9 Oct, 2019 [LOPSTR’19] Modeling and Reasoning in Event Calculus Using Goal-Directed Constraint Answer Set Programming Joaqu´ınArias1,2 Zhuo Chen3 Manuel Carro1,2 Gopal Gupta3 1IMDEA Software Institute, 2Universidad Politecnica´ de Madrid 3University of Texas at Dallas madrid institute for advanced studies in software development technologies www.software.imdea.org Introduction Example: Commonsense Reasoning (CR) • Requires modelling: • Non-monotonicity. • Continuous (i.e., non-discrete) characteristics of the world. • Event Calculus (EC): formalism that represents continuous change and captures law of inertia. • EC components: Narrative A description of the world we want to model. Assumes circumscription. A tap fills a vessel [Shanahan 1999]. Axioms A generic description of how the world behaves given a narrative. • Implementing EC: logic reasoning + continuous domains. madrid institute for advanced studies in software development technologies 1 / 12 www.software.imdea.org Introduction Example: Reasoning on EC: deduction / proving in first order logic (+ circumscription). Approaches • Non-interactive theorem prover: likely won’t always answer [12]. • Prolog: incomplete implementations [15; 10; 2]. • Constraint Answer Set Programming (CASP): requires grounding. • Has been used to model (discrete) EC [8; 9]. A tap fills a vessel [Shanahan 1999]. • Limited to variables ranging over discrete, finite domains. madrid institute for advanced studies in software development technologies 1 / 12 The program: Has two models: p :q. fp,:qg or fq,:pg q :p. • (Constraint) Answer Set Programming systems compute those stable models. • However, most (C)ASP systems require a grounding phase that restricts the domains and constraints supported. www.software.imdea.org (Constraint) Logic Programming + negation • A logic program P is a set of First Order Logic clauses. • Programs with negation in the body can have different, incomparable models: stable model semantics [Gelfond and Lifschitz 1988]. madrid institute for advanced studies in software development technologies 2 / 12 • (Constraint) Answer Set Programming systems compute those stable models. • However, most (C)ASP systems require a grounding phase that restricts the domains and constraints supported. www.software.imdea.org (Constraint) Logic Programming + negation • A logic program P is a set of First Order Logic clauses. • Programs with negation in the body can have different, incomparable models: stable model semantics [Gelfond and Lifschitz 1988]. The program: Has two models: p :q. fp,:qg or fq,:pg q :p. madrid institute for advanced studies in software development technologies 2 / 12 www.software.imdea.org (Constraint) Logic Programming + negation • A logic program P is a set of First Order Logic clauses. • Programs with negation in the body can have different, incomparable models: stable model semantics [Gelfond and Lifschitz 1988]. The program: Has two models: p :q. fp,:qg or fq,:pg q :p. • (Constraint) Answer Set Programming systems compute those stable models. • However, most (C)ASP systems require a grounding phase that restricts the domains and constraints supported. madrid institute for advanced studies in software development technologies 2 / 12 • Provides a constructive and sound default negation: • Make deductions in the absence of positive information. cross(T):- not train(T). • Dual rules (synthesized by s(CASP) compiler) infer conditions for goal to fail. • Provides support for classical negation. • Represent explicit (negative) knowledge. cross(T):--train(T) . • Allows rules with negated heads. -train(T):- not barrier(up,T). • Global constraints ensure consistency. :- train(T),-train(T) . www.software.imdea.org s(CASP) • Goal directed execution of CASP programs without grounding. • The execution starts with a query. ?-T #>5,T #<8, cross(T) . • Returns (partial) stable models [Gelfond and Lifschitz 1988] Only literals supporting the query. • For each successful top-down derivation, on backtracking returns: • A justification tree. Explanation for observations. • Bindings and constraints as part of the model. {T #> 5, T #< 7/3, cross(T), train(7.3)}. madrid institute for advanced studies in software development technologies 3 / 12 • Provides support for classical negation. • Represent explicit (negative) knowledge. cross(T):--train(T) . • Allows rules with negated heads. -train(T):- not barrier(up,T). • Global constraints ensure consistency. :- train(T),-train(T) . www.software.imdea.org s(CASP) • Goal directed execution of CASP programs without grounding. • The execution starts with a query. ?