Reductive logic and proof-search: proof theory, semantics, and control

2004 ◽  
Vol 42 (03) ◽  
pp. 42-1624-42-1624
Keyword(s):  
Proof Theory ◽  
Proof Search ◽  
And Control

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

scholarly journals A labelled sequent calculus for BBI: proof theory and proof search

2015 ◽  
Vol 28 (4) ◽  
pp. 809-872
Author(s):  
Zhé Hóu ◽  
Rajeev Goré ◽  
Alwen Tiu
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Proof Search ◽  
Labelled Sequent Calculus

scholarly journals Proof Search and Certificates for Evidential Transactions

2021 ◽  
pp. 234-251
Author(s):  
Vivek Nigam ◽  
Giselle Reis ◽  
Samar Rahmouni ◽  
Harald Ruess
Keyword(s):  
Logic Programming ◽  
Proof Theory ◽  
Knowledge Bases ◽  
Foreign Country ◽  
Proof System ◽  
Complete Proof ◽  
Proof Search ◽  
Logical Data

AbstractAttestation logics have been used for specifying systems with policies involving different principals. Cyberlogic is an attestation logic used for the specification of Evidential Transactions (ETs). In such transactions, evidence has to be provided supporting its validity with respect to given policies. For example, visa applicants may be required to demonstrate that they have sufficient funds to visit a foreign country. Such evidence can be expressed as a Cyberlogic proof, possibly combined with non-logical data (e.g., a digitally signed document). A key issue is how to construct and communicate such evidence/proofs. It turns out that attestation modalities are challenging to use established proof-theoretic methods such as focusing. Our first contribution is the refinement of Cyberlogic proof theory with knowledge operators which can be used to represent knowledge bases local to one or more principals. Our second contribution is the identification of an executable fragment of Cyberlogic, called Cyberlogic programs, enabling the specification of ETs. Our third contribution is a sound and complete proof system for Cyberlogic programs enabling proof search similar to search in logic programming. Our final contribution is a proof certificate format for Cyberlogic programs inspired by Foundational Proof Certificates as a means to communicate evidence and check its validity.

scholarly journals Focused Proof-search in the Logic of Bunched Implications

2021 ◽  
pp. 247-267
Author(s):  
Alexander Gheorghiu ◽  
Sonia Marin
Keyword(s):  
Data Structure ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Operational Semantics ◽  
Search Space ◽  
Difficult Problem ◽  
Cut Elimination ◽  
Proof Search ◽  
Nested Sequents

AbstractThe logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. The focusing principle is a restriction of the proof-search space that can capture various goal-directed proof-search procedures. In this paper we show that focused proof-search is complete for BI by first reformulating the traditional bunched sequent calculus using the simpler data-structure of nested sequents, following with a polarised and focused variant that we show is sound and complete via a cut-elimination argument. This establishes an operational semantics for focused proof-search in the logic of Bunched Implications.

scholarly journals First-order Answer Set Programming as Constructive Proof Search

2018 ◽  
Vol 18 (3-4) ◽  
pp. 673-690
Author(s):  
ALEKSY SCHUBERT ◽  
PAWEŁ URZYCZYN
Keyword(s):  
Logic Programming ◽  
Intuitionistic Logic ◽  
Proof Theory ◽  
Answer Set Programming ◽  
Theory And Practice ◽  
Close Similarity ◽  
Constructive Proof ◽  
First Order ◽  
Proof Search ◽  
Answer Set

AbstractWe propose an interpretation of the first-order answer set programming (FOASP) in terms of intuitionistic proof theory. It is obtained by two polynomial translations between FOASP and the bounded-arity fragment of the Σ1 level of the Mints hierarchy in first-order intuitionistic logic. It follows that Σ1 formulas using predicates of fixed arity (in particular unary) is of the same strength as FOASP. Our construction reveals a close similarity between constructive provability and stable entailment, or equivalently, between the construction of an answer set and an intuitionistic refutation. This paper is under consideration for publication in Theory and Practice of Logic Programming

Control of Metal Alloy Oxidation with Electric Fields

1973 ◽  
Vol 31 ◽  
pp. 164-165
Author(s):  
R. R. Dils ◽  
P. S. Follansbee
Keyword(s):  
Electric Fields ◽  
Oxidation Process ◽  
Base Alloy ◽  
Oxide Surface ◽  
Platinum Electrodes ◽  
Insulating Layer ◽  
Iron Base Alloy ◽  
Alloy Oxidation ◽  
Aluminum Oxide Layer ◽  
And Control

Electric fields have been applied across oxides growing on a high temperature alloy and control of the oxidation of the material has been demonstrated. At present, three-fold increases in the oxidation rate have been measured in accelerating fields and the oxidation process has been completely stopped in a retarding field.The experiments have been conducted with an iron-base alloy, Pe 25Cr 5A1 0.1Y, although, in principle, any alloy capable of forming an adherent aluminum oxide layer during oxidation can be used. A specimen is polished and oxidized to produce a thin, uniform insulating layer on one surface. Three platinum electrodes are sputtered on the oxide surface and the specimen is reoxidized.

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