Logical Seminar of POMI. 11 January 1999
Speakers: Evgeny Dantsin (St.Petersburg) and Andrei Voronkov (Uppsala)
Title: "Expressive power and data complexity of nonrecursive logic
programming."
ABSTRACT
In the context of databases, first-order logic without function symbols
is viewed as a query language over relational databases. This language is
equivalent to nonrecursive Datalog$^\neg$ in the sense that they express
the same set of queries. First-order logic with function symbols is a query
language over databases with complex values, namely trees. This language
is equivalent to nonrecursive logic programming. The expressiveness and
complexity of nonrecursive Datalog$^\neg$ are well-studied. We characterize
the expressiveness and complexity of nonrecursive logic programming. Our
results show that these two languages express the same set of queries
(under a natural representation of inputs). Thus, the use of recursive
data structures, namely trees, in nonrecursive Datalog$^\neg$ gives no
gain in the expressive power. It also follows from our results that
nonrecursive logic programming, like nonrecursive Datalog$^\neg$, has
polynomial data complexity. This contrasts with a huge difference between
these query languages in the program complexity. All of our results are
based on a technical theorem that plays the same role as quantifier
elimination in term algebras.
Logical Seminar of POMI. 25 January 1999
Speakers: Vladimir Grebinski.
Title: "Combinatorial Search: from coin weighing to
reconstruction of graphs"
ABSTRACT
In this talk we will consider the \emph{additive} model of
Combinatorial Search
The main subject of this talk is the so-called Additive Combinatorial
Search Model. We study the power of this model by considering two
problems: reconstruction of bounded weight vectors and reconstruction
of $k$--degenerated graphs. We discuss also a connection between these
problems and a problem of a coin weighing with a spring scale.
V ponedel'nik 27 sentjabrja na ocherednom zasedanii seminara
laboratorii matematicheskoj logiki POMI sostoitsja doklad
Michal Mnuk (Technische Universitaet Muenchen,
Munich, Germany)
Computing with Polynomial Ideals: Algorithms and Complexity
Nachalo v 14.30, komnata 203.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ABSTRAKT
Many tasks in a diversity of areas of science and technology
may be formulated in the language of polynomial ideals. In our talk we
describe some problems which naturally arise when performing
computations in polynomial ideals, sketch their solutions and estimate
the complexity. The technique of Groebner bases as a basic algorithmic
tool will be explained in more detail and applied to problems like:
ideal membership, computing dimension, radical, and others.
Logical Seminar of POMI. 27 September 1999
Speaker: E.A.Hirsch (POMI)
Title: "News on SAT"
ABSTRACT
The talk will consists of three (independent) parts.
I. A new result [Hirsch,99] on (exact) solution of MAX-2-SAT.
We describe a relatively simple algorithm which solves MAX-2-SAT in time
of the order 2^{K/4}, where K is the number of 2-clauses. This improves
recent results 2^{K/2.66} [Niedermeier,Rossmanith,98]
and 2^{K/3.76} [implicit from Bansal,Raman,99],
and drastically improves the algorithm.
The key idea of the new algorithm is a reformulation in terms of maximum
symmetric flow (this idea was introduced by Yannakakis for an approximate
algorithm).
II. A recent result of Uwe Sch\"oning on 3-SAT solution in time of the order
(4/3)^N (N is the number of variables) using a randomized algorithm.
The algorithm is very simple (the whole paper consists of 3 pages),
which drastically differs from the previous known ones
(for the bounds 1.36^N [Paturi,Pudlak,Saks,Zane,98: 20 pp.] and
1.5^N [Kullmann,93: 70 pp.]).
III. News on SAT: informal survey.
The last year brought an explosion of new results related to SAT;
now some well-known people work in the subject.
Logical Seminar of POMI. 27 December 1999 Speaker: D.Yu.Grigorev Title: "Complexity lower bounds for algebraic proofs"
Logical Seminar of POMI. 25 January 1999
Speakers: E.Dantsin, E.A.Hirsch
Title: "Algorithms for k-SAT based on covering codes"
ABSTRACT
We show that for any $k$ and $\epsilon$, satisfiability of
propositional formulas in $k$-CNF can be checked by a deterministic
algorithm running in time $poly(n) (2k/(k+1) + \epsilon)^n$, where
$n$ is the number of variables in the input formula. This is the best
known worst-case upper bound for deterministic $k$-SAT algorithms.
Our algorithm can be viewed as a derandomized version of Sch\"oning's
recent algorithm (FOCS'99) whose bound $poly(n) (2(k-1)/k)^n$ is the
best known bound for probabilistic 3-SAT algorithms. The key point of
our derandomization is the use of covering codes.
Like Sch\"oning's algorithm, our algorithm is quite simple. We show
how to obtain slightly improved bounds by using a more thorough
(but a more intricate) version of the algorithm. For example, for
3-SAT the modified algorithm gives the bound $poly(n) 1.490^n$.
V ponedel'nik, 3 aprelja, na ocherednom zasedanii seminara
laboratorii matematicheskoj logiki POMI sostoitsja doklad
E.A.Girsha o poezdke na konferentsii STACS-2000 i
DIMACS Workshop on Faster Exact Solutions for NP-hard problems
Nachalo v 14.30, komnata 203.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ABSTRAKT
Osnovu doklada sostavit obzor novostej s upomjanutyh konferentsij i
drugih rezul'tatov, uznannyh (i/ili poluchennyh) po doroge i
neposredstvenno posle. Primernyj spisok tem:
-- nedeterminirovannaja slozhnost' primera (instance complexity)
i trudnodokazuemye tavtologii;
-- shemy protiv derev'ev v algebraicheskih sistemah dok-va;
-- trudnye primery trudnyh jazykov;
-- optimal'nye sistemy dokazatel'stva i "redkie" (sparse) jazyki;
-- \lambda-raskraski;
-- trudnye (hard-core) predikaty;
-- rezul'taty o "slabosti" (lowness) slozhnostnyh klassov
(slozhnostnoj klass A slab dlja B, esli B^A=B);
-- uluchshennye verhnie otsenki dlja MAX-2-SAT;
-- ispol'zovanie lokal'nogo poiska dlja priblizhennogo reshenija MAX-k-SAT;
-- mozhno li reshit' k-SAT bystree, chem SAT?;
-- ...
