Declarative Solver Development: Case Studies by krr

Take-home message

Building a solver should not be hard (anno 2016)
A high-level description of semantics should suffice

If you can describe your semantics in second-order logic,

you get a solver for free

Abstract argumentation

An interpretation S is conflict-free if S attacks no argument in S

\mathcal{T}_{CF} = \left\{ \lnot \exists N, M: r(N,M) \land s(N) \land s(M).\right\}

\mathcal{T}_{CF} = \left\{ \lnot \exists N, M: r(N,M) \land s(N) \land s(M).\right\}

Abstract argumentation

A node x is attacked by S if S contains at least one of x's attackers
A node x is defended by S if S attacks all x's attackers

\begin{array}{l}\mathcal{T}_{AD} =\\ \left\{\begin{array}{l} \forall N: att(N) {\Leftrightarrow} ( a(N) \wedge \exists M: r(M,N) \wedge s(M) ). \\ \forall N: def(N) {\Leftrightarrow} ( a(N) \wedge \forall M: r(M,N) {\Rightarrow} att(M) ). \end{array} \right\} \end{array}

\begin{array}{l}\mathcal{T}_{AD} =\\ \left\{\begin{array}{l} \forall N: att(N) {\Leftrightarrow} ( a(N) \wedge \exists M: r(M,N) \wedge s(M) ). \\ \forall N: def(N) {\Leftrightarrow} ( a(N) \wedge \forall M: r(M,N) {\Rightarrow} att(M) ). \end{array} \right\} \end{array}

Abstract argumentation

Stable semantics An interpretation S is a stable extension if S consists exactly of all arguments not attacked by S

\mathcal{T}_{ST} = \left\{\begin{array}{l} \mathcal{T}_{AD}. \\ \forall N: a(N)\Rightarrow (s(N)\Leftrightarrow \lnot att(N)). \end{array} \right\}

\mathcal{T}_{ST} = \left\{\begin{array}{l} \mathcal{T}_{AD}. \\ \forall N: a(N)\Rightarrow (s(N)\Leftrightarrow \lnot att(N)). \end{array} \right\}

Abstract argumentation

Different reasoning modes:

Credulous inference: Does x hold in some extension
Skeptical inference: Does x hold in all extensions

\begin{array}{l} \mathcal{T}_{ST}^{cred} = \{\exists s:\, \mathcal{T}_{ST}\, \wedge\, s(x).\} \\ \mathcal{T}_{ST}^{skep} = \{\forall s:\, \mathcal{T}_{ST} \,\Rightarrow \,s(x).\} \end{array}

\begin{array}{l} \mathcal{T}_{ST}^{cred} = \{\exists s:\, \mathcal{T}_{ST}\, \wedge\, s(x).\} \\ \mathcal{T}_{ST}^{skep} = \{\forall s:\, \mathcal{T}_{ST} \,\Rightarrow \,s(x).\} \end{array}

Abstract argumentation

Implementation: We only give the solver an ASCII representation of the second-order theory

\begin{array}{l} \mathcal{T}_{ST}^{cred} = \{\exists s:\, \mathcal{T}_{ST}\, \wedge\, s(x).\} \end{array}

\begin{array}{l} \mathcal{T}_{ST}^{cred} = \{\exists s:\, \mathcal{T}_{ST}\, \wedge\, s(x).\} \end{array}

? s: 
    (?N: in_test(N) & s(N)) & 
     ! N: in_node(N) => (~s(N) <=> ( ? M: in_attack(M,N) & s(M) ) ).

Logic Programming

Disjunctive logic program: set of rules of the form

h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \wedge \neg b_1\wedge \cdots \wedge \neg b_m

h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \wedge \neg b_1\wedge \cdots \wedge \neg b_m

An interpretation I (a set of atoms) is a stable model if it is a (subset)minimal model of the reduct

\left\{\begin{array}{l} h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \mid\\ \quad h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \wedge \neg b_1\wedge \cdots \wedge \neg b_m \in \mathcal{P} \\ \qquad\wedge\, b_i \notin I \text{ for all } 1\leq i\leq m \end{array}\right\}

\left\{\begin{array}{l} h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \mid\\ \quad h_1\vee \cdots \vee h_l \leftarrow a_1\wedge \cdots \wedge a_n \wedge \neg b_1\wedge \cdots \wedge \neg b_m \in \mathcal{P} \\ \qquad\wedge\, b_i \notin I \text{ for all } 1\leq i\leq m \end{array}\right\}

Logic Programming

An interpretation I (a set of atoms) is a stable model if it is a (subset)minimal model of the reduct

\left\{ \begin{array}{l} \forall A: i(A)\Rightarrow a(A).\\ \forall R: r(R) \wedge \\\quad (\forall A: pb(R,A)\Rightarrow i(A)) \wedge (\forall B: nb(R,B)\Rightarrow \neg i(B)) \\ \qquad\Rightarrow (\exists H: h(R,H)\wedge i(H)). \\ \neg \exists i':\\ \quad (\forall A: i'(A)\Rightarrow i(A))\wedge (\exists A: i(A)\wedge \neg i'(A))\,\wedge\\ \quad\forall R: r(R)\wedge\\ \qquad (\forall A: pb(R,A)\Rightarrow i'(A)) \wedge (\forall B: nb(R,B)\Rightarrow \neg i(B)) \\ \quad\qquad\Rightarrow (\exists H: h(R,H)\wedge i'(H)). \end{array} \right\}

\left\{ \begin{array}{l} \forall A: i(A)\Rightarrow a(A).\\ \forall R: r(R) \wedge \\\quad (\forall A: pb(R,A)\Rightarrow i(A)) \wedge (\forall B: nb(R,B)\Rightarrow \neg i(B)) \\ \qquad\Rightarrow (\exists H: h(R,H)\wedge i(H)). \\ \neg \exists i':\\ \quad (\forall A: i'(A)\Rightarrow i(A))\wedge (\exists A: i(A)\wedge \neg i'(A))\,\wedge\\ \quad\forall R: r(R)\wedge\\ \qquad (\forall A: pb(R,A)\Rightarrow i'(A)) \wedge (\forall B: nb(R,B)\Rightarrow \neg i(B)) \\ \quad\qquad\Rightarrow (\exists H: h(R,H)\wedge i'(H)). \end{array} \right\}

S2S-CoQuiAAS	Some Ext	All Ext	Cred	Skep
Complete	192 - 192	191 - 191	576 - 575	576 - 576
Preferred	192 - 191	190 - 189	576 - 576	576 - 576
Grounded	192 - 192	192 - 192	576 - 576	576 - 576
Stable	192 - 192	192 - 191	576 - 576	576 - 576


paper	webpage

3.1

Take-home message Building a solver should not be hard (anno 2016) A high-level description of semantics should suffice If you can describe your semantics in second-order logic, you get a solver for free

Declarative Solver Development: Case Studies

By krr

Declarative Solver Development:
Case Studies

Knowledge Representation (and Reasoning): Status

Abstract Argumentation

KR 2016

Why is no-one building solvers?

Take-home message

Second-order logic

What's next

1. Applications

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Abstract argumentation

Logic Programming

Logic Programming

Logic Programming

Logic Programming

Logic Programming

Propositional logics

Propositional logics

2. Efficiency

Efficiency

3. Under the Hood

Under the Hood

SAT-to-SAT

Grounding to SAT-to-SAT

4. Conclusion & Future Work

Declarative solver development is feasable

Language should be extended

Giving the user more control

Declarative Solver Development: Case Studies