Yuri Gurevich

University of Michigan | U-M · Division of Computer Science and Engineering

Recent analysis of classical algorithms resulted in their axiomatization as transition systems satisfying some simple postulates, and in the formulation of the Abstract State Machine Theorem, which assures us that any classical algorithm can be emulated step-by-step by a most general model of computation, called an ``abstract state machine''. We re...

The Fall 2024 Logic in Computer Science column of the Bulletin of EATCS is a little discussion on intelligence, measuring intelligence, and related issues, provoked by a fascinating must-read article ``On the measure of intelligence'' by Fran\c{c}ois Chollet. The discussion includes a modicum of critique of the article.

A hundred years ago, logic was almost synonymous with foundational studies. The ongoing AI revolution raises many deep foundational problems involving neuroscience, philosophy, computer science, and logic. The goal of the following dialog is to provoke young logicians with a taste for foundations to notice the foundational problems raised by the AI...

According to Dirac's bra-ket notation, in an inner-product space, the inner product $\langle x\,|\,y\rangle$ of vectors $x,y$ can be viewed as an application of the bra $\langle x|$ to the ket $|y\rangle$. Here $\langle x|$ is the linear functional $|y\rangle \mapsto \langle x\,|\,y\rangle$ and $|y\rangle$ is the vector $y$. But often -- though not...

Primal logic arose in access control; it has a remarkably efficient (linear time) decision procedure for its entailment problem. But primal logic is a general logic of information. In the realm of arbitrary items of information (infons), conjunction, disjunction, and implication may seem to correspond (set-theoretically) to union, intersection, and...

Model theory was born and developed as a part of mathematical logic. It has various application domains but is not beholden to any of them. A priori, the research area known as finite model theory would be just a part of model theory but didn't turn out that way. There is one application domain -- relational database management -- that finite model...

We establish the following \emph{input independence} principle. If a quantum circuit C computes a unitary transformation along a computation path, then the probability that computation of C follows the path is independent of the input.

Being asked to write about Asan D. Taimanov, I had a little problem. I knew Taimanov. I liked and respected him, and so I wanted to write something. But I didn't know him well. I didn't live in the Novosibirsk Akademgorodok where he lived and worked; I just visited Novosibirsk a number of times. So I decided to widen the topic and write about my ow...

We show that, on the abstraction level of quantum circuit diagrams, quantum circuit algorithms belong to the species of interactive sequential algorithms that we studied in earlier work. This observation leads to a natural specification language for quantum circuit algorithms.

We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the simulation proof to the quantum case.

We illustrate the glorious history of logical foundations and discuss the uncertain future.

In article "Sequential abstract state machines capture sequential algorithms", one of us axiomatized sequential algorithms by means of three postulates: sequential time, abstract state, and bounded exploration postulates. Here we give a more abstract version of the bounded exploration postulate which is closer in spirit to the abstract state postul...

We define syntax and semantics of quantum circuits, allowing measurement gates and classical channels. We define circuit-based quantum algorithms and prove that, semantically, any such algorithm is equivalent to a single measurement that depends only on the underlying quantum circuit. Finally, we use our formalization of quantum circuits to state p...

In every state of a quantum particle, Wigner's quasidistribution is the unique quasidistribution on the phase space with the correct marginal distributions for position, momentum, and all their linear combinations.

To an extent, the 1966 congress was a hole in the iron curtain. At least that how a young Soviet mathematician saw it.

We construct reversible Boolean circuits efficiently simulating reversible Turing machines. Both the circuits and the simulation proof are rather simple. Then we give a fairly straightforward generalization of the circuits and the simulation proof to the quantum case.

We define syntax and semantics of quantum circuits, allowing measurement gates and classical channels. We define circuit-based quantum algorithms and prove that, semantically, any such algorithm is equivalent to a single measurement that depends only on the underlying quantum circuit. Finally, we use our formalization of quantum circuits to state p...