Logical Seminar of POMI. 3 April 2000
Speaker: E.A.Hirsch
Title: "News from STACS-2000 and
DIMACS Workshop on Faster Exact Solutions for NP-hard Problems"
ABSTRACT
In the talk I will survey news from the conferences mentioned above
and other results about which I was informed (+obtained)
during (and right after) the trip. Tentative list of subjects:
-- Nondeterministic instance complexity and hard-to-prove tautologies;
-- Circuits vs trees in algebraic complexity;
-- Hard instances of hard problems;
-- Optimal proof systems and sparse sets;
-- \lambda-coloring;
-- Hard-core predicates;
-- Lowness results (a complexity class A is low for B if B^A=B).
-- Better upper bounds for MAX-2-SAT;
-- Using local search for approximating MAX-k-SAT;
-- Is k-SAT solvable faster than SAT?;
-- ...
V ponedel'nik 29 maja na ocherednom zasedanii seminara laboratorii
matematicheskoj logiki POMI sostoitsja doklad S.S. Lavrova
O nachalah i granitsah matematiki
(aprel'skie tezisy)
Nachalo v 14.30, komnata 203.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ABSTRAKT
U matematiki net drugih kornej i nachal (osnovanij) krome estestvennogo jazyka.
V teorii mnozhestv net logicheskih paradoksov, a tol'ko semanticheskie.
Kazhdyj matematik imeet pravo na lichnuju tochku zrenija o predmete matematiki.
Matematicheskij ekstremizm vreden, kak i vsjakij drugoj.
Dlja menja kontinuum - eto "pamjat'", raspredeljaemaja po hodu rassuzhdenij
pod znachenija, nazyvaemye "veschestvennymi chislami".
Popytka postroit' na osnove etogo predstavlenija veschestvennoe chislo, ne
prinadlezhaschee kontinuumu, mozhet pokazat'sja ubeditel'noj tol'ko ee avtoru.
Diagonal'nym metodom nel'zja dokazat' neschetnost' kontinuuma.
Ne pomogaet on i oprovergnut' suschestvovanie algorifma reshenija tipichnoj
massovoj problemy.
Abstraktnye mashiny i modeli algorifmov pervoj poloviny XX veka
ustareli v komp'juternuju epohu.
Neplohoj abstraktnoj mashinoj mozhet posluzhit' bazovyj Lisp (prikladnoe
lambda-ischislenie).
Suschestvovanie universal'nogo algorifma ochevidno, ne stoit tratit' sily
na ego postroenie.
Esli algorifm - predpisanie, to nalichie mashiny, ispolnjajuschej algorifmy,
samo soboj podrazumevaetsja.
V Lispe est' vozmozhnost' vo vremja ispolnenija algorifma obratit'sja k
Lisp-mashine dlja ispolnenija drugogo algorifma, zaranee ili tol'ko chto
postroennogo.
Traditsionnyj sposob postroenija algorifma, pro kotoryj nel'zja
utverzhdat' ni chto on obadaet nekotorym netrivial'nym svojstvom, ni chto
on ne obladaet im, privodit lish' k algorifmu, ispolnenija kotorogo
nikogda ne zavershaetsja.
Eta nezavershaemost' dostatochno prosto obnaruzhivaetsja formal'nymi
metodami.
Pervaja teorema Klini o rekursii predlagaet v kachestve reshenija
rassmatrivaemogo v nej funktsional'nogo uravnenija aktual'no-beskonechnyj
ob"ekt.
V drugoj teoreme o rekursii vsego lish' dokazyvaetsja, chto predlagaemaja
funktsija javljaetsja nepodvizhnoj tochkoj rassmatrivaemogo funktsionala.
Net effektivnogo metoda postroenija etoj funktsii.
V obschem sluchae proverit', chto algorifm, predlagaemyj dlja reshenija
nekotoroj massovoj problemy, obladaet trebuemym svojstvom, mozhno
tol'ko uspeshno ispolniv etot algorifm.
Universal'nyj metod postroenija denotatsionnoj semantiki programm - mif.
Vozmozhnost' avtomaticheskogo sinteza programm po ee formal'noj
spetsifikatsii - takoj zhe mif.
--
Sviatoslav S. Lavrov
Phone: +7(812)2729279
mailto:ssl@lvr.usr.pu.ru
============================================================================
Logical Seminar of POMI. 04 September 2000
Speaker: E.A.Hirsch
Referat of Razborov & Rudich's paper "Natural Profs"
ABSTRACT
We are interested in lower bounds on the complexity of Boolean functions.
For a complexity class K, a "natural" proof against K is a proof using
a combinatorial property $C_n$ of Boolean functions that
(1) is efficiently checkable;
(2) defines a relatively large set of functions;
(3) $C_n(f)$ implies that $f \not\in K$.
It can be shown that (the most part of?) known proofs are "natural".
The main two theorems of the paper are:
THEOREM 1. Under a certain cryptographic assumption,
there is no "natural" proof against P/poly.
THEOREM 2. (Unconditionally) there is no "natural" proof that
Discrete Logarithm has no subexponential-size circuits.
The talk will contain (at least) the proof of the first theorem.
Logical Seminar of POMI. 18 September 2000
Speakers: Volker Diekert
Title: "The existential theory of equations with rational
constraints in free groups is $\PSPACE$--complete"
ABSTRACT
Famous results of Makanin state that the existential theory of
equations is decidable in free monoids and in free groups. The
original algorithms of Makanin are extremely complex, but recently
Plandowski invented a new method for solving word equations and he
showed that the satisfiability problem for word equations can be
solved in polynomial space. Shortly afterwards GutiIrrez extended his
method to the case of free groups.
In my lecture I will show that Plandowski's method also copes with
rational constraints, i.e., we can specify rational subsets for the
solutions. This more general form is necessary, e.g. for solving
equations in graph groups. The result is quite close to the borderline
of decidability, because the positive theory of equations with rational
constraints in free groups becomes undecidable (in contrast to the
case without constraints).
