular class of sets with low information content in nondeterministic double exponential time is included in the corresponding nondeterministic class, then p-optimal proof systems exist.

We also show some complexity theoretical consequences that follow from the assumption of the existence of p-optimal systems. We prove that if p-optimal systems exist the the class UP (an some other related complexity classes) have many-one complete languages, and that many-one complete sets for NP $\cap$ SPARSE follow from the existence of optimal proof systems.

Finite variable logics have played a prominent role in finite model theory, for a variety of reasons. We discuss some results and open questions on finite variable logics of a model theoretic nature, concentrating on their existential fragments. In particular, many concepts from classical model theory have natural finite variable analogs, so that one can ask whether versions of classical theorems hold in a finite variable context. For example, the Los-Tarski theorem states that every first order sentence that defines a class of models that is closed under extensions is equivalent to an existential first order sentence. On the other hand, Andreka, van Benthem, and Nemeti proved that the finite variable version of this statement fails to hold.

Another line of investigation considers questions about the finite axiomatizability of theories of finite structures. For example, although the L^k theory of every finite structure is finitely axiomatized, this is not true of existential L^k theories. We present some related results and open problems.

When it comes to extracting constructive or computational information from classical proofs, variations of Gentzen's cut-elimination procedure play a central role in most proof-theoretic analyses to date. Cut elimination involves applying a sequence of transformations to the proof in question, and hence has a very syntactic flavor. In this talk I will discuss a semantic approach to the subject, which is nonetheless finitistic and computationally oriented. (This work is still in progress.)

There have been a number of striking applications of the Craig interpolation to bounded arithmetic and to propostional proof complexity. Razborov proved that, under a cryptographic assumption, certain weak fragments of arithmetic cannot prove NP is unequal to P, and his proof is based on the fact that a certain kind of interpolation theorem holds for a fragment of arithmetic. On the other hand, interpolation does not hold for Peano arithmetic. Krajicek-Pudlak have shown that if interpolation holds for certain fragments of bounded arithmetic, then commonly used number-theoretic cryptographic schemes (eg, RSA) are not secure.

The interpolation theorem holds for many propositional proof systems as well and there are close connections between feasible interpolation for propositional proof systems and interpolation for bounded arithmetic.

This talk will give an introduction to Craig interpolation for bounded arithmetic and propositional proof systems, and its applications to complexity lower bounds and to cryptography.

In the automata-theoretic approach to verification, we model programs and specifications by automata on infinite words. Correctness of a program with respect to a specification can then be reduced to the language-containment problem. In a concurrent setting, the program is typically a parallel composition of many coordinating processes, and the language-containment problem that corresponds to verification is

(*) $L(P_1) \cap L(P_2) \cap ... \cap L(P_n) \subseteq L(T)$,

where $P_1,P_2,...,P_n$ are automata that model the underlying coordinating processes, and $T$ is the task they should perform. In 1994, Kurshan suggested the heuristic of Reduction Hierarchies for circumventing the exponential blow-up introduced by conventional methods that solve the problem (*). In the reduction-hierarchy heuristic, we solve the problem (*) by solving a sequence of easier problems, which involve only automata of tractable sizes. Complexity-theoretic conjectures imply that there are settings in which the heuristic cannot circumvent the exponential blow-up. In this talk, we demonstrate the strength of the heuristic, study its properties, characterize settings in which it performs effectively, and suggest a method for searching for reduction hierarchies.

This is a joint work with Robert Kurshan and Mihalis Yannakakis.

Large software design projects often begin with a requirements capture and analysis phase. So far, there are few tools that a designer can rely on to support these tasks. We show that a portion of the requirements, relating the architecture and the temporal behavior of the new system, can be captured in formalized message sequence charts. In this talk, a set of small, light-weight, design tools will be presented. THese tools support the requirements engineering phase with standardized descriptions, and automated checks. The design of these tools focusses on the use of graphical interfaces to avoid special formalisms and hide the actual algorithms that are used by these tools.

In this paper we define {\it timed regular expressions}, an extension of regular expressions for specifying sets of dense-time discrete-valued signals. We show that this formalism is equivalent in expressive power to the timed automata of Alur and Dill by providing a translation procedure from expressions to automata and vice versa. The result is extended to omega-regular expressions.

It is demonstrated how a Hyperset Theory (satisfying P.Aczel's Anti-Foundation Axiom) naturally arises from the idea of World-Wide Web (WWW). Alternatively, Web serves as an illustration and possible application of the abstract notion of antifounded sets. A $\Delta$-language of Bounded Hyperset Theory is presented as a query language to the Web or, more generally, to Web-like Data Bases (WDB). It is shown that it describes exactly all abstract (or generic, up to bisimulation) PTIME-computable queries with respect to (possibly cyclic) graph encoding of hypersets. This result is based (i) on reducing of the $\Delta$-language for hypersets to the language FO+LFP ( = First-Order Logic with the Least Fixed-Point operator) over graphs considered up to bisimulation relation and (ii) on definability in FO+LFP of a linear ordering on any strongly extensional finite graph by a single formula (cf.\ also DIMACS TR-97-05). The case of finitely branching, possibly infinite graphs and corresponding hypersets is also discussed. It corresponds to finitely-branching, but infinite Web. However, it deserves special further investigation.

The talk is based on DIMACS TR-97-21 and also 97-05 by Alexei Lisitsa and Vladimir Sazonov. Cf.

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Document last modified on July 2, 1997.