Let $H_1, H_2$ be Hilbert spaces of the same finite dimension $\ge2$, and $C$ an arbitrary quantum circuit with (principal) input state in $H_1$ and (principal) output state in $H_2$. $C$ may use ancillas and produce garbage which is traced out. $C$ may employ classical channels and measurement gates. If $C$ computes, for each computation path $\mu...

This is an attempt to illustrate the glorious history of logical foundations and to discuss the uncertain future.

Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of t...

Boolean and quantum circuits have commonalities and differences. To formalize the syntactical commonality we introduce syntactic circuits where the gates are black boxes. Syntactic circuits support various semantics. One semantics is provided by Boolean circuits, another by quantum circuits. Quantum semantics is a generalization of Boolean but, bec...

The Church-Turing thesis asserts that if a partial strings-to-strings function is effectively computable then it is computable by a Turing machine. In the 1930s, when Church and Turing worked on their versions of the thesis, there was a robust notion of algorithm. These traditional algorithms are known also as classical or sequential. In the orig...

Boolean and quantum circuits have commonalities and differences. To formalize the syntactical commonality we introduce syntactic circuits where the gates are black boxes. Syntactic circuits support various semantics. One semantics is provided by Boolean circuits, another by quantum circuits. Quantum semantics is a generalization of Boolean but, bec...

The Church-Turing thesis asserts that if a partial strings-to-strings function is effectively computable then it is computable by a Turing machine. In the 1930s, when Church and Turing worked on their versions of the thesis, there was a robust notion of algorithm. These traditional algorithms are known also as classical or sequential. In the origin...

Computation models and specification methods seem to be worlds apart. The project on abstract state machines (in short ASMs, also known as evolving algebras) started as an attempt to bridge the gap by improving on Turing's thesis. We sought more versatile machines which would be able to step-for-step simulate arbitrary algorithms on their natural a...

In category-theoretic models for the anyon systems proposed for topological quantum computing, the essential ingredients are two monoidal structures, $\oplus$ and $\otimes$. The former is symmetric but the latter is only braided, and $\otimes$ is required to distribute over $\oplus$. What are the appropriate coherence conditions for the distributiv...

In computer science, category theory remains a contentious issue, with enthusiastic fans and a skeptical majority. Categories were introduced by Samuel Eilenberg and Saunders Mac Lane as an auxiliary notion in their general theory of natural equivalences. Here we argue that something like categories is needed on a more basic level. As you work with...

Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of t...

A signed probability distribution may extend a given traditional probability from observable events to all events. We formalize and illustrate this approach. We also illustrate its limitation. We argue that the right question is not what negative probabilities are but what they are for.

No-go theorems prove that, under reasonable assumptions, classical hidden-variable theories cannot reproduce the predictions of quantum mechanics. Traditional no-go theorems proved that hidden-variable theories cannot predict correctly the values of observables. Recent expectation no-go theorems prove that hidden-variable theories cannot predict th...

Hidden-variable (HV) theories allege that a quantum state describes an ensemble of systems distinguished by the values of hidden variables. No-go theorems assert that HV theories cannot match the predictions of quantum theory. The present work started with repairing flaws in the literature on no-go theorems asserting that HV theories cannot predict...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the eighth publication in the Perspectives in Logic series, brings together several directi...

While p-values are widely used and discussed, we do not know a careful definition of the concept in the literature. We sought to fill in the gap. In the process we discovered that the traditional notion of test statistic is too narrow, and this article indicates a natural generalization.

We explain the concept of p-values presupposing only rudimentary probability theory. We also use the occasion to introduce the notion of p-function, so that p-values are values of a p-function. The explanation is restricted to the discrete case with no outcomes of zero probability. We are going to address the general case elsewhere.

In 2014, the Development in Language Theory conference took place in the city of Yekaterinburg, in the Ural mountains of Russia. I used to live there. In Soviet times, there were no international conferences in the city. The whole region of the Urals was closed to foreigners. As I walked the streets of Yekaterinburg, I thought of my friends there a...

We explain the use of category theory in describing certain sorts of anyons . Yoneda’s lemma leads to a simplification of that description. For the particular case of Fibonacci anyons , we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.