Logical Seminar of POMI. 18 December 2000
Speakers: Romas Alonderis (Lithuania, Vilnius)
Title: "Proof-Theoretical Investigation of Intuitionistic Temporal
Logic with Time Gaps"
ABSTRACT
A first-order intuitionistic temporal logic TBJ with temporal
operators "next" and "always" is considered. A sequent calculus
LBJ for TBJ is constructed. Invertibility of some LBJ rules and
admissibility of the structural rules and the cut rule in LBJ are
proved. Also, Harrop's theorem, Craig interpolation theorem and
Gentzen mid-sequent theorem are proved for LBJ. It is given a
solution to the LBJ loop rule specialization problem. LBJ
soundness for TBJ is proved. Purely Glivenko sigma-classes and
Glivenko sigma-classes with respect to LBJ and its classical
counterpart LB are given, necessary and sufficient conditions being
proved. It is showed that LBJ is incomplete for TBJ and that purely
Glivenko and Glivenko sigma-classes can be regarded as LBJ
completeness for TBJ classes.
Logical Seminar of POMI. 16 April 2001
Speakers:
Andrei Muchnik (Moscow)
Title: "Comparison of different approaches to definition of concept
of randomness"
ABSTRACT
A.N. Kolmogorov proposed two methods of definition of
randomness of infinite binary sequence. In the first case
required that frequencies of zeros and units in subsequences
selected with the aid of computable rules of selection
converges to 1/2. In the second case required all onsets of
sequence have Kolmogorov's complexity near the length.
Kolmogorov proved that it is sequence with logarithmic growth
of complexity of onsets for which every subsequence selected
with the aid of monotonic computable rules of selection have
limit of frequencies of 1/2. In the same paper Kolmogorov
supposed that his result would be true for selecting of
subsequences with no monotonic computable rules of selection. I
disproved this supposition. Really, the truth of the law of
large numbers for every subsequence selected with the aid of
computable rules of selection means at least linear growth of
onsets.
Logical Seminar of POMI. 21 May 2001
Speakers: Michael Gelfond (Comp. Science Dept.
Texas Tech University)
Title: "Programming in A-Prolog,"
ABSTRACT
A-Prolog is a language for knowledge representation
and reasoning based on the stable model semantics of
logic programs.
In this talk I'll define the syntax and semantics of
A-Prolog, give several examples of its use, and discuss
some mathematical properties of A-Prolog knowledge bases.
Logical Seminar of POMI. 4J une 2001
Speakers: Regimantas Pliuskevicius
Institute of Mathematics and Informatics
Akademijos 4, Vilnius 2600, LITHUANIA
email: regis@ktl.mii.lt
Title: "Saturation method for a linear first-order temporal logic"
ABSTRACT
The aim of this report is to present a saturation method for fragments
of first order linear temporal logic (FOLT, in short). The proposed
saturation method is aimed for the verification of loop properties. This
verification is carried out automatically by means of some deductive
procedures that replace an infinitary omega-type rule for the induction-type
temporal operator. In the report this method is demonstrated on the
following fragments of FOLT:
--- miniscoped fragment of FOLT
--- Horn-like and non-Horn-like fragments of Fisher--Orevkov normal
form.
The main attention in the report is devoted to Horn-like fragment of Fisher--
Orevkov normal form.
It is shown that the proposed saturation method for fragments of FOLT
allows:
--- to prove the completeness and decidability of these fragments
--- to present a rather efficient method for verification of deducibility.
Logical Seminar of POMI. 17 June 2001
Speakers: G. Mints (Stanford)
Title: "Epsilon substitution: past and future"
ABSTRACT
Epsilon substitution method introduced by Hilbert provides numerical
realizations of existential sentences. It attracted interest of J. von
Neumann, H. Weyl, W. Ackermann and other logicians. After original
setbacks and successes in number-theoretic setting before 1941
further progress was made in 1990-s for mathematical analysis (second
order arithmetic). We describe original formulation in the framework
of Hilbert's program, results obtained for predicative subsystems of
analysis and the most recent progress for impredicative part.
Logical Seminar of POMI. 5 July 2001
Speakers: Denis Therien
Title: "Definability of regular languages in first-order logic"
ABSTRACT:
First-order logic offers a convenient framework to
describe regular languages. I will present old and new results
about the model: in particular, deep connections with algebra
will be emphasized and shown to often yield decision procedures
to test if a language is expressible within a given class of formulas.
Interesting open questions will also be discussed.
Logical Seminar of POMI. 5 July 2001
Speakers: Edward Hirsch (Joint work with D.Grigoriev and D.Pasechnik.)
Title: "PROPOSITIONAL PROOF SYSTEMS WORKING WITH INEQUALITIES"
ABSTRACT:
We consider propositional proof systems working with inequalities.
For example, well-known Cutting Planes proof system allows to work
with linear inequalities instead of Boolean formulas; new inequalities
are derived using taking nonnegative linear combination and rounding:
e.g., from x+2y>=0.5 it follows that x+2y>=1. Many of these systems arise
from optimization theory.
The talk contains results concerning the relations between these systems.
We also show that these systems have short proofs of some well-known
tautologies. We pose numerous open questions and discuss some approaches
to their solution.
Logical Seminar of POMI. 15 October 2001
Speakers: Edward Hirsch (Joint work with D.Grigoriev and D.Pasechnik.)
Title: "PROPOSITIONAL PROOF SYSTEMS WORKING WITH INEQUALITIES"
ABSTRACT:
We consider propositional proof systems working with inequalities.
For example, well-known Cutting Planes proof system allows to work
with linear inequalities instead of Boolean formulas; new inequalities
are derived using taking nonnegative linear combination and rounding:
e.g., from x+2y>=0.5 it follows that x+2y>=1. Many of these systems arise
from optimization theory.
The talk contains results concerning the relations between these systems.
We also show that these systems have short proofs of some well-known
tautologies. We pose numerous open questions and discuss some approaches
to their solution.
