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ddl2thf

DDL2THF -- A preprocessor for translating problems in Dyadic Deontic Logic into THF problems

⚠️ This repository is archived, its functionality is superseded by the logic embedding tool at github.com/leoprover/logic-embedding.

Synopsis

The tool translates a problem statement formulated in Carmo/Jones' Dyadic Deontic Logic [1] into the standard THF format [2] for higher-order theorem provers. This translation employs a semantical embedding approach by Benzmüller et al. [3]. It can be used to pre-process a problem in such a way that standard THF provers can be used for the reasoning process.

Leo-III

This tool has been integrated into the higher-order automated theorem prover Leo-III [4]. See the GitHub page of Leo-III for more details.

Language

The syntax adapts a TPTP-like problem specification, similar to the FOF syntax [5]. Each input statement is of the form

ddl(name,role,formula).

where

  • name is an arbitrary alphanumerical name,
  • role is either axiom, localAxiom, conjecture or localConjecture, and
  • formula is a DDL formula (see below).

DDL formulas are given by

Syntax (Formulas) Interpretation
$true Propositional truth
$false Propositional falsehood
p where p is any lower-case string Propositional variable
~ p Negation of p
p & q Conjuction of p and q
p | q Disjunction of p and q
p => q Implication from p to q
p <=> q Equivalence of p and q
$box_a(p) In all actual worlds q
$box_p(p) In all possible worlds p
$box(p) In all worlds p
$O(p/q) It ought to be p given q (dyadic deontic obligation)
$O_a(p) Actual monadic deontic obligation p
$O_p(p) Primary monadic deontic obligation p
$O(p) It ought to be p (shorthand for $O(p/$true))

Example

ddl(a1, axiom, $O(processDataLawfully)).  % -- an "axiom" makes the formula globally valid
                                          % as in 'valid $O(processDataLawfully)'
ddl(a2, axiom, (~processDataLawfully) => $O(eraseData)). % same here. globally valid.
ddl(a3, localAxiom, ~processDataLawfully). % only valid at current world

ddl(c1, conjecture, $O(eraseData)). % all ok. conjecture is whether $O(eraseData) is globally valid.

A simple invocation of Leo-III verifies the conjecture (assuming that the problem is stored as gdpr.p)*:

[lex@gaunab]$ leo3 gdpr.p --ddl
% Axioms used in derivation (2): a3, a2
% No. of inferences in proof: 13
[...]
% SZS status Theorem for src/test/resources/gdpr.p : 2248 ms resp. 1008 ms w/o parsing

*: Leo-III natively include te ddl2thf tool so that no pre-processing into THF is necessary.

Installation

DDL2THF requires the Scala 2.12.X.

Using a pre-compiled release

tba

Building from source

If you have the Scala Build Tool (SBT) installed, simply run

make

This will create a .jar in the bin directory. You can create/install an executable ddl2thf via

make install

This will create the binary in $USER/bin/.

Usage

DDL2THF -- Encoding of Carmo/Jones DDL into HOL
(c) 2018 Alexander Steen

Usage: ddl2thf <infile> [<outfile>]

Parameters:
<infile>   - The file that contains a problem in DDL syntax.
             Use '-' (without quotes) to read from stdin.
<outfile>  - The file name in which the encoded result will be written
             (all parent directorys must already exist).
             If omitted, the result is printed to stdout.

References

[1] J. Carmo and A.J.I. Jones. Deontic logic and contrary-to-duties. In D. M. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic: Volume 8, pages 265–343. Springer Netherlands, Dordrecht, 2002.

[2] Automated Reasoning in Higher-Order Logic using the TPTP THF Infrastructure (Geoff Sutcliffe, Christoph Benzmüller), In Journal of Formalized Reasoning, volume 3, number 1, pp. 1-27, 2010.

[3] Faithful Semantical Embedding of a Dyadic Deontic Logic in HOL (Christoph Benzmüller, Ali Farjami, Xavier Parent), Technical report, CoRR, 2018. (https://arxiv.org/abs/1802.08454, submitted )

[4] The Higher-Order Prover Leo-III (Alexander Steen, Christoph Benzmüller), In Automated Reasoning --- 9th International Joint Conference, IJCAR 2018, Oxford, UK, July 14-17, 2018, Proceedings (Didier Galmiche, Stephan Schulz, Roberto Sebastiani, eds.), Springer, LNCS, 2018. (forthcoming)

[5] Sutcliffe, G. (2017). The TPTP Problem Library and Associated Infrastructure: From CNF to TH0, TPTP v6.4.0. Journal of Automated Reasoning, 1-20. DOI: 10.1007/s10817-017-9407-7