Contents

A set of tools to semantically embed (multi-modal) higher-order modal logic problems formulated in the TPTP [1] dialect THF into higher-order logic formulated in THF. It contains

• a standalone tool embed for the semantical embedding.
• a conversion tool convert_qmltp_to_thf for translating first-order modal logic problems formulated in the dialect of the QMLTP [2] to higher-order logic problems formulated in THF. A translated version of the QMLTP can be found on Github [3].
• a library embedding_lib you can include in your project to semantically embed problems. Contains wrapper functions for easier handling.

This software is based on the theoretical work of Benzmüller and Paulson [4].

Details of the embedding's procedure, a theoretical overview and some performance tests are available in Gleißner, Steen and Benzmüller [5].

Requirements

• Maven 3.5 or higher
• Java 8 or higher
• Make

Build

In the root directory invoke

mvn clean package

Usage

Converting a problem

To convert a modal problem run

java -jar embed/target/embed-1.0-SNAPSHOT-shaded.jar 
-i <INPUT_FILE>
-o <OUTPUT_FILE>

Converting a directory

To recursively convert a directory containing modal problems run

java -jar embed/target/embed-1.0-SNAPSHOT-shaded.jar 
-i <INPUT_DIRECTORY>
-o <OUTPUT_DIRECTORY>
-diroutput <OUTPUT_OPTIONS>

This will preserve the directory structure in the output. If more than semantic is specified the choices for <OUTPUT_OPTIONS> become relevant:

• joint: exactly one duplicate directory structure and filenames contain a string representing the semantics used during the embedding
• splitted: one duplicate directory structure for every semantic

Adding Semantics

Specify semantics (which overrides any semantics specified in the problem) for mono-modal problems using

-consequences <CONSEQUENCES>
-constants <CONSTANTS>
-domains <DOMAINS>
-systems <SYSTEMS>

Valid choices for the semantics parameters are

• <CONSEQUENCES>: global,local
• <CONSTANTS>: rigid
• <DOMAINS>: constant, cumulative, decreasing, varying
• <SYSTEMS>: K, KB, K4, K5, K45, KB5, KB5_KB5, KB5_KB4, D, DB, D4, D5, D45, T, B, S4, S5, S5_KT5, S5_KTB5, S5_KT45, S5_KT4B, S5_KD4B, S5_KD4B5, S5_KDB5, S5U

Note that systems with the same prefix are different axiomatizations of the same semantics.

Specify multiple semantics that creates multiple embedded problems by separating the parameters with a comma e.g.

-domains constant,cumulative

This will create embeddings for all semantics contained by the cross-product of <CONSEQUENCES> x <CONSTANTS> x <DOMAINS> x <SYSTEMS>

Transformation Parameters

Specify special parameters using

-t <OPTIONS>

Valid choices for <OPTIONS> are

• semantical_modality_axiomatization: embed modality semantics as frame conditions
• syntactical_modality_axiomatization: embed modality semantics as axioms about modal operators

References

[1] Geoff Sutcliffe - The TPTP Problem Library and Associated Infrastructure. From CNF to TH0, TPTP v6.4.0 - 2017 - Journal of Automated Reasoning - http://www.tptp.org

[2] Thomas Raths and Jens Otten - The QMLTP Problem Library for First-Order Modal Logics - 2012 - Automated Reasoning, IJCAR 2012 - http://www.jens-otten.de/papers/qmltp_ijcar12.pdf - http://www.iltp.de/qmltp

[3] Tobias Gleißner - QMLTP Mirror and QMLTP in THF - https://github.com/TobiasGleissner/QMLTP

[4] Christoph Benzmüller and Lawrence C. Paulson - Quantified Multimodal Logics in Simple Type Theory - 2013 - Logica Universalis (Special Issue on Multimodal Logics) - http://christoph-benzmueller.de/papers/J23.pdf

[5] Tobias Gleißner, Alexander Steen and Christoph Benzmüller - Theorem Provers For Every Normal Modal Logic - 2017 - LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning - https://easychair.org/publications/paper/340346