Logical Seminar of POMI. 26 November 2001
Speakers: Boris Jacovlevich Solon
(Ivanovo State University of Technology and Chemistry)
e-mail: solon@icti.ivanovo.su
Title: "Non-total enumeration degrees"
ABSTRACT:
Let subsets $A$ and $B$ of the positive integers and some computer which works
by given program and has an input and an output be given (for example, a
modification of the Turing machine with exterior tape). This computer with the
way of action written below is called {\it an enumeration operator}. From time
to time a computation is interrupted for peceipt of "input" integer (and is
continued afterwards), and from time to time is yielded "output" integer. When
an input is requested, any integer, or no integer, may be supplies. If the
members of $A$ are supplied as inputs and the output integers form the set $B$
where this takes place independent of any order in which is supplied the
members of $A$ as inputs of the computer then $B$ {\it is enumeration
reducible to} $A$ (notation: $B\leq_eA$). In addition it is assumed that the
order in which the members of $B$ appear may vary dependent on the way of
presenting of members of $A$ on the input and also it is assumed repetitions
in the listing of $A$ and in the listing of $B$. The class of sets
$d_e(A)=\{B:B\leq_eA\land A\leq_eB\}$ is called {\it the enumeration degree
of} $A$ or {\it the e-degree} of $A$.
An e-degree is called {\it total} if it contains a graph of some total
function. Total e-degrees have the following characteristic property: by any
listing of any set from the total e-degree can get with help a suitable
enumeration operator the listing of this set in a fixed order. Denote by {\bf
Tot} a partially ordered set of all total e-degrees. It is well known that the
upper semilattice of Turing degrees and {\bf Tot} are isomorphic.
The first serious result about the e-reducibility consisted in a proof of the
existence of non-total e-degrees. In this work are studied the properties of
non-total e-degrees, is given some classificaion of non-total e-degrees. This
work is the first attempt of the studing of non-total e-degrees as a special
class of e-degrees.
Logical Seminar of POMI. 15 April 2002
Speakers: Andrey Bovykin
Title: "New statements independent of Peano arithmetic."
ABSTRACT:
In the first half of the talk I will show that the statement (*): \forall m
k \exists n [ n=> (m)^k ]
is unprovable in PA (though provable in second-order arithmetic) where [ n=>
(m)^k ] means:
if we assign to each {x_0Power(x_0)
then there is a subset H\subset n of size m such that
for all x_0 PA+Phi_i does not prove Phi_j.
If we have time I will also talk about connections with
PA-provably recursive functions and large cardinals in
ZFC.
Logical Seminar of POMI. 1 July 2002
Speakers: Vladimir Korepin
Title: "Introduction to Quantum Comuting"
ABSTRACT:
I will consider several subjects:
1) Brief introduciton to Quantum Mechanics
2) Grover Algorithm
3) Deutch Algorithm
4) Other Algorithms
Logical Seminar of POMI. 4 July 2002
Speakers: G.V Davydov, I.M. Davydova
Title: "Construction of hard examples for discrete optimizaion problems."
ABSTRACT:
Hard CNF can be interpreted by discrete optimization problem
in such a way that it becomes possible to constuct hard examples
for optimization problem. The interpretation of Modified Pigeon Hole formula
by the knapsack problem gives ability to find such conditions
for initial data that make the knapsack to have the unique optimal packing.
The search of this packing becomes impossible by the branch and bound method
due to the time deficit when the number of object is a mere 56.
In the same time, the random geheration of the initial data
gives the detectable calculation time only starting from about 8000 objects.
Logical Seminar of POMI. 23 August 2002
Speaker: Vladik Kreinovich (University of Texas at El Paso)
Title: "Probabilities, intervals, what next?
Extension of interval computations to situations
with partial information about probabilities"
ABSTRACT:
When we have only interval ranges [xi-,xi+] of sample values
x1,...,xn, what is the interval [V-,V+] of possible values for the
variance V of these values? We prove that the problem of computing the
upper bound V+ is NP-hard. We provide a feasible (quadratic time)
algorithm for computing the lower bound V- on the variance of interval
data. We also provide a feasible algorithm that computes V+ under
reasonable easily verifiable conditions.
We also extend the main formulas of interval arithmetic for different
arithmetic operations x1*x2 to the case when, for each input xi, in
addition to the interval [xi]=[xi-,xi+] of possible values, we also
know its mean Ei (or an interval [Ei] of possible values of the mean),
and we want to find the corresponding bounds for x1*x2 and its mean.
Logical Seminar of POMI. 23 December 2002
Speakers: Zakharov V.K., Bunina E.I.,Andreev P.V.,Mikhalev A.V. (Moskow)
Title: "MakLane's problem about set-theoretical foundation for category theory and
local theory of sets."
ABSTRACT:
In 1945 Eilenberg and MacLane introduced a new mathematical notion of the
category. However from the very beginning of its origin category theory
encountered with the unpleasant circumstance that it did not go in the
theory of sets ZF and the theory of classes and sets NBG.
By this reason in 1959 MacLane raised the general problem of constructing a
new and more flaxible axiomatic set theory that could serve as an adequate
logical foundation for all naive category theory.
>From thet time some axiomatic set theories were proposed in the capacity of
foundations for category theory. In 2000 V.K. Zakharov proposed a local
theory of classes and sets(LTS).In 2002 a comparison of theories LTS and ZF
was done by the collective of mathematicians indicated above, ans some
results on relative consistency of LTS and ZF with additional axioms about
strongly inaccessible cardinals were obtained.
Logical Seminar of POMI. 6 October 2003
Speakers: Igor Lavrov
Title: "Polureshetka m-stepenei recursivno-perechislimich mnogestv."
ABSTRACT:
IZdes' I nameren rasskazat' neskol'ko rezul'tatov o stroenii etoi
polureshetki, kotorie nosiat otnositel'no okonchatel'nii character,
otlichnii ot T-stepenei.Takge nameren rasskazat' ob opredelimosti
kreativnich mnogestv.