An item of your personal information is inversely private if some party has access to it but you do not. We analyze the provenance of inversely private information and its rise to dominance over other kinds of personal information. In a nutshell, the inverse privacy problem is unjustified inaccessibility to you of your inversely private information...

No-go theorems assert that hidden-variable theories, subject to appropriate hypotheses, cannot reproduce the predictions of quantum theory. We examine two species of such theorems, value no-go theorems and expectation no-go theorems. The former assert that hidden-variables cannot match the predictions of quantum theory about the possible values res...

In a 2008 paper, Spekkens improved the traditional notions of non-negativity of Wigner-style quasi-probability distributions and non-contextuality of observations. He showed that the two improved notions are equivalent to each other. Then he proved what he called an even-handed no-go theorem. The paper contains some minor inaccuracies and one false...

This article was written for the Logic in Computer Science column in the February 2015 issue of the Bulletin of the European Association for Theoretical Computer Science. The intended audience is general computer science audience. The uncertainty principle asserts a limit to the precision with which position x and momentum p of a particle can be kn...

We explain the use of category theory in describing certain sorts of anyons. Yoneda's lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.

Alan Turing pioneered semantics-to-syntax analysis of algorithms. It is a kind of analysis where you start with a large semantically defined species of algorithms, and you finish up with a syntactic artifact, typically a computation model, that characterizes the species. The task of analyzing a large species of algorithms seems daunting if not impo...

We address the problem of optimal representation of single-qubit rotations in a certain unitary basis consisting of the so-called V gates and Pauli matrices. The V matrices were proposed by Lubotsky, Philips, and Sarnak [Commun. Pure Appl. Math. 40, 401-420 (1987)] as a purely geometric construct in 1987 and recently found applications in quantum c...

Primal infon logic was proposed by Gurevich and Neeman as an efficient yet expressive logic for policy and trust management. It is a propositional multimodal subintuitionistic logic decidable in linear time. However in that logic the principle of the replacement of equivalents fails. For example, x∧y→z does not entail y∧x→z, and similarly w→x∧y∧z d...

We consider the transformations of quantum states obtainable by a process of the following sort. Combine the given input state with a specially prepared initial state of an auxiliary system. Apply a unitary transformation to the combined system. Measure the state of the auxiliary subsystem. If (and only if) it is in a specified final state, conside...

In the first part of the paper, we develop a general method for converting derivability problems, from a broad range of deductive systems, into the derivability problem in a quite specific system, namely the Datalog fragment of universal Horn logic. In this generality, the derivability problems may not be recursively solvable, let alone feasible; i...

Most modern applications are empowered by online services, so application developers frequently implement authentication and authorization. Major online providers, such as Facebook and Microsoft, provide SDKs for incorporating authentication services. This paper considers whether those SDKs enable typical developers to build secure apps. Our work f...

Primal infon logic (PIL) was introduced in 2009 in the framework of policy and trust management. In the meantime, some generalizations appeared, and there have been some changes in the syntax of the basic PIL. This article is on the basic PIL, and one of our purposes is to ‘institutionalize’ the changes. We prove a small-model theorem for the propo...

A state component saves a present state of a program or model. This state component can be invoked by the program or model itself, thereby making state a first-class citizen. As the state of the program evolves from the saved state, the saved state remains for reflection and recall, for example, for testing, verification, transaction processing, et...

Primal infon logic was introduced in 2009 in connection with access control. In addition to traditional logic constructs, it contains unary connectives p said indispensable in the intended access control applications. Propositional primal infon logic is decidable in linear time, yet suffices for many common access control scenarios. The most obviou...

http://drops.dagstuhl.de/opus/volltexte/2013/4174/

We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis. The $V$ basis is an alternative universal basis to the more commonly studied $\{H,T\}$ basis. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in prec...

Many prior trust management frameworks provide authorization logics for specifying policies based on distributed trust. However, to implement a security protocol using these frameworks, one usually resorts to a general-purpose programming language. To reason about the security of the entire system, one must study not only policies in the authorizat...