-T #>5,T #<8, cross(T) . • Returns (partial) stable models [Gelfond and Lifschitz 1988] Only literals supporting the query. • For each successful top-down derivation, on backtracking returns: • A justification tree. Explanation for observations. • Bindings and constraints as part of the model. {T #> 5, T #< 7/3, cross(T), train(7.3)}. • Provides a constructive and sound default negation: • Make deductions in the absence of positive information. cross(T):- not train(T). • Dual rules (synthesized by s(CASP) compiler) infer conditions for goal to fail. madrid institute for advanced studies in software development technologies 3 / 12 www.software.imdea.org s(CASP) • Goal directed execution of CASP programs without grounding. • The execution starts with a query. ?-T #>5,T #<8, cross(T) . • Returns (partial) stable models [Gelfond and Lifschitz 1988] Only literals supporting the query. • For each successful top-down derivation, on backtracking returns: • A justification tree. Explanation for observations. • Bindings and constraints as part of the model. {T #> 5, T #< 7/3, cross(T), train(7.3)}. • Provides a constructive and sound default negation: • Make deductions in the absence of positive information. cross(T):- not train(T). • Dual rules (synthesized by s(CASP) compiler) infer conditions for goal to fail. • Provides support for classical negation. • Represent explicit (negative) knowledge. cross(T):--train(T) . • Allows rules with negated heads. -train(T):- not barrier(up,T). • Global constraints ensure consistency. :- train(T),-train(T) . madrid institute for advanced studies in software development technologies 3 / 12 • State constraints are introduced to capture restrictions on the model: • Ensure consistency of the narrative w.r.t. the axioms. :- holds(F,T),-holds(F,T) . • Trajectories: a fluent depends on the time elapsed since another fluent: • The level of water level(L2) correspond directly to the time elapsed T2-T1. L2 = L1 + T2-T1 • Support for non-monotonic reasoning • EC theory includes negation in rules and infers negative knowledge. • Therefore, alternative worlds can appear. Vessel size either max_level(10) or max_level(16). www.software.imdea.org Event Calculus • EC uses a universal theory (axioms) to reason about scenarios (narrative). • An event happens at a time point. tapOn: The tap opens. • A fluent is a time-varying property of the world. filling: The vessel is being filled. • Time and/or fluents may have continuous quantities associated. level(X) Level of water. madrid institute for advanced studies in software development technologies 4 / 12 • Trajectories: a fluent depends on the time elapsed since another fluent: • The level of water level(L2) correspond directly to the time elapsed T2-T1. L2 = L1 + T2-T1 • Support for non-monotonic reasoning • EC theory includes negation in rules and infers negative knowledge. • Therefore, alternative worlds can appear. Vessel size either max_level(10) or max_level(16). www.software.imdea.org Event Calculus • EC uses a universal theory (axioms) to reason about scenarios (narrative). • An event happens at a time point. tapOn: The tap opens. • A fluent is a time-varying property of the world. filling: The vessel is being filled. • Time and/or fluents may have continuous quantities associated. level(X) Level of water. • State constraints are introduced to capture restrictions on the model: • Ensure consistency of the narrative w.r.t. the axioms. :- holds(F,T),-holds(F,T) . madrid institute for advanced studies in software development technologies 4 / 12 • Support for non-monotonic reasoning • EC theory includes negation in rules and infers negative knowledge. • Therefore, alternative worlds can appear. Vessel size either max_level(10) or max_level(16). www.software.imdea.org Event Calculus • EC uses a universal theory (axioms) to reason about scenarios (narrative). • An event happens at a time point. tapOn: The tap opens. • A fluent is a time-varying property of the world. filling: The vessel is being filled. • Time and/or fluents may have continuous quantities associated. level(X) Level of water. • State constraints are introduced to capture restrictions on the model: • Ensure consistency of the narrative w.r.t. the axioms. :- holds(F,T),-holds(F,T) . • Trajectories: a fluent depends on the time elapsed since another fluent: • The level of water level(L2) correspond directly to the time elapsed T2-T1. L2 = L1 + T2-T1 madrid institute for advanced studies in software development technologies 4 / 12 www.software.imdea.org Event Calculus • EC uses a universal theory (axioms) to reason about scenarios (narrative). • An event happens at a time point. tapOn: The tap opens. • A fluent is a time-varying property of the world. filling: The vessel is being filled. • Time and/or fluents may have continuous quantities associated. level(X) Level of water. • State constraints are introduced to capture restrictions on the model: • Ensure consistency of the narrative