Logical Seminar of POMI. 13 October 2003
Speakers: Alexey Chernov (Moscow, MSU).
Title: "Realizability-like semantics for the logic of the weak law
of excluded middle"
ABSTRACT:
To clarify foundations of intuitionistic logic Kolmogorov proposed
to consider connectives as operations on problems: substituting
some original problems in a propositional formula, we get new,
combined problem. Kolmogorov said that a formula is intuitionistically
valid if there is a common solution of the combined problem for
all substitutions. Kleene's realizability and Medvedev's calculus
of finite problems were refinements of this idea.
The present talk is devoted to a modification of this approach.
Namely, we consider the set L of formulas such that to solve the
corresponding combined problem we need a small amount of additional
information, much smaller than to solve the original problems.
This set may depend on the formalization of the notion "problem",
on information measure, and on the estimation used instead of
the word "small". It turns out that for realizability-like operations
on sets and for operations on finite problems in many cases the set L
coincides with the logic of the weak law of excluded middle
(intuitionistic logic with one extra axiom (~x or ~~x)).
Logical Seminar of POMI. 20 October 2003
Speakers: Konev B.Yu.
Title: "O monodicheskom fragmente vremennoi logiki pervogo poryadka"
ABSTRACT:
Nesmotrya na bol'shuyu vyrazitel'niyu silu yazyka vremennoi logika pervogo poryadka,
eta logika redko ispol'zuetsia dlya mehanizatsii rassuzhdenij vsilu svoei
nepolnoty. Bolee togo, do nedavnego vremeni ne bylo izvestno netrivial'nyh
perechislimyh fragmentov etoi logiki. V rabote Voltera i Zaharyascheva
byla pokazana perechislimost' monodicheskogo framenta vremennoi logiki pervogo
poryadka, a tazhe razreshimost' nekotoryh ego podfragmentov. V doklade budet
rasskazano o monodicheskom fragmente, ego svoistvah, a takzhe ob avtomatizatsii
poiska dokazatel'stv v nem.
PS ne putat' monOdicheskij i monAdicheskij fragmenty. Monadicheskij fragment
ne imeet rekursivnoi aksiomatizatsii.
==========================================================================
Logical Seminar of POMI. 9 February 2004
Speakers: Vladimir Lifschitz (University of Texas)
Title: "Why Are There So Many Loop Formulas?"
ABSTRACT:
A theorem by Lin and Zhao shows how to turn any logic program, understood
in accordance with the answer set semantics, into an equivalent set of
propositional formulas. The set of formulas generated by this process can
be significantly larger than the original program. We show (assuming a
conjecture from the theory of computational complexity that is widely
believed to be true) that this is inevitable: any equivalent translation
from logic programs to propositional formulas involves a significant
increase in size.
This is joint work with Alexander Razborov.
=========================================================================
Logical Seminar of POMI. 16 February 2004
Speakers:
Arist Kojevnikov
Title: "Intuitionistic Frege systems are polynomially equivalent"
ABSTRACT
In [Cook&Reckhow 79] it is shown that any two classical Frege systems
polynomially simulate each other. The same proof does not work for the
intuitionistic Frege systems, since they can have non-derivable admissible
rules. (The rule A/B is _derivable_ if formula A -> B is derivable. The
rule A/B is _admissible_ if for all substitutions s if s(A) is derivable
then s(B) is derivable). We show how to polynomially simulate one
admissible rule. Therefore any two intuitionistic Frege system
polynomially simulate each other.
This is joint work with Grigori Mints.
=========================================================================
-------------------------------------------------------------------------------
Logical Seminar of POMI. 8 April 2004
Speakers: Thomas Sturm
Title: "Complexity and Applicability of Generalized Constraint Solving"
ABSTRACT
The REDLOG package of the computer algebra system REDUCE extends the
idea of symbolic computation from computer algebra to first order
logic. Development started around 1992 with real quantifier
elimination algorithms. Over the years the system has been
continuously extended by further domains and algorithms: real numbers,
complex numbers, p-adic numbers, and quantified propositional
calculus. The integration of Presburger arithmetic (additive theory of
the integers) is in progress, and that of free term algebras is under
consideration. Furthermore there are both the theoretical foundation
and an experimental implementation available for combining the various
domains within the framork of constraint logic programming. The talk
gives and overview on the various REDLOG domains, their application
range, and complexity issues. This will be supplemented by a brief
introduction of REMIS, a database of REDLOG examples, which is
available online since 2000.
-----
http://www.fmi.uni-passau.de/~sturm/
Logical Seminar of POMI. 24 May 2004
Speakers:
Arist Kojevnikov
Title: "Intuitionistic Frege systems are polynomially equivalent"
ABSTRACT
In [Cook&Reckhow 79] it is shown that any two classical Frege systems
polynomially simulate each other. The same proof does not work for the
intuitionistic Frege systems, since they can have non-derivable admissible
rules. (The rule A/B is _derivable_ if formula A -> B is derivable. The
rule A/B is _admissible_ if for all substitutions s if s(A) is derivable
then s(B) is derivable). We show how to polynomially simulate one
admissible rule. Therefore any two intuitionistic Frege system
polynomially simulate each other. This is joint work with Grigori Mints.
We are fixed error, that was found during the previous talk.
Logical Seminar of POMI. 20 September 2004
Speaker: Boris Konev
Title: "On dynamic topological and metric logics"
(joint work with Roman Kontchakov, Frank Wolter, and
Michael Zakharyaschev)
ABSTRACT:
Dynamic topological logics were first introduced in 1997 in a series of works
by Artemov, Davoren, Nerode, Kremer, and Mints as a logical formalism for
describing the behaviour of dynamical systems. Dynamical systems are usually
represented by some `mathematical' space W (modelling possible system states)
and a function f on W (modelling the evolution of the system), with one of the
main research problems being the study of iterations of f, in particular, the
orbits O(w) = {w,f(w),f^2(w),..} of states $w\in W$.