Our main goal is to put Datalog into a proper logic perspective. It may be too early to put Datalog into a proper perspective from the point of view of applications; nevertheless we discuss why Datalog pops up so often in applications.

How can one possibly analyze computation in general? The task seems daunting if not impossible. There are too many different kinds of computation, and the notion of general computation seems too amorphous. As in quicksand, one needs a rescue point, a fulcrum. In computation analysis, a fulcrum is a particular viewpoint on computation that clarifies...

The logic core of Distributed Knowledge Authorization Logic, DKAL, is constructive logic with a quotation construct said . This logic is known as the logic of infons. The primal fragment of infon logic is amenable to linear time decision algorithms when policies and queries are ground. In the presence of policies with variables and implicit univers...

Knowledge and information are central notions in DKAL, a logic based authorization language for decentral-ized systems, the most expressive among such languages in the literature. Pieces of information are called infons. Here we present DKAL 2, a surprisingly simpler version of the language that expresses new important scenarios (in addi-tion to th...

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn't particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenari...

Machines and Recursive Definitions2.1 Abstract MachinesThe best-known model of mechanical computation is (still) the first, introducedby Turing [18], and after half a century of study, few doubt thetruth of the fundamental Church-Turing Thesis : A function f : N#Non the natural numbers (or, more generally, on strings from a finite alphabet) is comp...

Gurevich and Neeman introduced Distributed Knowledge Authorization Language (DKAL). The world of DKAL consists of communicating principals computing their own knowledge in their own states. DKAL is based on a new logic of information, the so-called infon logic, and its efficient subsystem called primal logic. In this article, we simplify Kripkean s...

John organized a state lottery, and his wife won the main prize. One may feel that the event of her winning isn’t particularly random, but is it possible to convincingly impugn the alleged randomness, in cases like this, in a fair court of law? We develop an approach to do just that. We start with a principle that bridges between probability theor...

Quisani: I’ve been thinking about zero-one laws for first-order logic. I know it’s a rather old topic, but I noticed something in the literature that I’d like to understand better. The first proof [6] established that, for any first-order sentence σ in a finite relational vocabulary ϒ, the proportion of models of σ among all ϒ-structures with base...

We propose an extension of the behavioral theory of interactive sequential algorithms to deal with the following situation. A query is issued during a certain step, but the step ends before any reply is received. Later, a reply arrives, and later yet the algorithm makes use of this reply. By a persistent query, we mean a query for which a late repl...

Infons are statements viewed as containers of information (rather then representations of truth values). The logic of infons turns out to be a conservative extension of logic known as constructive or intuitionistic. Distributed Knowledge Authorization Language uses additional unary connectives “p said” and “p implied” where p ranges over principals...

Propositional primal logic, as defined by Gurevich and Neeman, has two kinds of quotations: p said f, and p implied f. Note 1. The derivation problem for propositional primal logic with one kind of quotations is solvable linear time. Note 2. In the Hilbertian calculus for propositional primal logic, the shortest derivation of a formula f from hypot...

The logic core of Distributed Knowledge Authorization Logic, DKAL, is constructive logic with a quotation construct "said". This logic is known as the logic of infons. The primal fragment of infon logic is amenable to linear time decision algorithms when policies and queries are ground. In the presence of policies with variables and implicit univer...

Consider interaction of principals where each principal has its own policy and different principals may not trust each other. In one scenario the principals could be pharmaceutical companies, hospitals, biomedical labs and health related government institutions. In another scenario principals could be navy fleets of different and not necessarily fr...

Recent analysis of sequential algorithms resulted in their ax- iomatization and in a representation theorem stating that, for any se- quential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted from details of intra-step computation, and the ASM, pro- du...

When a file is to be transmitted from a sender to a recipient and when the latter already has a file somewhat similar to it, remote differential compression seeks to determine the similarities interactively so as to transmit only the part of the new file not already in the recipient's old file. Content-dependent chunking means that the sender and r...

The tower-of-Babel problem is rather general: How to enable a collaboration among experts speaking different languages? A computer security version of the tower-of-Babel problem is rather important. A recent Microsoft solution for that security problem, called Security Assessment Sharing, is based on this idea: A tiny common language goes a long wa...

