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### An Analytic Propositional Proof System on Graphs [article]

Matteo Acclavio, Ross Horne, Lutz Strassburger
2021 arXiv   pre-print
In this paper we present a proof system that operates on graphs instead of formulas.  ...  For our proof system we show the admissibility of cut and a generalization of the splitting property.  ...  The main challenge is to design an analytic propositional proof system.  ...

### Reflection principles, propositional proof systems, and theories [article]

Pavel Pudlák
2020 arXiv   pre-print
In this paper we will revisit some results about the reflection principles for propositional proofs systems using a finer scale of reflection principles.  ...  We will also survey some results about arithmetical theories and proof systems associated with them.  ...  Basic concepts 2.1 Propositional proof systems In this paper "a proof system" will always mean a proof system for classical propositional logic.  ...

### Propositional proof systems and fast consistency provers [article]

Joost J. Joosten
2020 arXiv   pre-print
Krajíček and Pudlák proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover.  ...  A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of the finite consistency statements of any other poly-time axiomatized theory.  ...  A pps P is an optimal propositional proof system, an opps for short, if P ≥ Q for any propositional proof system Q. It is easy to see that a pps is optimal whenever it is super.  ...

### A Propositional Proof System for Log Space [chapter]

Steven Perron
2005 Lecture Notes in Computer Science
The proof system G * 0 of the quantified propositional calculus corresponds to N C 1 , and G * 1 corresponds to P , but no formula-based proof system that corresponds log space reasoning has ever been  ...  The proof system G * is a tree-like proof system for the quantified propositional calculus based on Gentzen's LK [2] .  ...  The Proof System The proof system P K is the Gentzen-style sequent calculus for propositional logic [5, 6] . The initial sequents are ⊥ →, → , and A → A, for any propositional formula A.  ...

### Disjoint NP-Pairs from Propositional Proof Systems [chapter]

Olaf Beyersdorff
2006 Lecture Notes in Computer Science
We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P ).  ...  For a proof system P we introduce the complexity class DNPP(P ) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P .  ...  By using Proposition 24 we can also establish a version of Proposition 30 for other line based proof systems which admit efficient deduction. Proposition 32 . 32 Let P be a proof system.  ...

### Propositional Proof Systems and Fast Consistency Provers

Joost J. Joosten
2007 Notre Dame Journal of Formal Logic
Krajíček and Pudlák proved in [5] that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover.  ...  A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of the finite consistency statements of any other poly-time axiomatized theory.  ...  A pps P is an optimal propositional proof system, an opps for short, if P ≥ Q for any propositional proof system Q. It is easy to see that a pps is optimal whenever it is super.  ...

### The relative efficiency of propositional proof systems

Stephen A. Cook, Robert A. Reckhow
1979 Journal of Symbolic Logic (JSL)
We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology.  ...  The smallest upper bound known for this function is exponential, no matter what the proof system.  ...  PROPOSITION. If a proof system f2 for L p-simulates a polynomially bounded proof system f1 for L, then f2 is also polynomially bounded.  ...

### A Non-wellfounded, Labelled Proof System for Propositional Dynamic Logic [article]

Simon Docherty, Reuben N. S. Rowe
2019 arXiv   pre-print
We additionally investigate proof-search strategies in the cyclic system for the fragment of PDL without tests.  ...  A finitarily representable cyclic system, G3PDL^ω, is then given. We show that both are sound and complete with respect to standard models of PDL and, further, that G3PDL^∞ is cut-free complete.  ...  This proof system has two important features. The first is that it is a labelled proof system.  ...

### Canonical Disjoint NP-Pairs of Propositional Proof Systems [chapter]

Christian Glaßer, Alan L. Selman, Liyu Zhang
2005 Lecture Notes in Computer Science
A proof system is optimal if it simulates every other propositional proof system.  ...  Although it is an open question whether optimal proof systems exist, as we stated above, Razborov showed that if there exists an optimal propositional proof system f , then its canonical pair is a complete  ...  Specifically, Razborov [Raz94] defined the canonical disjoint NP-pair, (SAT * , REF f ), for every propositional proof system f , and he showed that if there exists an optimal propositional proof system  ...

### The Deduction Theorem for Strong Propositional Proof Systems

Olaf Beyersdorff
2008 Theory of Computing Systems
even polynomially bounded proof systems.  ...  This paper focuses on the deduction theorem for propositional logic.  ...  Preliminaries Propositional Proof Systems.  ...

### The Model-Theoretic Expressiveness of Propositional Proof Systems

Erich Grädel, Benedikt Pago, Wied Pakusa
2017 Leibniz International Proceedings in Informatics
The Model-Theoretic Expressiveness of Propositional Proof Systems in such systems, and usually have completeness for relevant fragments of propositional logic, such as Horn-logic or 2-CNF.  ...  , provably require proofs of super-polynomial size even in quite strong proof systems.  ...  Since Cook and Reckhow [14] made the notion of an efficient propositional proof system precise, a huge body of research on the power of various propositional proof system has been established.  ...

### Canonical disjoint NP-pairs of propositional proof systems

Christian Glaßer, Alan L. Selman, Liyu Zhang
2007 Theoretical Computer Science
A proof system is optimal if it simulates every other propositional proof system.  ...  Although it is an open question whether optimal proof systems exist, as we stated above, Razborov showed that if there exists an optimal propositional proof system f , then its canonical pair is a complete  ...  Specifically, Razborov [Raz94] defined the canonical disjoint NP-pair, (SAT * , REF f ), for every propositional proof system f , and he showed that if there exists an optimal propositional proof system  ...

### The Deduction Theorem for Strong Propositional Proof Systems [chapter]

Olaf Beyersdorff
2007 Lecture Notes in Computer Science
even polynomially bounded proof systems.  ...  This paper focuses on the deduction theorem for propositional logic.  ...  Preliminaries Propositional Proof Systems.  ...

### Optimal proof systems for propositional logic and complete sets [chapter]

Jochen Messner, Jacobo Torán
1998 Lecture Notes in Computer Science
whose range is the set of tautologies in Propositional Logic (TAUT), is called a proof system.  ...  proof systems exist.  ...  A propositional proof system (or just proof system) is a polynomial time computable function h : !  ...

### Propositional Proofs in Frege and Extended Frege Systems (Abstract) [chapter]

Sam Buss
2015 Lecture Notes in Computer Science
We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász  ...  Frege systems are arguably the most important fully expressive, sound and complete proof system for propositional proofs: Frege proofs are "textbook" propositional proof systems usually formulated with  ...  for propositional logic and as an underpinning for stronger systems such as SMT solvers, modal logics and first-order logics.  ...
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