A natural logical formalism for speaking about such iterations is a variant of
temporal logic. To speak about the structure of the underlying space W as well
as the type of the intended functions f, one may require different non-temporal
operators. Important examples of such space W are (subspaces of) the Euclidean
spaces R^n, general topological spaces, metric spaces, and measure spaces. The
resulting combinations of temporal and topological/metric logics are of a clear
`two-dimensional character' which makes it very difficult to analyse their
computational properties.
We provide answers to some of the open problems. First, we show that some
dynamic topological logics introduced in [KremerMintsRybakov97] and interpreted
in various topological spaces with homeomorphisms are not recursively
enumerable (and so are not recursively axiomatisable). Second, we prove the
non-elementary decidability of the dynamic metric logic with distance operators
interpreted in arbitrary metric spaces with distance preserving automorphisms.
A survey of related results will also be given.
Logical Seminar of POMI. 27 September 2004
Speaker: Arist Kojevnikov
Title: "Intuitionistic Frege systems are polynomially equivalent"
ABSTRACT:
In [Cook&Reckhow 79] it is shown that any two classical Frege systems
polynomially simulate each other. The same proof does not work for the
intuitionistic Frege systems, since they can have non-derivable admissible
rules. (The rule A/B is _derivable_ if formula A -> B is derivable. The
rule A/B is _admissible_ if for all substitutions s if s(A) is derivable
then s(B) is derivable). We show how to polynomially simulate one
admissible rule. Therefore any two intuitionistic Frege system
polynomially simulate each other. This is joint work with Grigori Mints.
We are fixed error, that was found by V.V.Rybakov.
The theme of the talk was changed.
"Intuitionistic natural deduction systems are polynomially equivalent"
ABSTRACT
In [Cook&Reckhow 79] it is shown that any two classical natural deduction
systems polynomially simulate each other. The same proof does not work for
the intuitionistic natural deduction systems, since they can have
non-derivable admissible rules. (The rule A/B is _derivable_ if formula A
-> B is derivable. The rule A/B is _admissible_ if for all substitutions s
if s(A) is derivable then s(B) is derivable). We show how to polynomially
simulate one admissible rule. Therefore any two intuitionistic natural
deduction system polynomially simulate each other.
We are fixed gaps, that was found by V.V.Rybakov.
Kozhevnikov's talk at the logic seminar was canceled.
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Logical Seminar of POMI. 18 April 2005
Speaker: Gregory Gutin (Department of Computer Science Royal
Holloway, University of London Egham, Surrey, TW20 0EX, UK)
Title: "Upper Bounds on ATSP Neighborhood Size"
ABSTRACT:
Abstract: We consider the Asymmetric Traveling Salesman Problem (ATSP) =
and use
the definition of neighborhood by Deineko and Woeginger ( Math. Program. =
87
(2000) 519-542). Let $\mu(n)$ be the maximum cardinality of polynomial =
time
searchable neighborhood for the ATSP on $n$ vertices. Deineko and =
Woeginger
conjectured that $\mu (n)< \beta (n-1)!$ for any constant $\beta >0$ =
provided
P$\neq$NP. We prove that $\mu(n) < \beta (n-k)!$ for any fixed integer =
$k\ge 1$
and constant $\beta >0$ provided NP$\not\subseteq$P/poly, which (like =
P$\neq$NP)
is believed to be true. We also give upper bounds for the size of an =
ATSP
neighborhood depending on its search time.
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Logical Seminar of POMI. 18 April =
2005
Speaker:=20
Gregory Gutin (Department of Computer Science Royal
Holloway, =
University of=20
London Egham, Surrey, TW20 0EX, UK)
Title: "Upper Bounds on ATSP =
Neighborhood=20
Size"
&nbs=
p; =
; =20
ABSTRACT:
Abstract: We consider the Asymmetric =
Traveling=20
Salesman Problem (ATSP) and use
the definition of neighborhood by =
Deineko and=20
Woeginger ( Math. Program. 87
(2000) 519-542). Let $\mu(n)$ be the =
maximum=20
cardinality of polynomial time
searchable neighborhood for the ATSP =
on $n$=20
vertices. Deineko and Woeginger
conjectured that $\mu (n)< \beta =
(n-1)!$=20
for any constant $\beta >0$ provided
P$\neq$NP. We prove that =
$\mu(n) <=20
\beta (n-k)!$ for any fixed integer $k\ge 1$
and constant $\beta =
>0$=20
provided NP$\not\subseteq$P/poly, which (like P$\neq$NP)
is believed =
to be=20
true. We also give upper bounds for the size of an ATSP
neighborhood=20
depending on its search time.
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Logical Seminar of POMI. 26 August 2005
Speaker: Boris Konev
Title: "Transitive logics over transitive states"
ABSTRACT:
We consider the computational behaviour of propositional temporal logics
over N that are capable of reasoning about states with transitive relations.
This is a joint work with F.Wolter and M.Zakharyaschev
Logical Seminar of POMI. 19 December 2005
Speaker: N.V.Krupski. (Moscow)
Title: " Some algorithmic questions about the formal systems
with the internalization property of proofs."
ABSTRACT:
Decidable formal systems that can internalize
arguments about their own proofs are considered:
Logic of Proofs $LP$ and Reflective Combinatory Logic $RCL$.
The reflected fragment of such logic, i.e. the set of all derivable
formulas of the form ``$t$ proves $F$'', already contains
(in some special form) the formulations
of all theorems of the logic, but is more simple than the logic itself.
This fact can be formalized in terms of complexity of corresponding
decision procedures. We prove the upper complexity bound for
the reflected fragment of $LP$ that is better than the known
bound for $LP$ itself. We establish the decidability and
$PSPACE$-completeness of $RCL$ and provide the polynomial time
decision procedure for its reflected fragmend. It is also proved
that the well-formedness of formulas and the typability of terms
in $RCL$ can be tested in polynomial time.
Logical Seminar of POMI. 23 January 2006
Speaker: Sergei Soloviev (IRIT, Toulouse)
Title: "Otnocheniya ekvivalentnosti na vyvodah lineinoi logiki
i nesvobodnye kategornye modeli."
