Discretely ordered modules as a first-order extension of the cutting planes proof system

1998 ◽  
Vol 63 (4) ◽  
pp. 1582-1596 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bound ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Maximum Size ◽  
Interpolation Theorem ◽  
Proof System ◽  
First Order ◽  
First Order Theory ◽  
Sparse Polynomials ◽  
Order Extension

AbstractWe define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities ∑i ai · xi ≥ b (xi's variables, ai's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP-inequalities, we prove a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coefficients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials.The proof relies on a universal interpolation theorem for semantic derivations [16, Theorem 5.1].LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantifier elimination for such theories.We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP).

scholarly journals Randomized feasible interpolation and monotone circuits with a local oracle

2018 ◽  
Vol 18 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Interpolation Theorem ◽  
Proof System ◽  
Symbolic Logic ◽  
Approximate Counting ◽  
Small Width ◽  
Randomized Protocols ◽  
Monotone Circuits

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

scholarly journals Constructing weak simulations from linear implications for processes with private names

2019 ◽  
Vol 29 (8) ◽  
pp. 1275-1308 ◽  
Author(s):  
Ross Horne ◽  
Alwen Tiu
Keyword(s):  
Expressive Power ◽  
Dual Pair ◽  
Proof System ◽  
Branching Time ◽  
First Order ◽  
Order Extension

AbstractThis paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π -calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π -calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic

1997 ◽  
Vol 62 (2) ◽  
pp. 457-486 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Interpolation Theorem ◽  
Alternative Proof ◽  
Bounded Arithmetic ◽  
Proof Systems ◽  
Restricted Form ◽  
Arithmetic Theory ◽  
Equational Calculus

AbstractA proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1)Feasible interpolation theorems for the following proof systems:(a)resolution(b)a subsystem of LK corresponding to the bounded arithmetic theory (α)(c)linear equational calculus(d)cutting planes.(2)New proofs of the exponential lower bounds (for new formulas)(a)for resolution ([15])(b)for the cutting planes proof system with coefficients written in unary ([4]).(3)An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Double-exponential inseparability of Robinson subsystem Q+

2011 ◽  
Vol 76 (1) ◽  
pp. 94-124
Author(s):  
Lavinia Egidi ◽  
Giovanni Faglia
Keyword(s):  
Lower Bound ◽  
Turing Machine ◽  
Lower Bounds ◽  
Order Theory ◽  
Exponential Time ◽  
First Order ◽  
First Order Theory ◽  
Double Exponential

AbstractIn this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q+. The theory, subset of Presburger theory of addition S+, is the additive fragment of Robinson system Q. We prove that every set that separates Q+ from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models.The result implies also that any theory of addition that is consistent with Q+—in particular any theory contained in S+—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories.Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S+. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable.

Locally finite theories

1986 ◽  
Vol 51 (1) ◽  
pp. 59-62 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Main Idea ◽  
Order Theory ◽  
Interpolation Theorem ◽  
Point Of View ◽  
Finite Model ◽  
Consistent Theory ◽  
Locally Finite ◽  
First Order ◽  
First Order Theory ◽  
Finite Theory

We say that a first order theoryTislocally finiteif every finite part ofThas a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theoryTa locally finite theory FIN(T) which is syntactically (in a sense) isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of [5], [6] and [7]. This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.)The first mathematically strong locally finite theory, called FIN, was defined in [1] (see also [2]). Now we get much stronger ones, e.g. FIN(ZF).From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilbert's second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects.In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.The results of this paper were announced in [3].

First-order-theory-based thermoelastic stability of functionally graded material circular plates

AIAA Journal ◽  
2002 ◽  
Vol 40 ◽  
pp. 1444-1450 ◽  
Author(s):  
M. Najafizadeh ◽  
M. R. Eslami
Keyword(s):  
Functionally Graded Material ◽  
Order Theory ◽  
Functionally Graded ◽  
Circular Plates ◽  
First Order ◽  
First Order Theory ◽  
Graded Material

scholarly journals On equations and first-order theory of one-relator monoids

2021 ◽  
pp. 104745
Author(s):  
Albert Garreta ◽  
Robert D. Gray
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory
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