DKAL is a new expressive high-level authorization language. It has been successfully tried at Microsoft which led to further improvements of the language itself. One improvement is the separa- tion of concerns between static core policies and dynamic workflow; im- portant safety properties can be proved from the core policies alone, independently f...

Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. If x is an element of X, let Bn (x) be the ball of radius n around x. Suppose that X is infinite, connected and of bounded degree. A first-order sentence ϕ in the language of X is almost surely true (resp. a.s. false) for finite substructures...

Abstract State Machines (ASMs) allow modeling system behaviors at any desired level of abstraction, including a level with rich data types, such as sets, sequences, maps, and user-deflned data types. The availability of high-level data types allow state elements to be represented both abstractly and faithfully at the same time. In this paper we loo...

Infons are pieces of information. In our work on the Dis-tributed Knowledge Authorization Language (DKAL), we discovered that the logic of infons is a conservative exten-sion of intuitionistic logic by means of connectives p said and p put where p ranges over principals. We investigate infon logic and a primal fragment of it. In both cases, we de-v...

Infons are pieces of information. In our work DKAL (Distributed Knowledge Authorization Language), we discovered that logic of infons is a conservative extension of intuitionistic logic by means of connectives "p said" and "p put" where p ranges over principals. Here we investigate infon logic and a primal fragment of it. In both cases, we formulat...

We share our experience of using abstract state machines for teaching computation theory at the University of Michigan.

A natural liberalization of Datalog is used in the Distributed Knowledge Authorization Language (DKAL). We show that the expressive power of this liberal Datalog is that of existential fixed-point logic. The exposition is self-contained. 1. Prologue Existential fixed point logic (EFPL) differs from first-order logic by prohibiting universal quantif...

In connection with machine arithmetic, we are interested in systems of constraints of the form x + k \leq y + k'. Over integers, the satisfiability problem for such systems is polynomial time. The problem becomes NP complete if we restrict attention to the residues for a fixed modulus N.

People usually regard algorithms as more abstract than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue that no such equivalence relation exists.

We propose a syntax and semantics for interactive abstract state machines to deal with the following situation. A query is issued during a certain step, but the step ends before any reply is received. Later, a reply arrives, and later yet the algorithm makes use of this reply. By a persistent query, we mean a query for which a late reply might be u...

Existential fixed point logic (EFPL) is a natural fit for some applications, and the purpose of this talk is to attract attention to EFPL. The logic is also interesting in its own right as it has attractive properties. One of those properties is rather unusual: truth of formulas can be defined (given appropriate syntactic apparatus) in the logic. W...

The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church's...

Existential fixed point logic (EFPL) is a natural fit for some applications, and the purpose of this talk is to attract attention to EFPL. The logic is also interesting in its own right as it has attractive properties. One of those properties is rather unusual: truth of formulas can be defined (given appropriate syntactic apparatus) in the logic. W...

DKAL is a new declarative authorization language for distributed systems. It is based on existential fixed-point logic and is considerably more expressive than existing authorization languages in the literature. Yet its query algorithm is within the same bounds of computational complexity as e.g. that of SecPAL. DKAL's communication is targeted whi...

We consider parallel algorithms working in sequential global time, for example circuits or parallel random access machines (PRAMs). Parallel abstract state machines (parallel ASMs) are such par- allel algorithms, and the parallel ASM thesis asserts that every parallel algorithm is behaviorally equivalent to a parallel ASM. In an earlier paper, we a...

The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many well-founded relations U 1 , . . . , U n then R is well-founded. A question arises how to bound the ordinal height |R| of the relation R in terms of the ordinals α i = |U i |. We introduce the notion of the stature P...

Data streams are modeled as infinite or finite sequences of data elements coming from an arbitrary but fixed universe. The uni- verse can have various built-in functions and predicates. Stream queries are modeled as functions from streams to streams. Both timed and un- timed settings are considered. Issues investigated include abstract defini- tion...