ABSTRACT:
V doklade ispolzuyutsya rezultaty, polutchennye v dissertatsii
L. Mehats'a (Toulouse), vypolnennoi pod rukovodstvom S. Solovieva.
Otpravnoi tochkoi slujit fakt, chto intuitsionistskaya
multiplikativnaya lineinaya logika (IMLL) mojet rassmatrivatsya
kak svobodnaya simmetricheskaya monoidalnaya
zamknutaya kategoriya (SMCC). Morfizmami slujat pri etom
klassy ekvivalentnosti vyvodov po onosheniyu k
"basovomu" kategornomu otnoshenoyu ekvivalentnosti.
Rashireniya etoi ekvivalentnosti, udovletvotyaushie
estestvennym usloviyam, budut sootvetstvovat ravenstvu morphismov
v nesvobodnyh modelyah. Eto pozvolyaet primenit dlya
izucheniya kommutativnosti diagramm v etih modelyah
teoriyu dokazatelstv. Osnovnye rezultaty: "kanonichaskaya
forma" axiomatizatsii raschirennyh otnochenii ekvivalentnosti
kriticheskimi parami i identichnost basovogo otnocheniya
otnochenoyu porojdennomu perestanovkami pravil na gentzenovskih
vyvodah.
Logical Seminar of POMI. 6 February 2006
Speaker: Yi Zhang (Sun Yat-sen university,
Giangzhou, P.R. China)
Title: "MAXIMAL COFINITARY GROUPS"
ABSTRACT:
A permutation $g \in Sym(N)$ is cofinitary iff $g$ has only finitely
many fixed-points. A group $G\le Sym(N)$ is cofinitary iff
$\forall g \in (G\{id})$, $g$ is a cofinitary permutation.
In my talk I'll first outline my answers to a question which was asked
by P.Cameron and P.Neumann about maximal cofinitary groups. Then I will
discuss some set-theoretical problems around this area.
Logical Seminar of POMI. 9 Oktober 2006
Speaker: Sergey Dudakov (Tver).
Title: " Some algorithmic questions about the formal systems
with the internalization property of proofs."
ABSTRACT:
We investigate the expressive power of first-order languages for
'stand-alone' finite structures and finite structures wich are
embedded into infinite universes. We demonstrate conditions for
universes and its theories when the expressive power grows and when
the expressive power remains the same (collapse result property). We
answer some open questions from papers of M.Taitslin, O.Belegradek,
A.Stolboushkin, J.Baldwin, M.Benedikt. For example we show
1. There are decidable theories without the collapse result
property (it refuses the hypothesys of M.Taitslin, O.Belegradek,
A.Stolboushkin)
2. We have found an effective method to prove the collapse result
(it proves the hypothesys of M.Taitslin, O.Belegradek, A.Stolboushkin)
for many universes
3. There are universes with the collapse result which have an
independent formula on any infinite subset (it refuses the hypothesys
of J.Baldwin and M.Benedikt)
Logical Seminar of POMI. 23 Oktober 2006 Speaker: N.A.Shanin. Title: "Logic of finitary mathematics."
Logical Seminar of POMI. 30 Oktober 2006 Speaker: N.A.Shanin. Title: "Logic of finitary mathematics (continued)."
Logical Seminar of POMI. 15 January 2007
Speaker: Yury Lifshits.
Title: "Algorithms and computational complexity analysis for
processing compressed texts."
ABSTRACT:
How should we store data in order to minimize space and preserve fast
query processing at the same time? For the first objective many
compression algorithms were invented. For the second one we use
various data structures that allows fast processing for some specific
types of queries. But can we combine these two approaches? We present
new algorithms and complexity analysis for processing compressed texts
without unpacking. Finaly, we suggest a new compression scheme based
on tiling periodicity.
Logical Seminar of POMI. 28 May 2007 Speaker: Sy David Friedman (Austria). Title: "About Turing machines that run for infinitely many steps."
Logical Seminar of POMI. 28 May 2007
Speaker: Sy-David Friedman (Kurt Goedel
Research Center University of Vienna).
Title: "Infinite time Turing Machines."
Abstract.
Infinite time Turing machines (ITTM's) use the
classical Turing machine model but allow computations to
continue into the transfinite. The length of time for a
computation to converge is measured by a countable ordinal
number. In joint work with Philip Welch, I describe a
universal ITTM, called the Theory Machine, which (on input 0)
writes on its tape encoded versions of theories of initial
segments of Goedel's L-hierarchy. The Theory Machine easily
yields a description of the supremum of the possible lengths
of converging ITTM computations as well as a
characterisation of those infinite sequences of 0's and 1's
which can appear on the tape of some ITTM. The construction
of the Theory Machine uses ideas from Jensen's fine
structure theory of L.
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Logical Seminar of POMI. 17 September 2007, 11:00
Speaker: Oscar H. Ibarra (Department of Computer Science,=20
University of California, Santa Barbara, CA 93106, USA)
Title: "On Counter Machines, Reachability Problems,=20
and Diophantine Equations"
ABSTRACT:
We give a brief survey of results that use "reversal-bounded counters" =
in
studying reachability problems for various classes of transition =
systems.
We also discuss the connection between the decidability of reachability
in counter machines and the solvability of certain quadratic Diophantine
equations. Finally, we report on some preliminary results concerning the
reachability/emptiness problem for "stateless" automata.
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Logical Seminar of POMI. 17 September =
2007,=20
11:00
Speaker: Oscar H. Ibarra (Department of Computer=20
Science,
University of California, Santa =
Barbara, CA 93106,=20
USA)
Title: "On Counter Machines, Reachability Problems, =
and Diophantine Equations"
&nbs=
p; =
; =20
ABSTRACT:
We give a brief survey of results that use "reversal-bounded =
counters"=20
in
studying reachability problems for various classes of transition=20
systems.
We also discuss the connection between the decidability of=20
reachability
in counter machines and the solvability of certain =
quadratic=20
Diophantine
equations. Finally, we report on some preliminary results =
concerning the
reachability/emptiness problem for "stateless"=20
automata.
