|
LowerUnits
In proof compression LowerUnits (LU) is an algorithm used to compress propositional logic resolution proofs. The main idea of LowerUnits is to exploit the following fact:Fontaine, Pascal; Merz, Stephan; Woltzenlogel Paleo, Bruno. ''Compression of Propositional Resolution Proofs via Partial Regularization''. 23rd International Conference on Automated Deduction, 2011. Theorem: Let \varphi be a potentially redundant proof, and \eta be the redundant proof , redundant node. If \eta’s clause In language, a clause is a constituent that comprises a semantic predicand (expressed or not) and a semantic predicate. A typical clause consists of a subject and a syntactic predicate, the latter typically a verb phrase composed of a verb with ... is a unit clause, then \varphi is redundant. The algorithm targets exactly the class of global redundancy stemming from multiple resolutions with unit clauses. The algorithm takes its name from the fact that, when this rewriting is done an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Compression
In proof theory, an area of mathematical logic, proof compression is the problem of algorithmically compressing formal proofs. The developed algorithms can be used to improve the proofs generated by automated theorem proving tools such as SAT solvers, SMT-solvers, first-order theorem provers and proof assistants. Problem Representation In propositional logic a resolution proof of a clause \kappa from a set of clauses ''C'' is a directed acyclic graph (DAG): the input nodes are axiom inferences (without premises) whose conclusions are elements of ''C'', the resolvent nodes are resolution inferences, and the proof has a node with conclusion \kappa. The DAG contains an edge from a node \eta_ to a node \eta_ if and only if a premise of \eta_ is the conclusion of \eta_. In this case, \eta_ is a child of \eta_, and \eta_ is a parent of \eta_. A node with no children is a root. A proof compression algorithm will try to create a new DAG with fewer nodes that represents a valid proof of \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" ( conjunction), "or" (disjunction), "not" (negation) and "if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redundant Proof
In mathematical logic, a redundant proof is a proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ... that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof \psi of \kappa is considered redundant if there exists another proof \psi^ of \kappa^ such that \kappa^\subseteq\kappa (i.e. \kappa^ \;\text\; \kappa) and , \psi^, to denote a proof-context \psi\left \right/math> with a single placeholder replaced by the subproof \eta. Global redundancy A proof : \psi eta_\odot_p_\eta_1_.html" ;"title="\psi_1 [\eta \odot_p \eta_1 ">\psi_1 [\eta \odot_p \eta_1 \odot \psi_2 [\eta\odot_ \eta_]\text \psi [ \psi_1 [ \eta\odot_p ( \eta_1 \odot \psi_2 [ \eta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clause (logic)
In logic, a clause is a propositional formula formed from a finite collection of literals (atoms or their negations) and logical connectives. A clause is true either whenever at least one of the literals that form it is true (a disjunctive clause, the most common use of the term), or when all of the literals that form it are true (a conjunctive clause, a less common use of the term). That is, it is a finite disjunction or conjunction of literals, depending on the context. Clauses are usually written as follows, where the symbols l_i are literals: :l_1 \vee \cdots \vee l_n Empty clauses A clause can be empty (defined from an empty set of literals). The empty clause is denoted by various symbols such as \empty, \bot, or \Box. The truth evaluation of an empty disjunctive clause is always false. This is justified by considering that false is the neutral element of the monoid (\, \vee). The truth evaluation of an empty conjunctive clause is always true. This is related to the conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Clause
Unit propagation (UP) or Boolean Constraint propagation (BCP) or the one-literal rule (OLR) is a procedure of automated theorem proving that can simplify a set of (usually propositional) clauses. Definition The procedure is based on unit clauses, i.e. clauses that are composed of a single literal, in conjunctive normal form. Because each clause needs to be satisfied, we know that this literal must be true. If a set of clauses contains the unit clause l, the other clauses are simplified by the application of the two following rules: # every clause (other than the unit clause itself) containing l is removed (the clause is satisfied if l is); # in every clause that contains \neg l this literal is deleted (\neg l can not contribute to it being satisfied). The application of these two rules lead to a new set of clauses that is equivalent to the old one. For example, the following set of clauses can be simplified by unit propagation because it contains the unit clause a. : \ Since a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redundant Proof
In mathematical logic, a redundant proof is a proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ... that has a subset that is a shorter proof of the same result. In other words, a proof is redundant if it has more proof steps than are actually necessary to prove the result. Formally, a proof \psi of \kappa is considered redundant if there exists another proof \psi^ of \kappa^ such that \kappa^\subseteq\kappa (i.e. \kappa^ \;\text\; \kappa) and , \psi^, to denote a proof-context \psi\left \right/math> with a single placeholder replaced by the subproof \eta. Global redundancy A proof : \psi eta_\odot_p_\eta_1_.html" ;"title="\psi_1 [\eta \odot_p \eta_1 ">\psi_1 [\eta \odot_p \eta_1 \odot \psi_2 [\eta\odot_ \eta_]\text \psi [ \psi_1 [ \eta\odot_p ( \eta_1 \odot \psi_2 [ \eta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Theorem Proving
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science. Logical foundations While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics. Frege's '' Begriffsschrift'' (1879) introduced both a complete propositional calculus and what is essentially modern predicate logic. His ''Foundations of Arithmetic'', published 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Russell and Whitehead in their influential '' Principia Mathematica'', first published 1910–1913, and with a revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
