<p>Some rewriting Rules <math display="inline"><semantics> <mi mathvariant="script">R</mi> </semantics></math> for DECLARE<span class="html-italic">d</span> (<a href="#sec4dot3-logics-02-00004" class="html-sec">Section 4.3</a>).</p> Full article ">Figure 1 Cont.
<p>Some rewriting Rules <math display="inline"><semantics> <mi mathvariant="script">R</mi> </semantics></math> for DECLARE<span class="html-italic">d</span> (<a href="#sec4dot3-logics-02-00004" class="html-sec">Section 4.3</a>).</p> Full article ">Figure 2
<p>Comparing the specification reducer’s running time with the ones of Lydia and AALTAF running over <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> </semantics></math> vs. running over <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mo>′</mo> <mi>c</mi> </mrow> </msubsup> </semantics></math> (LYDIA(R) and AALTAF(R) respectively).</p> Full article ">Figure 3
<p>Comparing different running times of KnoBAB over <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> </msub> </semantics></math> (False) vs. <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> <mo>′</mo> </msubsup> </semantics></math> (True).</p> Full article ">Figure 4
<p>Comparing the specification reducer’s running time with the ones of Lydia and AALTAF running over <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> </semantics></math> vs. running over <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mo>′</mo> <mi>c</mi> </mrow> </msubsup> </semantics></math> (LYDIA(R) and AALTAF(R) respectively) and the grounded representation <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mrow> <mo>↓</mo> </mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> </mrow> </msub> </semantics></math> (LYDIA+AX and AALTAF+AX respectively).</p> Full article ">Figure 5
<p>Comparing different running times of KnoBAB over <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> </msub> </semantics></math> (False) vs. <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> <mo>′</mo> </msubsup> </semantics></math> (True).</p> Full article ">Figure A1
<p>Representation of the NFA associated to <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mrow> <mo>{</mo> <mi mathvariant="monospace">ChainResponse</mi> <mrow> <mo>(</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mspace width="0.277778em"/> <mo form="prefix">mod</mo> <mspace width="0.277778em"/> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>}</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>}</mo> </mrow> </msub> </mrow> </semantics></math> before minimisation for Lemma 1.</p> Full article ">Figure A2
<p>Representation of the NFA associated to <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Φ</mi> <mo>=</mo> <msub> <mo>⋀</mo> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>3</mn> </mrow> </msub> <mo>□</mo> <mrow> <mo>(</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mi>i</mi> </msub> <mo>⇒</mo> <mo>◯</mo> <mo>◊</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mspace width="0.277778em"/> <mo form="prefix">mod</mo> <mspace width="0.277778em"/> <mn>3</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> before minimisation for Lemma A1.</p> Full article ">Figure A3
<p>Representation of the NFA associated to <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Φ</mi> <mo>=</mo> <msub> <mrow> <mo>{</mo> <mi mathvariant="monospace">AltResponse</mi> <mrow> <mo>(</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi mathvariant="sans-serif">c</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mspace width="0.277778em"/> <mo form="prefix">mod</mo> <mspace width="0.277778em"/> <mn>4</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>}</mo> </mrow> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>4</mn> </mrow> </msub> </mrow> </semantics></math> before minimisation for Lemma A2.</p> Full article ">Figure A4
<p>Running times for rewriting <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> </msub> </semantics></math> as <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>d</mi> <mo>′</mo> </msubsup> </semantics></math>.</p> Full article ">Figure A5
<p>Running times for Lydia for both <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> </semantics></math> (LYDIA), <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mo>′</mo> <mi>c</mi> </mrow> </msubsup> </semantics></math> (LYDIA(R)) and <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>↓</mo> <mo>Σ</mo> </mrow> </msub> </semantics></math> (LYDIA+AX).</p> Full article ">Figure A6
<p>Running times for AALTAF for both <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> </semantics></math> (AALTAF), <math display="inline"><semantics> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mo>′</mo> <mi>c</mi> </mrow> </msubsup> </semantics></math> (AALTAF(R)) and <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msubsup> <mi mathvariant="sans-serif">Φ</mi> <mi>i</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>g</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>↓</mo> <mo>Σ</mo> </mrow> </msub> </semantics></math> (AALTAF+AX).</p> Full article ">