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=20
Logical Seminar of POMI. 15 December 2008, 14:00
Speaker: Grigori Mints (Stanford)
Title: "New Trends in the Foundations of Mathematics"
ABSTRACT:
We discuss the turn in foundations of mathematics happening now under
the slogans of HARD ANALYSIS (T. Tao) and PROOF
MINING. (U. Kohlenbach). It follows Kreisel's program of unwinding
proofs and comes from new emphasis on finitist methods and results =
caused
by mathematical needs. New tools mathematicians needed turned out to
be instances of well-known constructions of proof theory.
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Logical Seminar of POMI. 15 December =
2008,=20
14:00
Speaker: Grigori Mints (Stanford)
Title: "New Trends in the Foundations of=20
Mathematics"
&nbs=
p; =
; =20
ABSTRACT:
We discuss the turn in foundations of mathematics happening now=20
under
the slogans of HARD ANALYSIS (T. Tao) and PROOF
MINING. (U.=20
Kohlenbach). It follows Kreisel's program of unwinding
proofs and =
comes from=20
new emphasis on finitist methods and results caused
by mathematical=20
needs. New tools mathematicians needed turned out to
be =
instances of=20
well-known constructions of proof =
theory.
------=_NextPart_000_001D_01C95B21.ABAF4A90--
This is a multi-part message in MIME format. ------=_NextPart_000_00FD_01C9EA97.FAFA2AB0 Content-Type: text/plain; charset="koi8-r" Content-Transfer-Encoding: quoted-printable Logical Seminar of POMI. 15 May 2009, 14:00 Speaker: Grigori Mints (Stanford) Title: "Logical Equations in Monadic Logic" ------=_NextPart_000_00FD_01C9EA97.FAFA2AB0 Content-Type: text/html; charset="koi8-r" Content-Transfer-Encoding: quoted-printable------=_NextPart_000_00FD_01C9EA97.FAFA2AB0--Logical Seminar of POMI. 15 May 2009,=20 14:00
Speaker: Grigori Mints (Stanford)
Title: "Logical Equations in Monadic Logic"
This is a multi-part message in MIME format. ------=_NextPart_000_0028_01C9EA9B.E1DE1BF0 Content-Type: text/plain; charset="koi8-r" Content-Transfer-Encoding: quoted-printable Logical Seminar of POMI. June15 2009, 14:00 Speaker: Grigori Mints (Stanford) Title: "Logical Equations in Monadic Logic" ------=_NextPart_000_0028_01C9EA9B.E1DE1BF0 Content-Type: text/html; charset="koi8-r" Content-Transfer-Encoding: quoted-printable------=_NextPart_000_0028_01C9EA9B.E1DE1BF0--Logical Seminar of POMI. June15 2009,=20 14:00
Speaker: Grigori Mints (Stanford)
Title: "Logical Equations in Monadic Logic"
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------=_NextPart_000_009D_01CA4882.B7602400
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charset="koi8-r"
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"=F5=D2=C1=D7=CE=C5=CE=C9=D1 =CE=C1=C4 =
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=EE=C1=DE=C1=CC=CF =D7 14.00, =CB=CF=CD=CE=C1=D4=C1 203
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=D5=D2=C1=D7=CE=C5=CE=C9=CA =D7=C9=C4=C1 $X=3DY+Z$ =C9 $X=3Dconst$, =
=C7=C4=C5=20
=CE=C5=C9=DA=D7=C5=D3=D4=CE=D9=C5 --- =CD=CE=CF=D6=C5=D3=D4=D7=C1 =
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=D0=C5=D2=C9=CF=C4=C9=DE=C5=D3=CB=C9=C5=20
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S, n \in T\}$. =E5=D3=CC=C9 =CF=C7=D2=C1=CE=C9=DE=C9=D7=C1=D4=D8=D3=D1 =
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=D0=C5=D2=D7=D9=CA=20
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//=ED.: =E6=C9=DA=CD=C1=D4=CC=C9=D4, 2000.
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=D6=D5=D2=CE=C1=CC, =D4.38, ? 5, 1997, =D3.1034-1046. =20
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______________________________
1.=20
=F3.=EE.=F7=C1=D3=C9=CC=D8=C5=D7.=20
=ED=C5=D4=CF=C4=20
=D2=C5=C4=D5=CB=C3=C9=C9 =C9 =CB=C1=DE=C5=D3=D4=D7=C5=CE=CE=D9=CA =
=C1=CE=C1=CC=C9=DA =C4=C9=CE=C1=CD=C9=DE=C5=D3=CB=C9=C8 =
=D3=C9=D3=D4=C5=CD, I-II // =E9=DA=D7. =F2=E1=EE, =D3=C5=D2.=20
=F4=C5=CF=D2=C9=D1 =C9 =D3=C9=D3=D4=C5=CD=D9 =
=D5=D0=D2=C1=D7=CC=C5=CE=C9=D1, 2006, №1, =D3.21-29; =
№2, =D3.=20
5-17.=20
2. =
=F3.=EE.=F7=C1=D3=C9=CC=D8=C5=D7,=20
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=D3=C9=D3=D4=C5=CD=C1=CD=C9 //=ED.: =E6=C9=DA=CD=C1=D4=CC=C9=D4, =
2000.
=
3.=20
=F3.=EE.=F7=C1=D3=C9=CC=D8=C5=D7. =ED=C5=D4=CF=C4=20
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=CD=C1=D4=C5=CD=C1=D4=C9=DE=C5=D3=CB=C9=CA =
=D6=D5=D2=CE=C1=CC,=20
=D4.38,=20
№ 5, 1997, =D3.1034-1046. =
4. =
=F3.=EE.=F7=C1=D3=C9=CC=D8=C5=D7,=20
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=C4=CF=CF=D3=CE=C1=DD=C5=CE=C9=D1 //=F4=C5=DA=C9=D3=D9 =
=CB=CF=CE=C6=C5=D2=C5=CE=C3=C9=C9 "=ED=C1=CC=D8=C3=C5=D7=D3=CB=C9=C5 =
=DE=D4=C5=CE=C9=D1", =E9=ED=20
=F3=EF =F2=E1=EE, 2009.=20
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