The Full Deduction Rule and the Substance Over Form Doctrine

2018 ◽  
Author(s):  
Joseph Bankman ◽  
Darien Shanske
Keyword(s):  
Deduction Rule

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.

MP-CLOSED SUBSETS IN BASIC ALGEBRAS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250048
Author(s):  
I. Chajda ◽  
J. Krňávek
Keyword(s):  
Basic Algebra ◽  
Simple Condition ◽  
Modus Tollens ◽  
Deduction Rule ◽  
A Chain

A subset M of a given basic algebra [Formula: see text] is MP-closed if a ∈ M and a → b ∈ M always imply b ∈ M for b ≤ a. We get a characterization of MP-closed subsets for commutative basic algebras in terms of closedness in the derived operation ⊙. If, moreover, the basic algebra is a chain, then a particular simple condition is derived. For basic algebras which are not commutative we derive only a closedness with respect to the deduction rule Modus Tollens.

The deduction rule and linear and near-linear proof simulations

1993 ◽  
Vol 58 (2) ◽  
pp. 688-709 ◽  
Author(s):  
Maria Luisa Bonet ◽  
Samuel R. Buss
Keyword(s):  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Upper Bounds ◽  
Proof System ◽  
Proof Systems ◽  
Linear Speedup ◽  
Frege Systems ◽  
Deduction Rule ◽  
Gentzen Sequent Calculus

AbstractWe introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

scholarly journals A deduction rule for ${\rm VBTO}($ $)^n_{i=1}$.

1977 ◽  
Vol 18 (3) ◽  
pp. 510-512
Author(s):  
Richard Butrick
Keyword(s):  
Deduction Rule
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